The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Second Edition

de Gruyter Series in Logic and Its Applications 1 Editors: Wilfrid A. Hodges (London) Steffen Lempp (Madison) Menachem Magidor (Jerusalem)

W. Hugh Woodin

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

Second revised edition

De Gruyter

Mathematics Subject Classification 2010: 03-02, 03E05, 03E15, 03E25, 03E35, 03E40, 03E57, 03E60.

ISBN 978-3-11-019702-0 e-ISBN 978-3-11-021317-1 ISSN 1438-1893 Library of Congress Cataloging-in-Publication Data Woodin, W. H. (W. Hugh) The axiom of determinacy, forcing axioms, and the nonstationary ideal / by W. Hugh Woodin. ⫺ 2nd rev. and updated ed. p. cm. ⫺ (De Gruyter series in logic and its applications ; 1) Includes bibliographical references and index. ISBN 978-3-11-019702-0 (alk. paper) 1. Forcing (Model theory) I. Title. QA9.7.W66 2010 511.3⫺dc22 2010011786

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. 쑔 2010 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents

1 Introduction 1.1 The nonstationary ideal on !1 . . . . . . . . . . . 1.2 The partial order Pmax . . . . . . . . . . . . . . . . 1.3 Pmax variations . . . . . . . . . . . . . . . . . . . 1.4 Extensions of inner models beyond L.R/ . . . . . 1.5 Concluding remarks – the view from Berlin in 1999 1.6 The view from Heidelberg in 2010 . . . . . . . . .

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1 2 6 10 13 15 18

2 Preliminaries 2.1 Weakly homogeneous trees and scales 2.2 Generic absoluteness . . . . . . . . . 2.3 The stationary tower . . . . . . . . . 2.4 Forcing Axioms . . . . . . . . . . . . 2.5 Reflection Principles . . . . . . . . . 2.6 Generic ideals . . . . . . . . . . . . .

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3 The nonstationary ideal 51 3.1 The nonstationary ideal and ı12 . . . . . . . . . . . . . . . . . . . . . 51 3.2 The nonstationary ideal and CH . . . . . . . . . . . . . . . . . . . . 108 116 4 The Pmax -extension 4.1 Iterable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2 The partial order Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5 Applications 5.1 The sentence AC . . . . . . . . . . . 5.2 Martin’s Maximum and AC . . . . . 5.3 The sentence AC . . . . . . . . . . . 5.4 The stationary tower and Pmax . . . .  . . . . . . . . . . . . . . . . . . 5.5 Pmax 0 . . . . . . . . . . . . . . . . . . 5.6 Pmax 5.7 The Axiom  . . . . . . . . . . . . 5.8 Homogeneity properties of P .!1 /=INS

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184 184 187 192 199 221 232 238 274

6 Pmax variations 6.1 2 Pmax . . . . . . . . . . . . . . . . . . 6.2 Variations for obtaining !1 -dense ideals 6.2.1 Qmax . . . . . . . . . . . . . . 6.2.2 Qmax . . . . . . . . . . . . . .

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287 288 306 306 334

vi

Contents

6.3

6.2.3 2 Qmax . . . . . . . . . . . . . . . . 6.2.4 Weak Kurepa trees and Qmax . . . . 6.2.5 KT Qmax . . . . . . . . . . . . . . . 6.2.6 Null sets and the nonstationary ideal Nonregular ultrafilters on !1 . . . . . . . .

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370 377 383 403 421

7 Conditional variations 426 7.1 Suslin trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 7.2 The Borel Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 441 8 | principles for !1 493 8.1 Condensation Principles . . . . . . . . . . . . . . . . . . . . . . . . 496 |NS 8.2 Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 C CC 8.3 The principles, |NS and |NS . . . . . . . . . . . . . . . . . . . . . . 577 9 Extensions of L.; R/ 9.1 ADC . . . . . . . . . . . . . . . . . . 9.2 The Pmax -extension of L.; R/ . . . . 9.2.1 The basic analysis . . . . . . 9.2.2 Martin’s Maximum CC .c/ . . 9.3 The Qmax -extension of L.; R/ . . . . 9.4 Chang’s Conjecture . . . . . . . . . . 9.5 Weak and Strong Reflection Principles 9.6 Strong Chang’s Conjecture . . . . . . 9.7 Ideals on !2 . . . . . . . . . . . . . .

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609 610 617 618 622 633 637 651 667 683

10 Further results 10.1 Forcing notions and large cardinals . . . . . 10.2 Coding into L.P .!1 // . . . . . . . . . . . 10.2.1 Coding by sets, SQ . . . . . . . . . . 10.2.2 Q.X/ max . . . . . . . . . . . . . . . . .;/ 10.2.3 Pmax . . . . . . . . . . . . . . . . . .;;B/ . . . . . . . . . . . . . . . . 10.2.4 Pmax 10.3 Bounded forms of Martin’s Maximum . . . 10.4 -logic . . . . . . . . . . . . . . . . . . . 10.5 -logic and the Continuum Hypothesis . . 10.6 The Axiom ./C . . . . . . . . . . . . . . 10.7 The Effective Singular Cardinals Hypothesis

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694 694 701 703 708 739 768 784 807 813 827 835

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11 Questions

840

Bibliography

845

Index

849

Chapter 1

Introduction

As always I suppose, when contemplating a new edition one must decide whether to rewrite the introduction or simply write an addendum to the original introduction. I have chosen the latter course and so after this paragraph the current edition begins with the original introduction and summary from the first edition (with comments inserted in italics and some other minor changes) and then continues beginning on page 18 with comments regarding this edition. The main result of this book is the identification of a canonical model in which the Continuum Hypothesis (CH) is false. This model is canonical in the sense that G¨odel’s constructible universe L and its relativization to the reals, L.R/, are canonical models though of course the assertion that L.R/ is a canonical model is made in the context of large cardinals. Our claim is vague, nevertheless the model we identify can be characterized by its absoluteness properties. This model can also be characterized by certain homogeneity properties. From the point of view of forcing axioms it is the ultimate model at least as far as the subsets of !1 are concerned. It is arguably a completion of P .!1 /, the powerset of !1 . This model is a forcing extension of L.R/ and the method can be varied to produce a wide class of similar models each of which can be viewed as a reduction of this model. The methodology for producing these models is quite different than that behind the usual forcing constructions. For example the corresponding partial orders are countably closed and they are not constructed as forcing iterations. We provide evidence that this is a useful method for achieving consistency results, obtaining a number of results which seem out of reach of the current technology of iterated forcing. The analysis of these models arises from an interesting interplay between ideas from descriptive set theory and from combinatorial set theory. More precisely it is the existence of definable scales which is ultimately the driving force behind the arguments. Boundedness arguments also play a key role. These results contribute to a curious circle of relationships between large cardinals, determinacy, and forcing axioms. Another interesting feature of these models is that although these models are generic extensions of specific inner models (L.R/ in most cases), these models can be characterized without reference to this. For example, as we have indicated above, our canonical model is a generic extension of L.R/. The corresponding partial order we denote by Pmax . In Chapter 5 we  give a characterization for this model isolating an axiom  . The formulation of  does not involve Pmax , nor does it obviously refer to L.R/. Instead it specifies properties of definable subsets of P .!1 /.

2

1 Introduction

The original motivation for the definition of these models resulted from the discovery that it is possible, in the presence of the appropriate large cardinals, to force (quite by accident) the effective failure of CH. This and related results are the subject of Chapter 3. We discuss effective versions of CH below. Gdel was the first to propose that large cardinal axioms could be used to settle questions that were otherwise unsolvable. This has been remarkably successful particularly in the area of descriptive set theory where most of the classical questions have now been answered. However after the results of Cohen it became apparent that large cardinals could not be used to settle the Continuum Hypothesis. This was first argued by Levy and Solovay .1967/. Nevertheless large cardinals do provide some insight to the Continuum Hypothesis. One example of this is the absoluteness theorem of Woodin .1985/. Roughly this theorem states that in the presence of suitable large cardinals CH “settles” all questions with the logical complexity of CH. More precisely if there exists a proper class of measurable Woodin cardinals then †21 sentences are absolute between all set generic extensions of V which satisfy CH. The results of this book can be viewed collectively as a version of this absoluteness theorem for the negation of the Continuum Hypothesis (:CH).

1.1

The nonstationary ideal on !1

We begin with the following question. Is there a family ¹S˛ j ˛ < !2 º of stationary subsets of !1 such that S˛ \ Sˇ is nonstationary whenever ˛ ¤ ˇ? The analysis of this question has played (perhaps coincidentally) an important role in set theory particularly in the study of forcing axioms, large cardinals and determinacy. The nonstationary ideal on !1 is !2 -saturated if there is no such family. This statement is independent of the axioms of set theory. We let INS denote the set of subsets of !1 which are not stationary. Clearly INS is a countably additive uniform ideal on !1 . If the nonstationary ideal on !1 is !2 -saturated then the boolean algebra P .!1 /=INS is a complete boolean algebra which satisfies the !2 chain condition. Kanamori .2008/ surveys some of the history regarding saturated ideals, the concept was introduced by Tarski. The first consistency proof for the saturation of the nonstationary ideal was obtained by Steel and VanWesep .1982/. They used the consistency of a very strong form of the Axiom of Determinacy (AD), see .Kanamori 2008/ and Moschovakis .1980/ for the history of these axioms.

1.1 The nonstationary ideal on !1

3

Steel and VanWesep proved the consistency of ZFC C “The nonstationary ideal on !1 is !2 -saturated” assuming the consistency of ZF C AD R C “‚ is regular ”: AD R is the assertion that all real games of length ! are determined and ‚ denotes the supremum of the ordinals which are the surjective image of the reals. The hypothesis was later reduced by Woodin .1983/ to the consistency of ZF C AD. The arguments of Steel and VanWesep were motivated by the problem of obtaining a model of ZFC in which !2 is the second uniform indiscernible. For this Steel defined a notion of forcing which forces over a suitable model of AD that ZFC holds (i. e. that the Axiom of Choice holds) and forces both that !2 is the second uniform indiscernible and (by arguments of VanWesep) that the nonstationary ideal on !1 is !2 -saturated. The method of .Woodin 1983/ uses the same notion of forcing and a finer analysis of the forcing conditions to show that things work out over L.R/. In these models obtained by forcing over a ground model satisfying AD not only is the nonstationary ideal saturated but the quotient algebra P .!1 /=INS has a particularly simple form, P .!1 /=INS Š RO.Coll.!;
4

1 Introduction

more general problem of computing the effective size of the continuum. This problem has a variety of formulations, two natural versions are combined in the following:  Is there a (consistent) large cardinal whose existence implies that the length of any prewellordering arising in either of the following fashions, is less than the least weakly inaccessible cardinal? – The prewellordering exists in a transitive inner model of AD containing all the reals. – The prewellordering is universally Baire. The second of these formulations involves the notion of a universally Baire set of reals which originates in .Feng, Magidor, and Woodin 1992/. Universally Baire sets are discussed briefly in Section 10.3. We note here that if there exists a proper class of Woodin cardinals then a set A  R is universally Baire if and only if it is 1 -weakly homogeneously Suslin which in turn is if and only if it is 1 -homogeneously Suslin. Another relevant point is that if there exist infinitely many Woodin cardinals with measurable above and if A  R is universally Baire, then L.A; R/  AD and so A belongs to an inner model of AD. The converse can fail. More generally one can ask for any bound provided of course that the bound is a “specific” !˛ which can be defined without reference to 2@0 . 1 For example every † 2 prewellordering has length less than !2 and if there is a measurable cardinal then every  †13 prewellordering has length less than !3 . A much deeper theorem of .Jackson 1988/ is that if every projective set is determined then every projective prewellordering has length less than !! . This combined with the theorem of Martin and Steel on projective determinacy yields that if there are infinitely many Woodin cardinals then every projective prewellordering has length less than !! . The point here of course is that these bounds are valid independent of the size of 2@0 . The current methods do not readily generalize to even produce a forcing extension of L.R/ (without adding reals) in which ZFC holds and !3 < ‚L.R/ . Thus at this point it is entirely possible that !3 is the bound and that this is provable in ZFC. If a large cardinal admits an inner model theory satisfying fairly general conditions then most likely the only (nontrivial) bounds provable from the existence of the large cardinal are those provable in ZFC; i. e. large cardinal combinatorics are irrelevant unless the large cardinal is beyond a reasonable inner model theory. For example suppose that there is a partial order P 2 L.R/ such that for all transitive models M of ADC containing R, if G  P is M -generic then  .R/M ŒG D .R/M ,  .ı13 /M ŒG D .!3 /M ŒG ,  L.R/ŒG  ZFC,

1.1 The nonstationary ideal on !1

5

where  ı 13 is the supremum of the lengths of  13 prewellorderings of R. The axiom ADC is a technical variant of AD which is actually implied by AD in many instances. Assuming DC it is implied, for example, by AD R . It is also implied by AD if V D L.R/. By the results of .Woodin 2010b/ if inner model theory can be extended to the level of one supercompact cardinal then the existence of essentially all large cardinals is consistent with  ı 13 D !3 . It follows from the results of .Steel and VanWesep 1982/ and .Woodin 1983/ that such a partial order P exists in the case of ı12 , more precisely, assuming L.R/  AD; there is a partial order P 2 L.R/ such that for all transitive models M of ADC containing R, if G  P is M -generic then  .R/M ŒG D .R/M ,  .ı12 /M ŒG D .!2 /M ŒG ,  L.R/ŒG  ZFC. Thus if a large cardinal admits a suitable inner model theory then the existence of the large cardinal is consistent with ı12 D !2 . We shall prove a much stronger result in Chapter 3, showing that if ı is a Woodin cardinal and if there is a measurable cardinal above ı then there is a semiproper partial order P of cardinality ı such that V P  ı12 D !2 : This result which is a corollary of Theorem 1.1, stated below, and Theorem 2.64, due to Shelah, shows that this particular instance of the Effective Continuum Hypothesis is as intractable as the Continuum Hypothesis. Foreman and Magidor initiated a program of proving that  ı 12 < !2 from various combinatorial hypotheses with the goal of evolving these into large cardinal hypotheses, .Foreman and Magidor 1995/. By the (initial) remarks above their program if successful would have identified a critical step in the large cardinal hierarchy. Foreman and Magidor proved among other things that if there exists a (normal) ı 12 < !2 . In !3 -saturated ideal on !2 concentrating on a specific stationary set then  Chapter 9 we improve this result slightly showing that this restriction is unnecessary; if there is a measurable cardinal and if there is an !3 -saturated (uniform) ideal on !2 then  ı 12 < !2 . An early conjecture of Martin is that  ı 1n D @n for all n follows from reasonable 1 1n prewellorderings. hypotheses. ın is the supremum of the lengths of  The following theorem “proves” the Martin conjecture in the case of n D 2. Theorem 1.1. Assume that the nonstationary ideal on !1 is !2 -saturated and that there is a measurable cardinal. Then  ı 12 D !2 and further every club in !1 contains a club constructible from a real. t u

6

1 Introduction

As a corollary we obtain, Theorem 1.2. Assume Martin’s Maximum. Then  ı 12 D !2 and every club in !1 contains a club constructible from a real. t u Another immediate corollary is a refinement of the upper bound for the consistency strength of ZFC C “For every real x; x # exists.” C “!2 is the second uniform indiscernible.” Assuming in addition that larger cardinals exist then one obtains more information. For example, Theorem 1.3. Assume the nonstationary ideal on !1 is !2 -saturated and that there exist ! many Woodin cardinals with a measurable cardinal above them all. (1) Suppose that A  R, A 2 L.R/, and that there is a sequence hB˛ W ˛ < !1 i of borel sets such that A D [¹B˛ j ˛ < !1 º: 1 Then A is † 2 .

(2) Suppose that X is a bounded subset of ‚L.R/ of cardinality !1 . Then there exists t u a set Y 2 L.R/ of cardinality !1 in L.R/ such that X  Y . We note that assuming for every x 2 R, x # exists, the statement (1) of Theorem 1.3 1 implies that  ı 12 D !2 ; if ı12 < !2 then every † 3 set is an !1 union of borel sets.

1.2

The partial order Pmax

Theorem 1.3 suggests that if the nonstationary ideal is saturated (and if modest large cardinals exist) then one might reasonably expect that the inner model L.P .!1 // may be close to the inner model L.R/. However if the nonstationary ideal is saturated one can, by passing to a ccc generic extension, arrange that P .R/  L.P .!1 // and preserve the saturation of the nonstationary ideal. Nevertheless this intuition was the primary motivation for the definition of Pmax . The canonical model for :CH is obtained by the construction of this specific partial order, Pmax . The basic properties of Pmax are given in the following theorem. Theorem 1.4. Assume ADL.R/ and that there exists a Woodin cardinal with a measurable cardinal above it. Then there is a partial order Pmax in L.R/ such that; (1) Pmax is !-closed and homogeneous (in L.R/), (2) L.R/Pmax  ZFC.

1.2 The partial order Pmax

7

Further if  is a …2 sentence in the language for the structure hH.!2 /; 2; INS i and if hH.!2 /; 2; INS i   then Pmax

hH.!2 /; 2; INS iL.R/

 :

t u

The partial order Pmax is definable and thus, since granting large cardinals Th.L.R// is canonical, it follows that Th.L.R/Pmax / is canonical. Many of the open combinatorial questions at !1 are expressible as …2 statements in the structure hH.!2 /; 2; INS i and so assuming the existence of large cardinals these questions are either false, or they are true in L.R/Pmax . In some sense the spirit of Martin’s Axiom and its generalizations is to maximize the collection of …2 sentences true in the structure hH.!2 /; 2i Indeed MA!1 is easily reformulated as a …2 sentence for hH.!2 /; 2i. By the remarks above, assuming fairly weak large cardinal hypotheses, any such sentence which is true in some set generic extension of V is true in a canonical generic extension of L.R/. The situation is analogous to the situation of †12 sentences and L. By Shoenfield’s absoluteness theorem if a †12 sentence holds in V then it holds in L. The difference here is that the model analogous to L is not an inner model but rather it is a canonical generic extension of an inner model. This is not completely unprecedented. Mansfield’s theorem on †12 wellorderings can be reformulated as follows. Theorem 1.5 (Mansfield). Suppose that  is a …13 sentence which is true in V and there is a nonconstructible real. Then  is true in LP where P is Sacks forcing .defined in L/. t u Of course the …13 sentence also holds in L so this is not completely analogous to our situation. :CH is a (consistent) …2 sentence for hH.!2 /; 2i which is false in any of the standard inner models. Nevertheless the analogy with Sacks forcing is accurate. The forcing notion Pmax is a generalization of Sacks forcing to !1 . The following theorem, slightly awkward in formulation, shows that any attempt to realize in H.!2 / all suitably consistent …2 sentences, requires at least †12 Determinacy.

8

1 Introduction

Theorem 1.6. Suppose that there exists a model, hM; Ei, such that hM; Ei  ZFC and such that for each …2 sentence  if there exists a partial order P such that hH.!2 /; 2iV then

P

 ;

hH.!2 /; 2ihM;E i  :

Assume there is an inaccessible cardinal. Then V  †12 -Determinacy:

t u

One can strengthen Theorem 1.4 by expanding the structure hH.!2 /; 2; INS i by adding predicates for each set of reals in L.R/. This theorem requires additional large cardinal hypotheses which in fact imply ADL.R/ unlike the large cardinal hypothesis of Theorem 1.4. Theorem 1.7. Assume there are ! many Woodin cardinals with a measurable above. Suppose  is a …2 sentence in the language for the structure hH.!2 /; 2; INS ; X I X 2 L.R/; X  Ri and that hH.!2 /; 2; INS ; X I X 2 L.R/; X  Ri   Then

Pmax

hH.!2 /; 2; INS ; X I X 2 L.R/; X  RiL.R/

 :

t u

We note that since Pmax is !-closed, the structure Pmax

hH.!2 /; 2; INS ; X I X 2 L.R/; X  RiL.R/ is naturally interpreted as a structure for the language of hH.!2 /; 2; INS ; X I X 2 L.R/; X  Ri: The key point is that this strengthened absoluteness theorem has in some sense a converse. Theorem 1.8. Assume ADL.R/ . Suppose that for each …2 sentence in the language for the structure hH.!2 /; 2; INS ; X I X 2 L.R/; X  Ri if

Pmax

hH.!2 /; 2; INS ; X I X 2 L.R/; X  RiL.R/



then hH.!2 /; 2; INS ; X I X 2 L.R/; X  Ri  : Then L.P .!1 // D L.R/ŒG for some G  Pmax which is L.R/-generic.

t u

1.2 The partial order Pmax

9

If one assumes in addition that R# exists then Theorem 1.8 can be reformulated as follows. For each n 2 ! let Un be a set which is †1 definable in the structure hL.R/; hi W i < ni; 2i where hi W i < ni is an increasing sequence of Silver indiscernibles of L.R/, and such that Un is universal. Theorem 1.9. Assume ADL.R/ and that R# exists. Suppose that for each …2 sentence in the language for the structure hH.!2 /; 2; INS ; Un I n < !i if Pmax hH.!2 /; 2; INS ; Un I n < !iL.R/  then hH.!2 /; 2; INS ; Un I n < !i  : Then L.P .!1 // D L.R/ŒG for some G  Pmax which is L.R/-generic. t u Thus in the statement of Theorem 1.9 one only refers to a structure of countable signature. These theorems suggest that the axiom: ./ AD holds in L.R/ and L.P .!1 // is a Pmax -generic extension of L.R/; is perhaps, arguably, the correct maximal generalization of Martin’s Axiom at least as far as the structure of P .!1 / is concerned. However an important point is that we do not know if this axiom can always be forced to hold assuming the existence of suitable large cardinals. Conjecture. Assume there are ! 2 many Woodin cardinals. Then the axiom ./ holds in a generic extension of V . t u Because of the intrinsics of the partial order Pmax , this axiom is frequently easier to use than the usual forcing axioms. We give some applications for which it is not clear that Martin’s Maximum suffices. Another key point is:  There is no need in the analysis of L.R/Pmax for any machinery of iterated forcing. This includes the proofs of the absoluteness theorems. Further  The analysis of L.R/Pmax requires only ADL.R/ . For the definition of Pmax that we shall work with the analysis will require some iterated forcing but only for ccc forcing and only to produce a poset which forces MA!1 . In Chapter 5 we give three other presentations of Pmax based on the stationary tower forcing. The analysis of these (essentially equivalent) versions of Pmax require no local forcing arguments whatsoever. This includes the proof of the absoluteness theorems.

10

1 Introduction

Also in Chapter 5 we shall discuss methods for exploiting ./, giving a useful reformulation of the axiom. This reformulation does not involve the definition of Pmax . We shall also prove that, assuming ./, L.P .!1 //  AC: This we accomplish by finding a …2 sentence which if true in the structure, hH.!2 /; 2i; implies (in ZF C DC) that there is a surjection  W !2 ! R which is definable in the structure hH.!2 /; 2i from parameters. This sentence is a consequence of Martin’s Maximum and an analogous, but easier, argument shows that assuming ADL.R/ , it is true in L.R/Pmax . Thus the axiom ./ implies 2@0 D @2 . Actually we shall discuss two such sentences, AC and AC . These are defined in Section 5.1 and Section 5.3 respectively.

1.3 Pmax variations Starting in Chapter 6, we shall define several variations of the partial order Pmax . Interestingly each variation can be defined as a suborder of a reformulation of Pmax . The  and it is the subject of Section 5.5. A slightly more general reforreformulation is Pmax 0 mulation is Pmax and in Section 5.6 we prove a theorem which shows that essentially any possible variation, subject to the constraint that 2@0 D 2@1 0 in the resulting model, is a suborder of Pmax . The variations yield canonical models which can be viewed as constrained versions of the Pmax model. Generally the constrained versions will realize any …2 sentence in the language for the structure hH.!2 /; INS ; 2i which is (suitably) consistent with the constraint; i. e. unless one takes steps to prevent something from happening it will happen. This is in contrast to the usual forcing constructions where nothing happens unless one works to make it happen. One application will be to establish the consistency with ZFC that the nonstationary ideal on !1 is !1 -dense. This also shows the consistency of the existence of an !1 dense ideal on !1 with :CH. Further for these results only the consistency of ZFCAD is required. This is best possible for we have proved that if there is an !1 -dense ideal on !1 then L.R/  AD: More precisely we shall define a variation of Pmax , which we denote Qmax , and assuming ADL.R/ we shall prove that L.R/Qmax  ZFC C “The nonstationary ideal on !1 is !1 -dense”:

1.3 Pmax variations

11

Again ADL.R/ suffices for the analysis of L.R/Qmax and there are absoluteness theorems which characterize the Qmax -extension. Collectively these results suggest that the consistency of ADL.R/ is an upper bound for the consistency strength of many propositions at !1 , over the base theory, ZFC C “For all x 2 R, x # exists” C “ı12 D !2 ”: However there are two classes of counterexamples to this. Suppose that R# exists and that L.R# /  AD. For each sentence  such that L.R/  ; the following:  There exists a sequence, hB˛;ˇ W ˛ < ˇ < !1 i, of borel sets such that  [ \ R# D B˛;ˇ ; ˛

ˇ >˛

and  L.R/  AD C , can be expressed by a †2 sentence in hH.!2 /; 2i which can be realized by forcing with a Pmax variation over L.R# /. There must exist a choice of  such that this †2 sentence cannot be realized in the structure hH.!2 /; 2i of any set generic extension of L.R/. This is trivial if the extension adds no reals (take  to be any tautology), otherwise it is subtle in that if L.R/  AD then we conjecture that there is a partial order P 2 L.R/ such that L.R/P  ZFC C “R# exists”: The second class of counterexamples is a little more subtle, as the following example illustrates. If the nonstationary ideal on !1 is !1 -dense and if Chang’s Conjecture holds then there exists a countable transitive set, M , such that M  ZFC C “There exist ! C 1 many Woodin cardinals”; (and so M  ADL.R/ and much more). The application of Chang’s Conjecture is only necessary to produce X †2 H.!2 / such that X \ !2 has ordertype !1 . The subtle and interesting aspect of this example is that L.R/Qmax  Chang’s Conjecture; but by the remarks above, this can only be proved by invoking hypotheses stronger than ADL.R/ . In fact the assertion,  L.R/Qmax  Chang’s Conjecture, is equivalent to a strong form of the consistency of AD. This is the subject of Section 9.4.

12

1 Introduction

The statement that the nonstationary ideal on !1 is !1 -dense is a †2 sentence in hH.!2 /; 2; INS i: This is an example of a (consistent) †2 sentence (in the language for this structure) which implies :CH. Using the methods of Section 10.2 a variety of other examples can be identified, including examples which imply c D !2 . Thus in the language for the structure hH.!2 /; 2; INS i there are (nontrivial) consistent †2 sentences which are mutually inconsistent. This is in contrast to the case of …2 sentences. It is interesting to note that this is not possible for the structure hH.!2 /; 2i; provided the sentences are each suitably consistent. We shall discuss this in Chapter 8, (see Theorem 10.159), where we discuss problems related to the problem of the relationship between Martin’s Maximum and the axiom ./. The results we have discussed suggest that if the nonstationary ideal on !1 is !2 saturated, there are large cardinals and if some particular sentence is true in L.P .!1 // then it is possible to force over L.R/ (or some larger inner model) to make this sentence true (by a forcing notion which does not add reals). Of course one cannot obtain models of CH in this fashion. The limitations seem only to come from the following consequence of the saturation of the nonstationary ideal in the presence of a measurable cardinal:  Suppose C  !1 is closed and unbounded. Then there exists x 2 R such that ¹˛ < !1 j L˛ Œx is admissibleº  C: This is equivalent to the assertion that for every x 2 R, x # exists together with the assertion that every closed unbounded subset of !1 contains a closed, cofinal subset which is constructible from a real. Motivated by these considerations we define, in Chapter 7 and Chapter 8, a number of additional Pmax variations. The two variations considered in Chapter 7 were selected simply to illustrate the possibilities. The examples in Chapter 8 were chosen to highlight quite different approaches to the analysis of a Pmax variation, there we shall work in “L”-like models in order to prove the lemmas required for the analysis. It seems plausible that one can in fact routinely define variations of Pmax to reproduce a wide class of consistency results where c D !2 . The key to all of these variations is really the proof of Theorem 1.1. It shows that if the nonstationary ideal on !1 is !2 -saturated then H.!2 / is contained in the limit of a directed system of countable models via maps derived from iterating generic elementary embeddings and (the formation of) end extensions. Here again there is no use of iterated forcing and so the arguments generally tend to be simpler than their standard counterparts. Further there is an extra degree of freedom in the construction of these models which yields consequences not obviously

1.4 Extensions of inner models beyond L.R/

13

obtainable with the usual methods. The first example of Chapter 7 is the variation, Smax , which conditions the model on a sentence which implies the existence of a Suslin tree. The sentence asserts:  Every subset of !1 belongs to a transitive model M in which ˘ holds and such that every Suslin tree in M is a Suslin tree in V . If AD holds in L.R/ and if G  Smax is L.R/-generic then in L.R/ŒG the following strengthening of the sentence holds:  For every A  !1 there exists B  !1 such that A 2 LŒB and such that if T 2 LŒB is a Suslin tree in LŒB, then T is a Suslin tree. In L.R/ŒG every subset of !1 belongs to an inner model with a measurable cardinal (and more) and under these conditions this strengthening is not even obviously consistent. The second example of Chapter 7 is motivated by the Borel Conjecture. The first consistency proof for the Borel Conjecture is presented in .Laver 1976/. The Borel Conjecture can be forced a variety of different ways. One can iterate Laver forcing or Mathias forcing, etc. In Section 7.2, we define a variation of Pmax which forces the Borel Conjecture. The definition of this forcing notion does not involve Laver forcing, Mathias forcing or any variation of these forcing notions. In the model obtained, a version of Martin’s Maximum holds. Curiously, to prove that the Borel Conjecture holds in the resulting model we do use a form of Laver forcing. An interesting technical question is whether this can be avoided. It seems quite likely that it can, which could lead to the identification of other variations yielding models in which the Borel Conjecture holds and in which additional interesting combinatorial facts also hold.

1.4

Extensions of inner models beyond L.R/

In Chapter 9 we again focus primarily on the Pmax -extension but now consider extensions of inner models strictly larger than L.R/. These yield models of ./ with rich structure for H.!3 /; i. e. with “many” subsets of !2 . The ground models that we shall consider are of the form L.; R/ where   P .R/ is a pointclass closed under borel preimages, or more generally inner models of the form L.S; ; R/ where   P .R/ and S  Ord. We shall require that a particular form of AD hold in the inner model, the axiom is ADC which is discussed in Section 9.1. It is by exploiting more subtle aspects of the consequences of ADC that we can establish a number of combinatorially interesting facts about the corresponding extensions. Applications include obtaining extensions in which Martin’s Maximum holds for partial orders of cardinality c, this is Martin’s Maximum.c/, and in which !2 exhibits some interesting combinatorial features.

14

1 Introduction

Actually in the models obtained, Martin’s MaximumCC .c/ holds. This is the assertion that Martin’s MaximumCC holds for partial orders of cardinality c where Martin’s MaximumCC is a slight strengthening of Martin’s Maximum. These forcing axioms, first formulated in .Foreman, Magidor, and Shelah 1988/, are defined in Section 2.5. Recasting the Pmax variation for the Borel Conjecture in this context we obtain, in the spirit of Martin’s Maximum, a model in which the Borel Conjecture holds together with the largest fragment of Martin’s Maximum.c/ which is possibly consistent with the Borel Conjecture. Another reason for considering extensions of inner models larger than L.R/ is that one obtains more information about extensions of L.R/. For example the proof that L.R/Qmax  Chang’s Conjecture; requires considering the .Qmax /N -extension of inner models N such that .R \ N /# 2 N and much more. Finally any systematic study of the possible features of the structure hH.!2 /; INS ; 2i in the context of ZFC C ADL.R/ C “ı12 D !2 ” requires considering extensions of inner models beyond L.R/; as we have indicated, there are (†2 ) sentences which can be realized in the structure, hH.!2 /; INS ; 2i, of these extensions but which cannot be realized in any such structure defined in an extension of L.R/. The results of Chapter 9 suggest a strengthening of the axiom ./:  Axiom ./C : For each set X  !2 there exists a set A  R and a filter G  Pmax such that (1) L.A; R/  ADC , (2) G is L.A; R/-generic and X 2 L.A; R/ŒG. This is discussed briefly in Chapter 10 which explores the possible relationships between Martin’s Maximum and the axiom ./. One of the theorems we shall prove Chapter 10 shows that in Theorem 1.8, it is essential that the predicate, INS , for the nonstationary sets be added to the structure. We shall show that Martin’s Maximum CC .c/ C Strong Chang’s Conjecture together with all the …2 consequences of ./ for the structure hH.!2 /; Y; 2 W Y  R; Y 2 L.R/i does not imply ./. We shall also prove an analogous theorem which shows that “cofinally” many sets from P .R/ \ L.R/ must be added; for each set Y0 2 P .R/ \ L.R/, Martin’s Maximum CC .c/ C Strong Chang’s Conjecture

1.5 Concluding remarks – the view from Berlin in 1999

15

together with all the …2 consequences of ./ for the structure hH.!2 /; INS ; Y0 ; 2i does not imply ./. Finally, we shall also show in Chapter 10 that the axiom ./ is equivalent (in the context of large cardinals) with a very strong form of a bounded version of Martin’s MaximumCC .

1.5

Concluding remarks – the view from Berlin in 1999

The following question resurfaces with added significance.  Assume ADL.R/ . Is ‚L.R/  !3 ? The point is that if it is consistent to have ADL.R/ and ‚L.R/ > !3 then presumably this can be achieved in a forcing extension of L.R/. This in turn would suggest there are generalizations of Pmax which produce generic extensions of L.R/ in which c > !2 . There are many open questions in combinatorial set theory for which a (positive) solution requires building forcing extensions in which c > !2 . The potential utility of Pmax variations for obtaining models in which !3 < ‚L.R/ is either enhanced or limited by the following theorem of S. Jackson. This theorem is an immediate corollary of Theorem 1.3(2) and Jackson’s analysis of measures and ultrapowers in L.R/ under the hypothesis of ADL.R/ . Theorem 1.10 (Jackson). Assume the nonstationary ideal on !1 is !2 -saturated and that there exist ! many Woodin cardinals with a measurable cardinal above them all. Then either: (1) There exists < ‚L.R/ such that is a regular cardinal in L.R/ and such that is not a cardinal in V , or; (2) There exists a set A of regular cardinals, above !2 , such that a) jAj D @1 , b) jpcf.A/j D @2 .

t u

One of the main open problems of Shelah’s pcf theory is whether there can exist a set, A, of regular cardinals such that jAj < jpcf.A/j (satisfying the usual requirement that jAj < min.A/). Common to all Pmax variations is that Theorem 1.3(2) holds in the resulting models and so the conclusions of Theorem 1.10 applies to these models as well. Though,

16

1 Introduction

recently, a more general class of “variations” has been identified for which Theorem 1.3(2) fails in the models obtained. These latter examples are variations only in the sense that they also yield canonical models in which CH fails, cf. Theorem 10.185. I end with a confession. This book was written intermittently over a 7 year period beginning in early 1992 when the initial results were obtained. During this time the exposition evolved considerably though the basic material did not. Except that the material in Chapter 8, the material in the last three sections of Chapter 9 and much of Chapter 10, is more recent. Earlier versions contained sections which, because of length considerations, we have been compelled to remove. This account represents in form and substance the evolutionary process which actually took place. Further a number of proofs are omitted or simply sketched, especially in Chapter 10. Generally it seemed better to state a theorem without proof than not to state it at all. In some cases the proofs are simply beyond the scope of this book and in other cases the proofs are a routine adaptation of earlier arguments. Of course in both cases this can be quite frustrating to the reader. Nevertheless it is my hope that this book does represent a useful introduction to this material with few relics from earlier versions buried in its text. By the time (May, 1999) of this writing a number of papers have appeared, or are in press, which deal with Pmax or variations thereof. P. Larson and D. Seabold have each obtained a number of results which are included in their respective Ph. D. theses, some of these results are discussed in this book. Shelah and Zapletal consider several variations, recasting the absoluteness theorems in terms of “…2 -compactness” but restricting to the case of extensions of L.R/, .Shelah and Zapletal 1999/. More recently Ketchersid, Larson, and Zapletal .2007/ isolate a family of explicit Namba-like forcing notions which can, under suitable circumstances, change the value of  ı 12 even in situations where CH holds. These examples are really the first to be isolated which can work in the context of CH. Other examples have been discovered and are given in .Doebler and Schindler 2009/. Finally there are some very recent developments (as of 1999) which involve a generalization of !-logic which we denote -logic. Arguably -logic is the natural limit of the lineage of generalizations of classical first order logic which begins with !-logic and continues with ˇ-logic etc. We (very briefly) discuss -logic (updated to 2010) in Section 10.4 and Section 10.5. In some sense the entire discussion of Pmax and its variations should take place in the context of -logic and were we to rewrite the book this is how we would proceed. In particular, the absoluteness theorems associated to Pmax and its variations are more naturally stated by appealing to this logic. For example Theorem 1.4 can be reformulated as follows. Theorem 1.11. Suppose that there exists a proper class of Woodin cardinals. Suppose that  is a …2 sentence in the language for the structure hH.!2 /; 2; INS i and that ZFC C “hH.!2 /; 2; INS i  ”

1.5 Concluding remarks – the view from Berlin in 1999

is -consistent, then

Pmax

hH.!2 /; 2; INS iL.R/

 :

17

t u

In fact, using -logic one can give a reformulation of ./ which does not involve forcing at all, this is discussed briefly in Section 10.4. Another feature of the forcing extensions given by the (homogeneous) Pmax variations, this holds for all the variations which we discuss in this book, is that each provides a finite axiomatization, over ZFC, of the theory of H.!2 / (in -logic). For Pmax , the axiom is ./ and the theorem is the following. Theorem 1.12. Suppose that there exists a proper class of Woodin cardinals. Then for each sentence , either (1) ZFC C ./ ` “H.!2 /  ”, or (2) ZFC C ./ ` “H.!2 /  :”.

t u

This particular feature underscores the fundamental difference between the method of Pmax variations and that of iterated forcing. We note that it is possible to identify finite axiomatizations over ZFC of the theory of hH.!2 /; 2i which cannot be realized by any Pmax variation. Theorem 10.185 indicates such an example, the essential feature is that ı12 < !2 but still there is an effective failure of CH. Nevertheless it is at best difficult through an iterated forcing construction to realize in hH.!2 /; 2iV ŒG a theory which is finitely axiomatized over ZFC in -logic. The reason is simply that generally the choice of the ground model will influence, in possibly very subtle ways, the theory of the structure hH.!2 /; 2iV ŒG . There is at present no known example which works, say from some large cardinal assumption, independent of the choice of the ground model. -logic provides the natural setting for posing questions concerning the possibility of such generalizations of Pmax , to for example !2 , i. e. for the structure H.!3 /, and beyond. The first singular case, H.!!C /, seems particularly interesting. There is also the case of !1 but in the context of CH. One interesting result (but as of 2010, this is contingent on the ADC Conjecture), with, we believe, potential implications for CH, is that there are limits to any possible generalization of the Pmax variations to the context of CH; more precisely, if CH holds then the theory of H.!2 / cannot be finitely axiomatized over ZFC in -logic. Acknowledgments to the first edition. Many of the results of the first half of this book were presented in the Set Theory Seminar at UC Berkeley. The (ever patient) participants in this seminar offered numerous helpful suggestions for which I remain quite grateful. I am similarly indebted to all those willingly to actually read preliminary versions of this book and then relate to me their discoveries of mistakes, misprints and relics. I only wish that the final product better represented their efforts. I owe a special debt of thanks to Ted Slaman. Without his encouragement, advice and insight, this book would not exist.

18

1 Introduction

The research, the results of which are the subject of this book, was supported in part by the National Science Foundation through a succession of summer research grants, and during the academic year, 1997–1998, by the Miller Institute in Berkeley. Finally I would like to acknowledge the (generous) support of the Alexander von Humboldt Foundation. It is this support which enabled me to actually finish this book. Berlin, May 1999

1.6

W. Hugh Woodin

The view from Heidelberg in 2010

In the 10 years since what was written above as the introduction to the first edition of this book there have been quite a number of mathematical developments relevant to this book and I find myself again in Germany on sabbatical from Berkeley working on this book. This edition contains revisions that reflect these developments including the deletion of some theorems now not relevant because of these developments or simply because the proofs, sketched or otherwise, were simply not correct. Finally I stress that I make no claim that this revision is either extensive or thorough and I regret to say that it is not – I feel that the entire subject is at a critical crossroads and as always in such a situation one cannot be completely confident in which direction the future lies. But it is this future that dictates which aspects of this account should be stressed. First and most straightforward, the theorems related to ˘! .!2 /, such as the theorem that Martin’s Maximum implies ˘! .!2 /, have all been rendered irrelevant by a remarkable theorem of .Shelah 2008/ which shows that ˘! .!2 / is a consequence of 2!1 D !2 . Shelah’s result shows that assuming Martin’s Maximum.c/, or simply assuming that 2!1 D !2 , then the nonstationary ideal at !2 cannot be semi-saturated on the ordinals of countable cofinality. It does not rule out the possibility that there exists a uniform semi-saturated at !2 on the ordinals of countable cofinality. On the other hand, the primary motivation for obtaining such consistency results for ideals at !2 in the first edition was the search for evidence that the consistency strength of the theory ZF C ADR C “‚ is regular” was beyond that of the existence of a superstrong cardinals. Dramatic recent results .Sargsyan 2009/ have shown that this theory is not that strong, proving that the consistency of this theory follows from simply the existence of a Woodin cardinal which is a limit of Woodin cardinals. Therefore in this edition the consistency results for semisaturated ideals at !2 are simply stated without proof. The proofs of these theorems are sketched at length in the first edition but based upon an analysis in the context of ADC of HOD which is open without requiring that one work relative to the minimum model of ZF C ADR C “‚ is Mahlo” but of course the sketch in the case of obtaining the consistency that JNS is semisaturated is not correct – that error was due to a careless misconception regarding

1.5 The view from Heidelberg in 2010

19

iterations of forcing with uncountable support. As indicated in the first edition the analysis of HOD in the context of ADC is not actually necessary for the proofs, it was used only to provide a simpler framework for the constructions. Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work. The fundamental result of this book is the identification of a canonical axiom for :CH which is characterized in terms of a logical completion of the theory of H.!2 / (in logic of course). But the validation of this axiom requires a synthesis with axioms for V itself for otherwise it simply stands as an isolated axiom. This view is reinforced by the use of the  Conjecture to argue against the generic-multiverse view of truth .Woodin 2009/. I remain convinced that if CH is false then the axiom ./ holds and certainly there are now many results confirming that if the axiom ./ does hold then there is a rich structure theory for H.!2 / in which many pathologies are eliminated. But nevertheless for all the reasons discussed at length in .Woodin 2010b/, I think the evidence now favors CH. The picture that is emerging now based on .Woodin 2010b/ and .Woodin 2010a/ is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model. At present there seem to be two possibilities for this enlargement, as an extender model or as strategic extender model. There is a key distinction however between these two versions. An extender model in which there is a Woodin cardinal is a (nontrivial) generic extension of an inner model which is also an extender model whereas a strategic extender model in which there is a proper class of Woodin cardinals is not a generic extension of any inner model. The most optimistic generalizations of the structure theory of L.R/ in the context of AD to a structure theory of L.VC1 / in the context of an elementary embedding, j W L.VC1 / ! L.VC1 / with critical point below require that V not be a generic extension of any inner model which is not countably closed within V . Therefore these generalizations cannot hold in the extender models and this leave the strategic extender models as essentially the only option. Thus there could be a compelling argument that V is a strategic extender model based on natural structural principles. This of course would rule out that the axiom ./ holds though if V is a strategic extender model (with a Woodin cardinal) then the axiom ./ holds in a homogeneous forcing extension of V and so the axiom ./ has a special connection to V as an axiom which holds in a canonical companion to V mediated by an intervening model of ADC which is the manifestation of -logic. An appealing aspect to this scenario is that the relevant axiom for V can be explicitly stated now – and in a form which clarifies the previous claims – without knowing the detailed level by level inductive definition of a strategic extender model .Woodin 2010b/: in its weakest form the axiom is simply the conjunction of:

20

1 Introduction

(1) There is a supercompact cardinal. (2) There exist a universally Baire set A  R and  < ‚L.A;R/ such that V .HOD/L.A;R/ \ V for all …2 -sentences (equivalently, for all †2 -sentences). As with the previous scenarios this scenario could collapse but any scenario for such a collapse which leads back to the validation of the axiom ./ seems rather unlikely at present. Acknowledgments to the second edition. I am very grateful to all of those who sent me lists of errata for the first edition or otherwise offered valuable comments, I wish this edition better reflected their efforts. I would also like to thank Christine Woodin for an extremely useful python script for finding unbalanced parentheses in very large LATEX files. Heidelberg, March 2010

W. Hugh Woodin

Chapter 2

Preliminaries

We briefly review, without giving all of the proofs, some of the basic concepts which we shall require, .Foreman and Kanamori (Eds.) 2010/ covers most of what we need and obviously quite a bit more. In the course of this we shall fix some notation. As is the custom in Descriptive Set Theory, R denotes the infinite product space, ! ! . Though sometimes it is convenient to work with the Cantor space, 2! , or even with the standard Euclidean space, . 1; 1/. If at some point the discussion is particularly sensitive to the manifestation of R then we may be more careful with our notation. For example L.R/ is relatively immune to such considerations, but Wadge reducibility is not. We shall require at several points some coding of sets by reals or by sets of reals. There is a natural coding of sets in H.!1 / (the hereditarily countable sets) by reals. For example if a 2 H.!1 / then the set a can be coded by coding the structure hb [ !; a; 2i where b is the transitive closure of a. A real x codes a if x decodes sets A  ! and E  !  ! such that hb [ !; a; 2i Š h!; A; Ei; where again b is the transitive closure of a. Suppose that M 2 H.c C / and let N be the transitive closure of M . Fix a reasonable decoding of a set X  R to produce an element of P .R/  P .R  R/  P .R  R/: A set X  R codes M if X decodes sets A  R, E  R  R and   R  R such that  is an equivalence relation on R, A  R, E  R  R, A and E are invariant relative to , and such that hN; M; 2i Š hR=; A=; E=i: We shall be interested in sets M which are coded in this fashion by sets X  R such that X belongs to a transitive inner model in which the Axiom of Choice fails.

2.1

Weakly homogeneous trees and scales

For any set X , X
22

2 Preliminaries

(1) s 2 !
Tx D [¹Txjk j k 2 !º:

Thus for each x 2 ! , Tx is a tree on . We let !

ŒT  D ¹.x; f / j x 2 ! ! ; f 2 ! ; and for all k 2 !; .xjk; f jk/ 2 T º denote the set of infinite branches of T and we let pŒT  D ¹x 2 ! ! j .x; f / 2 ŒT  for some f 2 ! º: Thus pŒT   ! ! , it is the projection of T , and clearly pŒT  D ¹x 2 ! ! j Tx is not wellfoundedº: A set of reals, A, is Suslin if A D pŒT  for some tree T . Of course assuming the Axiom of Choice every set is Suslin. One can obtain a more interesting notion by restricting the choice of the tree. This can done two ways, by definability or by placing combinatorial constraints on the tree. The first route is the descriptive set theoretic one. A pointclass is a set   P .! ! /. Suppose that  is a pointclass and that for any continuous function F W !! ! !! if A 2  then F 1 ŒA 2 ; i. e. suppose  is closed under continuous preimages. Then  has an unambiguous interpretation as a subset of P .X / where X is any space homeomorphic with ! ! . The point of course is that this does not depend on the homeomorphism. We shall use this freely. Similarly if in addition,  is closed under finite intersections and contains the closed sets, then  has an unambiguous interpretation as a subset of P .X / where X is any space homeomorphic with a closed subset of ! ! . If  is a pointclass closed under preimages by borel functions then  has an unambiguous interpretation as a subset of P .X / where X is any space homeomorphic with a borel subset of ! ! . If the borel set is uncountable, i. e. if X is uncountable, then the pointclass, , is uniquely determined by this interpretation. More generally if X is a topological space for which there is an isomorphism  W hX; .Z.X //i ! h! ! ; B.! ! /i where .Z.X // is the -algebra generated by the zero sets of X , and B.! ! / is the -algebra of borel subsets of ! ! , then again  has an unambiguous interpretation as a subset of P .X / which again uniquely determines . This includes any space we shall ever need to interpret  in. We shall almost exclusively be dealing with pointclasses closed under preimages by borel functions.

2.1 Weakly homogeneous trees and scales

23

Suppose that  is a pointclass. Then : denotes the pointclass obtained from complementing the sets in , : D ¹! ! n A j A 2 º: Clearly if  is closed under continuous preimages then so is the dual pointclass, :. Moschovakis introduced the fundamental notion in descriptive set theory of a scale, (see .Moschovakis 1980/). We recall the definition. Definition 2.1. Suppose that  is a pointclass closed under continuous preimages. (1) Suppose that A 2 . The set A has a -scale if there is a sequence hi W i 2 !i of prewellorderings on A such that the following conditions hold. a) The set ¹hi; x; yi j i 2 !; x i yº belongs to . b) There exists Y 2 : such that Y  !  !!  !! and such that for all i < !, i D Yi \ .R  A/: where Yi D ¹.x; y/ j .i; x; y/ 2 Y º is the section given by i . c) Suppose that hxi W i < !i is a sequence of reals in A which converges to x. Suppose that for each i there exists i  such that xj i xi  and xi  i xj for all j i  . Then x 2 A and for all i < !, x i x i  : (2) The pointclass  has the scale property if every set in  has a -scale.

t u

The notion of a scale is closely related to Suslin representations. Remark 2.2. (1) If the pointclass  is a -algebra closed under continuous preimages and if  contains the open sets then a set A 2  has a -scale if and only if there is a sequence hi W i < !i of prewellorderings on A such that each belongs to  and the condition (c) of the definition holds. (2) If  is a -algebra closed under both continuous preimages and continuous images then a set A 2  has a -scale if and only if A D pŒT  for some tree T which is coded by a set in . t u

24

2 Preliminaries

Recall that a set A 2 P .R/ \ L.R/ is †21 -definable in L.R/ if and only if it is †1 definable in L.R/ with parameter R. Assuming the Axiom of Choice fails in L.R/, then it is easily verified that there must exist a set A 2 P .R/ \ L.R/, such that R n A is †21 -definable in L.R/ and such that A is not Suslin in L.R/. The following theorem of Martin and Steel .1983/ shows that assuming .AD/L.R/ , 2 L.R/ the pointclass .† has the scale property. By the remarks above this is best pos1 / sible. In fact it follows by Wadge reducibility that, assuming .AD/L.R/ , every set A 2 P .R/ \ L.R/ 2 L.R/ . which is Suslin in L.R/, is necessarily .† 1 / This theorem will play an important role in the analysis of the Pmax extension of L.R/. Theorem 2.3 (Martin–Steel). Suppose that L.R/  AD: 2 Then every set A  R which is †1 -definable in L.R/ has a scale which is †21 -definable in L.R/. t u Suppose that X is a nonempty set. We let m.X / denote the set of countably complete ultrafilters on the boolean algebra P .X /. Our convention is that is a measure on X if 2 m.X /. As usual for 2 m.X / and A  X , we write .A/ D 1 to indicate that A 2 . Suppose that X D Y
1 if k1 < k2 and, for all A  Y k1®, 1 .A/ D 1 if and only ¯ if 2 .A / D 1 where  k2 A D s 2 Y j sjk1 2 A : We write 1 < 2 to indicate that 2 projects to 1 . For each 2 m.X / there is a canonical elementary embedding j W V ! M where M is the transitive inner model obtained from taking the transitive collapse of V X = . Suppose that 1 2 m.Y
k1 < k2 . The tower, h k W k 2 !i, is countably complete if for any sequence hAk W k 2 !i such that for all k < !, k .Ak / D 1, there exists f 2 Y ! such that f jk 2 Ak for all k 2 !. It is completely standard that if h k W k 2 !i is a tower of measures on Y
2.1 Weakly homogeneous trees and scales

25

We come to the key notions of homogeneous trees and weakly homogeneous trees. These definitions are due independently to Kunen and Martin. Definition 2.4. Suppose that is an ordinal and ¤ 0. Suppose that T is a tree on !  . (1) The tree T is ı-weakly homogeneous if there is a partial function  W !
t u

Any tree on !  ! is ı-weakly homogeneous for all ı and similarly any tree on !  1 is ı-homogeneous for all ı. In each case the associated measures are principal. The definition of a weakly homogeneous tree has a simple reformulation which is frequently more relevant to the process of actually verifying that specific trees are weakly homogeneous. This reformulation is given in the following lemma which we leave as an exercise.

26

2 Preliminaries

Lemma 2.6. Suppose that T is a tree on !  . Then T is ı-weakly homogeneous if and only if there exists a countable set  m.
k .Txjk / D 1. Homogeneity is a rather restrictive condition on a tree, weak homogeneity, however, is not. For example if ı is a Woodin cardinal and T is a tree on !  for some then there exists an ordinal ˛ < ı such that if G  Coll.!; ˛/ is V -generic then in V ŒG, T is 32

2 Preliminaries

Lemma 2.27. Suppose T1 is a tree on !  1 , T2 is a tree on !  2 , and pŒT1  D pŒT2 : Suppose T1 and T2 are ı-weakly homogeneous. Then .pŒT1 /V ŒG D .pŒT2 /V ŒG where G  P is V -generic for a partial order P 2 Vı .

t u

Suppose A  R and let  D ¹B  R j B is projective in Aº: Suppose every set in  is ı-weakly homogeneously Suslin. Suppose .x1 ; x2 / is a formula in the language of the structure hH.!1 /; A; 2i and a 2 R. Let

B D ¹t 2 R j hH.!1 /; A; 2i  Œt; aº:

Thus B 2 . Suppose P 2 Vı is a partial order and that G  P is V -generic. Let AG and BG be the interpretations of A and B in V ŒG. Then ® ¯ BG D t 2 R j hH.!1 /V ŒG ; AG ; 2i  Œt; a : This is an easy consequence of Lemma 2.26 and Lemma 2.27. Alternate formulations are given in the next two lemmas. Lemma 2.28. Suppose A  R and let B  R be the set of reals which code elements of the first order diagram of the structure hH.!1 /; 2; Ai: Suppose S and T are trees on !  such that (1) S and T are ı-weakly homogeneous, (2) A D pŒS  and B D pŒT . Suppose P 2 Vı and G  P is V -generic. Let AG D pŒS  and let BG D pŒT , each computed in V ŒG. Then in V ŒG, BG is the set of reals which code elements of the first order diagram of the structure hH.!1 /V ŒG ; 2; AG i:

t u

Lemma 2.29. Suppose A  R and suppose that each set B  R which is projective in A, is ı-weakly homogeneously Suslin. Suppose Z  V˛ is a countable elementary substructure such that ı C ! < ˛, ı 2 Z and such that A 2 Z. Let MZ be the transitive collapse of Z and let ıZ be the image of ı under the collapsing map. Suppose P 2 .MZ /ıZ is a partial order and that g  P is MZ generic.

2.2 Generic absoluteness

33

Then (1) A \ MZ Œg 2 MZ Œg, (2) hV!C1 \ MZ Œg; A \ MZ Œg; 2i  hV!C1 ; A; 2i. Suppose further that A 2 .ıWH /V˛ : Then A \ MZ Œg 2 .ıWH /MZ Œg : Z

t u

Suppose that A  R and that every set B 2 P .R/ \ L.A; R/ is ı-weakly homogeneously Suslin. Then .A; R/# is ı-weakly homogeneously Suslin. This is easily verified by noting that .A; R/# is a countable union of sets in L.A; R/. This observation yields the following generic absoluteness theorem. Theorem 2.30. Suppose that A  R and that every set in P .R/\L.A; R/ is ı-weakly homogeneously Suslin. Suppose that T is a ı-weakly homogeneous tree such that A D pŒT  and that P 2 Vı is a partial order. Suppose that G  P is V -generic. Then there is a generic elementary embedding jG W L.A; R/ ! L.AG ; RG / such that (1) jG .A/ D AG D pŒT V ŒG , (2) RG D RV ŒG , (3) L.AG ; RG / D ¹jG .f /.a/ j a 2 RG ; f W R ! L.A; R/ and f 2 L.A; R/º. Further the properties (1)–(3) uniquely specify jG .

t u

One corollary of Theorem 2.30 is the following generic absoluteness theorem which we shall need. Theorem 2.31. Suppose that ı is a limit of Woodin cardinals and that there a measurable cardinal above ı. Suppose that GP is V -generic where P is a partial order such that P 2 Vı . Then L.R/V L.R/V ŒG : Proof. By Theorem 2.13, each set X 2 P .R/ \ L.R/ is
t u

2.3 The stationary tower

35

Definition 2.34 (Stationary Tower). Suppose a and b are stationary sets. Then a  b if [b  [a and ¹ \ .[b/ j 2 aº  b: (1) For each ordinal ˛, P such that: a) X  Y and jY j D !1 ; b) Z \ P!1 .Y / contains a set which is closed and unbounded in P!1 .Y /. u t Remark 2.55. (1) The principle WRP was introduced in .Foreman, Magidor, and Shelah 1988/ as Strong Reflection. It implies the (weaker) assertion that for any partial order P , P is semiproper if and only if forcing with P preserves stationary subsets of !1 , see .Foreman, Magidor, and Shelah 1988/. Interestingly, Todorcevic had previously proved that a special case of WRP implies that c  @2 . The results of .Foreman, Magidor, and Shelah 1988/ show that WRP is consistent with CH. (2) The principle SRP was formulated in .Todorcevic 1984/ and is based on Shelah’s proof that Martin’s Maximum is equivalent to SPFA. The precise formulation given in Definition 2.54(2) is the principle of Projective Stationary Reflection of Feng and Jech .1998/. Feng and Jech proved that Projective Stationary Reflection is actually equivalent to Todorcevic’s principle. (3) SRP implies WRP and many of the consequences of Martin’s Maximum follow from it. For example, SRP implies the nonstationary ideal on !1 is saturated and that 2@1  @2 see .Todorcevic 1984/. It will follow from the principal results of Chapter 3, that SRP implies that ı12 D !2 and so SRP implies that c D @2 . Theorem 9.79 shows that a fairly weak fragment of SRP suffices. (4) Both WRP and SRP follow from SPFA. (5) One can show that SRP is consistent with the existence of a Suslin tree on !1 and so SRP does not imply Martin’s Maximum. u t

2.6

Generic ideals

One of the main results of Chapter 3 is that if the nonstationary ideal on !1 is !2 saturated and if there is a measurable cardinal, then there is an effective failure of CH. The force of this result is greatly amplified by the results of .Foreman, Magidor, and Shelah 1988/ and Shelah .1987/ which show that if suitable large cardinals exist then there is a semiproper partial order P such that in V P , the nonstationary ideal on !1 is !2 -saturated.

44

2 Preliminaries

Combining these results yields that the effective version of the Continuum Hypothesis is as intractable a problem as the Continuum Hypothesis itself. We review briefly the results of .Foreman, Magidor, and Shelah 1988/ and .Shelah and Woodin 1990/. We begin with the key definition. Suppose that A  P .!1 / n INS is nonempty. Let PA denote the following partial order. Conditions are pairs .f; c/ such that (1) for some ˛ < !1 , f W ˛ ! A, (2) c  !1 is a countable closed subset such that for each ˇ 2 c, if ˇ 2 dom.f / then ˇ 2 f ./ for some  < ˇ, and such that c ¤ ;. The ordering on PA is by extension. Suppose that .f1 ; c1 / 2 PA and that .f2 ; c2 / 2 PA . Then .f2 ; c2 /  .f1 ; c1 / if f1  f2 and c1 D c2 \ .max.c1 / C 1/. We note that if .f; c/ 2 PA then necessarily sup.c/ 2 c. This is because c is closed in !1 and not cofinal. One of the key theorems of .Foreman, Magidor, and Shelah 1988/ is that if A  P .!1 / n INS is predense in .P .!1 / n INS ; / then forcing with PA preserves stationary subsets of !1 . It is not difficult to show that PA is proper if and only if there exists a sequence hA˛ W ˛ < !1 i of elements of A and a closed cofinal set C  !1 such that for all ˛ 2 C, ˛ 2 Aˇ for some ˇ < ˛. The question of when the partial order PA is semiproper is more interesting. This isolates a fundamental combinatorial condition on the predense set A which we define below. This condition is implicit in .Foreman, Magidor, and Shelah 1988/. Definition 2.56. Suppose that A  P .!1 / n INS : Then A is semiproper if for any transitive set M such that M P .H.!2 //  M; if X M is a countable elementary substructure such that A 2 X , then there exists a countable elementary substructure Y M

2.6 Generic ideals

45

such that (1) X  Y , (2) X \ !1 D Y \ !1 , (3) Y \ !1 2 S for some S 2 Y \ A.

t u

The selection of name semiproper in Definition 2.56 is explained in the following lemma. Lemma 2.57 (Foreman–Magidor–Shelah). Suppose that A  P .!1 / n INS is nonempty. Then the following are equivalent. (1) A is semiproper. (2) The partial order PA is semiproper.

t u

The nonstationary ideal on !1 is presaturated if for any A 2 P .!1 / n INS and for any sequence hAi W i < !i of maximal antichains in P .!1 / n INS there exists B  A such that B … INS and such that for each i < !, ¹X 2 Ai j X \ B … INS º has cardinality at most !1 . Theorem 2.58 (Foreman–Magidor–Shelah). Suppose that for each predense set A  P .!1 / n INS ; A is semiproper. Then the nonstationary ideal on !1 is precipitous.

t u

Theorem 2.59 (Foreman–Magidor–Shelah). Suppose that is a supercompact cardinal and that G  Coll.!1 ; < / is V -generic. Then in V ŒG, (1) each predense set A  P .!1 / n INS is semiproper, (2) the nonstationary ideal on !1 is presaturated.

t u

The large cardinal hypothesis of Theorem 2.59 can be reduced, this yields the following theorem. Theorem 2.60. Suppose that ı is a Woodin cardinal and that G  Coll.!1 ;
® ¯ .M !1 /M D f W !1M ! M j f 2 M :

Then hN; Ei  ZFC and the natural map j WM !N is an elementary embedding from the structure hM; 2i into hN; Ei.

t u

Let S be the set of stationary subsets of !1 . The partial order .S; / is not separative. It is easily verified that RO.S; / D RO.P .!1 /=INS /: Definition 3.3. Suppose M is a model of ZFC . (1) .P .!1 / n INS /M denotes the partial order .S; / computed in M . (2) A filter G  .P .!1 / n INS /M is M -generic if G \ D ¤ ; for all predense sets D 2 M.

3.1 The nonstationary ideal and ı12

53

(3) M  “The nonstationary ideal on !1 is !2 saturated ” if in M every predense subset of .P .!1 / n INS /M contains a predense subset of t u cardinality !1M in M . Remark 3.4. (1) “The nonstationary ideal is saturated” has several possible formulations within ZFC and they are not in general equivalent. (2) H.!2 /  “The nonstationary ideal on !1 is !2 saturated ”. (3) Suppose the nonstationary ideal on !1 is !2 -saturated, M is a transitive set, M  ZFC , and P .!1 /  M . Then M  “The nonstationary ideal on !1 is !2 saturated”: (4) Suppose that M is a transitive model of ZFC , .P .!1 //M 2 M; and that G  .P .!1 / n INS /M is a filter such that G \ D ¤ ; for all dense sets D 2 M . Then G is M -generic. t u Definition 3.5. Suppose that M is a countable model of ZFC . A sequence hMˇ ; G˛ ; j˛;ˇ W ˛ < ˇ <  i is an iteration of M if the following hold. (1) M0 D M . (2) j˛;ˇ W M˛ ! Mˇ is a commuting family of elementary embeddings. (3) For each  C 1 <  , G is M -generic for .P .!1 / n INS /M , M C1 is the M ultrapower of M by G and j ; C1 W M ! M C1 is the induced elementary embedding. (4) For each ˇ <  if ˇ is a (nonzero) limit ordinal then Mˇ is the direct limit of ¹M˛ j ˛ < ˇº and for all ˛ < ˇ, j˛;ˇ is the induced elementary embedding. If  is a limit ordinal then  is the length of the iteration, otherwise the length of the iteration is ı where ı C 1 D  . A model N is an iterate of M if it occurs in an iteration of M . The model M is iterable if every iterate of M is wellfounded. t u Remark 3.6. (1) In many instances a slightly weaker notion suffices. A model M is weakly iterable if for any iterate N of M , !1N is wellfounded. For elementary substructures of H.!2 / weak iterability is equivalent to iterability. (2) Suppose M is a countable iterable model of ZFC. Then: M  “The nonstationary ideal is precipitous ”:

54

3 The nonstationary ideal

(3) It will be our convention that the assertion,  j W M ! M  is an embedding given by an iteration of M of length  , abbreviates the supposition that there is an iteration hMˇ ; G˛ ; j˛;ˇ W ˛ < ˇ <  C 1i of M such that M D M  and such that j D j0; : (4) Suppose M is a countable model of ZFC . Then any iteration of M has length at most !1 . (5) The assertion that a countable transitive model M is iterable is a …12 statement about M and therefore is absolute. (6) Suppose M is iterable and N  M is an elementary substructure then in general N may not be iterable. This will follow from results later in this section. In fact here are two natural conjectures. a) Suppose there is no transitive inner model of ZFC containing the ordinals with a Woodin cardinal. Suppose M is a countable transitive model of ZFC and that M is iterable. Suppose X  M . Then the transitive collapse of X is iterable. b) Suppose there is no transitive inner model of ZFC containing the ordinals with a Woodin cardinal for which the sharp of the model exists. Suppose M is a countable transitive model of ZFC and that M is iterable. Suppose M  “The nonstationary ideal on !1 is !2 saturated”: Suppose X  M , NX 2 M and NX is countable in M where NX is the transitive collapse of X ,  2 M and where M  ZFC . Then NX is not iterable. t u Remark 3.7. We shall usually only consider iterations of M in the case that in M , INS is saturated. We caution that without this restriction it is possible that M be iterable but that H.!2 /M not be iterable. If in M , INS is saturated and if M is iterable then H.!2 /M is also iterable. This is a corollary of the next lemma. The correct notion of iterability for those transitive sets in which INS is not saturated is slightly different, see Definition 4.23. t u The next two lemmas record some basic facts about iterations that we shall use frequently. These are true in a much more general context. Lemma 3.8. Suppose that M and M  are countable models of ZFC such that 

(i) !1M D !1M , 

(ii) P .!1 /M D P .!1 /M .

3.1 The nonstationary ideal and ı12

55

Suppose that either 

(iii) P 2 .!1 /M D P 2 .!1 /M , or (iv) M   The nonstationary ideal on !1 is !2 saturated; and that hMˇ ; G˛ ; j˛;ˇ W ˛ < ˇ <  i is an iteration of M . Then there corresponds uniquely an iteration  W ˛ < ˇ < i hMˇ ; G˛ ; j˛;ˇ

of M  such that for all ˛ < ˇ <  : Mˇ

(1) !1

M

D !1 ˇ ; 

(2) P .!1 /Mˇ D P .!1 /Mˇ ; (3) G˛ D G˛ .  .M / 2 Mˇ and there is Suppose further that M 2 M  . Then for all ˇ <  , j0;ˇ an elementary embedding  .M / kˇ W Mˇ ! j0;ˇ  jM D kˇ ı j0;ˇ . such that j0;ˇ

Proof. This is immediate by induction on  .

t u

Remark 3.9. The Lemma 3.8 has an obvious interpretation for arbitrary models. We shall for the most part only use it for wellfounded models. t u For the second lemma we need to use a stronger fragment of ZFC. There are obvious generalizations of this lemma, see Remark 3.11. Lemma 3.10. Suppose M is a countable transitive model of ZFC C Powerset C AC C †1 -Replacement in which the nonstationary ideal on !1 is !2 -saturated. Suppose hMˇ ; G˛ ; j˛;ˇ W ˛ < ˇ <  i is an iteration of M such that   M \ Ord. Then Mˇ is wellfounded for all ˇ <  . Proof. Let .0 ; 0 ; 0 / be the least triple of ordinals in M such that: (1.1) M  “cof. 0 / > !1 ”; (1.2) 0 < 0 ;

56

3 The nonstationary ideal

(1.3) there is an iteration, of V0

hNˇ ; G˛ ; j˛;ˇ W ˛ < ˇ < 0 C 1i; \ M such that j0;0 .0 / not wellfounded.

Choose .0 ; 0 ; 0 / minimal relative to the lexicographical order. Thus 0 and 0 are limit ordinals. Let hNˇ ; G˛ ; j˛;ˇ W ˛ < ˇ < 0 C 1i be an iteration of V0 \ M of length 0 such that j0;0 .0 / is not wellfounded. Choose ˇ  < 0 and  such that  < j0;ˇ  .0 / and such that jˇ  ;0 . / is not wellfounded. Let hMˇ ; G˛ ; k˛;ˇ W ˛ < ˇ < 0 C 1i be the induced iteration of M . By the minimality of 0 it follows that Mˇ is wellfounded for all ˇ < 0 . The key point is that for any ˇ 2 M \ Ord if G  Coll.!; ˇ/ then the set M ŒG is 1 -correct. Thus .0 ; 0 ; 0 / can be defined in M . More precisely .0 ; 0 ; 0 / is least † 1 such that: (2.1) M  “cof. 0 / > !1 ”; (2.2) 0 < 0 ; (2.3) there exist an ordinal ˇ 2 M , an M -generic filter G  Coll.!; ˇ/, and an iteration,  W ˛ < ˇ < 0 C 1i 2 M ŒG; hNˇ ; G˛ ; j˛;ˇ of V0 \ M of length 0 such that j0;0 .0 / not wellfounded. Further since Mˇ  is wellfounded the same considerations apply to Mˇ  and so .j0;ˇ  .0 /; j0;ˇ  . 0 /; j0;ˇ  .0 // must be the triple as defined in V for Mˇ  . However the tail of the iteration hNˇ ; G˛ ; j˛;ˇ W ˛ < ˇ < 0 C 1i  starting at ˇ is an iteration of j0;ˇ  .V0 \ M / of length at most 0 and 0 C 1  j0;ˇ  .0 / C 1: Further the image of  by this iteration is not wellfounded. This is a contradiction t u since  < j0;ˇ  .0 /. Remark 3.11. Lemma 3.10 can be easily generalized to any iteration of generic elementary embeddings. A generic elementary embedding is an elementary embedding j WV !M VP where M is the transitive collapse of the ultrapower, Ult.V; E/ of V by E where E is a V -extender in V P . As usual, this ultrapower is computed using only functions in V . t u

3.1 The nonstationary ideal and ı12

57

Lemma 3.12. Let M be a transitive set such that M  ZFC and such that P .!1 /  M . Suppose the nonstationary ideal on !1 is !2 -saturated in M , X  M and that X is countable. Let ˛ D X \ !1 and let Y D ¹f .˛/ j f 2 X º: Let NX D collapse.X /, let NY D collapse.Y /, and let j W NX ! NY be the induced ® ¯ embedding. Finally let G D A j A 2 NX ; and !1NX 2 j.A/ . Then (1) Y  M . (2) j is an elementary embedding. (3) G is NX -generic for P .!1 / n INS .computed in NX /. (4) NY is the generic ultrapower of NX by G and j is the corresponding generic elementary embedding. Proof. This is straightforward. Since M  ZFC it follows that Y  M . The rest of the lemma follows provided we can show the following: Claim: Suppose A  P .!1 / is a set of stationary subsets of !1 which defines a maximal antichain in P .!1 / n INS . Suppose A 2 X . Then X \ !1 2 S for some S 2 X \ A. Since the nonstationary ideal is saturated in M , every antichain has cardinality at most !1 . Thus suppose A D ¹S˛ j ˛ < !1 º is a maximal antichain of stationary subsets of !1 and A 2 X . Since A is a maximal antichain, the diagonal union 5¹S˛ W ˛ < !1 º contains a set C which is a club in !1 . Since X  M , we can choose C such that t C 2 X in which case X \ !1 2 C . Therefore X \ !1 2 Sˇ for some ˇ < X \ !1 . u Corollary 3.13. Let M be a transitive set such that M  ZFC and such that P .!1 /  M . Suppose the nonstationary ideal on !1 is !2 -saturated in M , X  M and that X is countable. Let NX be the transitive collapse of X and let !1X D X \ !1 . Then there is a wellfounded iteration j W NX ! N of NX such that j.!1X / D !1 and such that for all A 2 X \ H.!2 / j.AX / D A where AX is the image of A under the collapsing map.

58

3 The nonstationary ideal

Proof. Define an !1 sequence hX˛ W ˛ < !1 i of countable elementary substructures of M by induction on ˛: (1.1) X0 D X ; (1.2) for each ˛ < !1 , X˛C1 D ¹f .X˛ \ !1 / j f 2 X˛ ºI (1.3) for each limit ordinal ˛ < !1 , X˛ D [¹Xˇ j ˇ < ˛º: Let X!1 D [¹X˛ j ˛ < !1 º. For each ˛  !1 let

N˛ D collapse.X˛ /

and for each ˛ < ˇ  !1 let j˛;ˇ W N˛ ! Nˇ the elementary embedding obtained from the collapse of the inclusion map X˛  Xˇ . Thus N0 D NX and by induction on ˛  !1 using Lemma 3.12, it follows that for each ˛ < !1 , N˛C1 is a generic ultrapower of N˛ and j˛;˛C1 W N˛ ! N˛C1 is the induced embedding. Therefore j0;!1 W N0 ! N!1 is obtained via an iteration of length !1 . Finally !1  X!1 . Hence j0;!1 .!1X / D !1 and j0;!1 .AX / D A for each set A 2 X \ H.!2 /.

t u

Lemma 3.14. Suppose that the nonstationary ideal on !1 is !2 -saturated. Let M be a transitive set such that M  ZFC and such that P .!1 /  M . Suppose M # exists. Then ¹X  M j X is countable and MX is iterable º contains a club in P!1 .M /. Here MX is the transitive collapse of X . Proof. Fix a stationary set S  P!1 .M /: It suffices to find a countable elementary substructure X  M such that X 2 S and such that MX is iterable. Fix a cardinal  such that M 2 V and such that V  ZFC : Thus M # 2 V . Let Y  V be a countable elementary substructure with M 2 Y and such that Y \ M 2 S . Let X D Y \ M . We claim that MX is iterable. To see this let

3.1 The nonstationary ideal and ı12

59

NY be the transitive collapse of Y and let  W Y ! NY be the collapsing map. X D Y \ M and M # 2 Y and so .M # / D .MX /# . NY  ZFC . Let G  Coll.!; MX / be NY -generic. Let xG 2 R be the code of MX given by G, this is the real given by ¹.i; j / j p.i / 2 p.j / for some p 2 Gº: # Thus xG 2 NY ŒG and so NY ŒG is correct in V for …12 statements about xG . Therefore if MX is not iterable then MX is not iterable in NY ŒG. Assume toward a contradiction that ˇ 2 NY and that there is an iteration in NY ŒG of MX of length ˇ which is not wellfounded. Then by Lemma 3.8 this defines an iteration of NY of length ˇ t u which is not wellfounded, a contradiction since ˇ 2 NY . The next lemma gives the key property of iterable models. For this we shall need some mild coding. There is a natural partial map  W R ! H.!1 / such that: (1)  is onto; (2) (definability)  is 1 -definable; (3) (absoluteness) If x 2 dom./ and .x/ D a then M  “.x/ D a” where M is any ! model of ZFC containing x and a; (4) (boundedness) if A  dom./ is  †11 then ¹rank..x// j x 2 Aº is bounded by the least admissible relative to the parameters for A. For example one can code a set X 2 H.!1 / by relations P  ! and E  !  ! where  h!; P; Ei Š hY [ !; X; 2i,  Y is the transitive closure of X . Lemma 3.15. Suppose M is an iterable countable transitive model of ZFC . Suppose N is an iterate of M by a countable iteration of length ˛. Suppose x is a real which codes M and ˛. Then rank.N / <  where  is least ordinal which is admissible for x. Proof. Let x 2 R code M and let y 2 R code ˛. Then by the properties of the coding map , the set of z 2 dom./ such that .z/ is an iteration of M of length ˛ is t u †11 .x; y/. The result now follows by boundedness. Theorem 3.16. Suppose that the nonstationary ideal on !1 is !2 -saturated. The following are equivalent. (1)  ı 12 D !2 . (2) There exists a countable elementary substructure X  H.!2 / whose transitive collapse is iterable.

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3 The nonstationary ideal

(3) For every countable X  H.!2 /, the transitive collapse of X is iterable. (4) If C  !1 is closed and unbounded, then C contains a closed unbounded subset which is constructible from a real. Proof. We fix some notation. Suppose x is a real and x # exists. For each ordinal  let M.x # ;  / be the  model of x # . (1 ) 3) Fix X  H.!2 /. Fix an ! sequence hi W i < !i of ordinals in X \ !2 which are cofinal in X \ !2 . For each i < ! let zi 2 X be a real such that i < rank.M.zi# ; !1 C 1//. Let N be the transitive collapse of X . For each i < ! let iN be the image of i under the collapsing map. Thus ¹iN j i < !º is cofinal in N \ Ord. Suppose j W .N; 2/ ! .M; E/ is an iteration of N . Then ¹j.iN / W i < !º is cofinal in OrdM . The first key point is the following. Suppose that j.!1N / is wellfounded. Then for each i < !, j.M.zi# ; !1N C 1// is wellfounded since by absoluteness: j.M.zi# ; !1N C 1// Š M.zi# ; j.!1N C 1//: Thus: (1.1) For any iterate .M; E/ of N if !1M is wellfounded then M is wellfounded. By assumption the nonstationary ideal on !1 is saturated. Thus if G  P .!1 / n INS is V -generic for the partial order .P .!1 / n INS ; / and if j W H.!2 / ! M is the induced embedding then j.!1 / D !2 D OrdH.!2 / . This is expressible in H.!2 / as a first order sentence. This is the second key point. Thus: (2.1) If M is a wellfounded iterate of N and if M  is a generic ultrapower of M then M  is wellfounded. From (1.1) and (2.1) it follows that N is iterable. (2 ) 4) Fix X  H.!2 / such that NX is iterable where NX is the transitive collapse of X . It suffices to show that if C 2 X and if C  !1 is closed and unbounded then C contains a closed unbounded subset which is constructible from a real. This is because if (4) fails then there must be a counterexample in X . Fix C 2 X such that C is a club in !1 . Let z be a real which codes NX . Let CX D C \ X . By Corollary 3.13 there is an iteration of length !1 j W NX ! N such that j.CX / D C .

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By Lemma 3.15, if ˛ is admissible relative to z and if k W NX ! M is any iteration of length ˛ then k.!1NX / D ˛. Therefore if ˛ < !1 is admissible relative to z then ˛ 2 C . Thus D D ¹˛ < !1 j L˛ Œz  L!1 Œzº is a closed unbounded subset of C and D 2 LŒz. (4 ) 1) This is a standard fact. The only additional hypothesis required is that for all x 2 R, x # exists and this is an immediate consequence of the assumption that the nonstationary ideal on !1 is saturated. Suppose !1 < ˛ < !2 . Fix a wellordering !1 . Thus Vı  ZFC and .N! ; kjN! / 2 Vı : Let X  Vı be a countable elementary substructure such that N! ; kjN! 2 X . We show that X \ H.!2 / is iterable. The relevant point is that .N! ; kjN! / is naturally a structure that can be iterated and further all of its iterates are wellfounded. Let k! D kjN! . The fact that .N! ; k! / is iterable is a standard fact. k  N and

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N contains the ordinals, therefore .N; k/ is iterable; i. e. any iteration of set length is wellfounded. The image of .N! ; k! / under an iteration of .N; k/ of length ˛ is simply the ˛ th iterate of .N! ; k! /. Let MX be the transitive collapse of X . Let NX! be the image of N! under the collapsing map and let kX! be the image of k! . We claim that .NX! ; kX! / is iterable. This too is a standard fact. Any iterate of .NX! ; kX! / embeds into an iterate of .N! ; k! / which is wellfounded since .N! ; k! / is iterable. The image of .NX! ; kX! / under any iteration of MX is an iterate of .NX! ; kX! /. This is an immediate consequence of the definitions and the hypothesis, (2), of the lemma. Therefore the image of !2 under any iteration of MX is wellfounded and so by Lemma 3.8, the transitive collapse of X \H.!2 / is iterable. But then by Theorem 3.16, ı12 D !2 . t u Combining Shelah’s theorem with Theorem 3.17 yields a new upper bound for the consistency strength of ZFC C “For every real x, x # exists” C “ı12 D !2 ”: With an additional argument the upper bound can be further refined to give the following theorem. One corollary is that one cannot prove significantly more than  12 -Determinacy from the hypothesis of Theorem 3.22. It is proved in .Koellner and Woodin 2010/ that 1  3 -Determinacy implies that there exists an inner model with two Woodin cardinals. Therefore Theorem 3.22 cannot be improved to obtain  13 -Determinacy. Theorem 3.25. Suppose ı is a Mahlo cardinal and that there exists ı  < ı such that: (i) ı  is a Woodin cardinal in L.Vı  /; (ii) Vı   Vı . Then there is a semiproper partial order P such that V P  ZFC C “For every real x, x # exists” C “ı12 D !2 ”: If in addition ı is a Woodin cardinal then V P  “INS is !2 -saturated”: Proof. The partial order is simply the partial order P defined by Shelah in his proof of Theorem 3.20. We shall need a little more information from this proof which we sketched in Section 2.4. The partial order P is obtained as an iteration of length ı, hP˛ W ˛ < ıi, such that: (1.1) hP˛ W ˛ <  i  V for all  < ı such that jV j D  ; (1.2) hP˛ W ˛ <  i is definable in V for all  < ı such that jV j D  ; (1.3) For each  < ı, if  is strongly inaccessible then P D [¹P˛ j ˛ <  º:

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3 The nonstationary ideal

By (1.2) the definition of hP˛ W ˛ <  i for suitable  is absolute, more precisely suppose that N is a transitive model of ZFC,  < ı and that N D V : Suppose that jV j D  . Then hP˛ W ˛ <  i is the iteration of length  as defined in N . We note by (1.3), since ı is a Mahlo cardinal, the partial order P is ı-cc. Thus if GP is V -generic then H.!2 /V ŒG D Vı ŒG: Let ı  < ı be least such that ı  is a Woodin cardinal in L.Vı  / and such that Vı   Vı . Since Vı# exists it follows that ı  has cofinality !. Therefore we can construct in V an L.Vı  /-generic filter H  Q where Q is the poset for adding a generic subset of ı  . The point is that since ı  is a Woodin cardinal in L.Vı  / it follows that 76

3 The nonstationary ideal

By Lemma 3.33, the embedding jX is given by an iteration of MX . Therefore there exists a sequence hXi W i < !i of elements of X such that for all i < !, .Xi ; jXi ; NXi / 2 XiC1 \ X and such that X D [¹Xi j i < !º: We now fix a countable elementary substructure X  H.!2 / such that hX; A \ X; 2i  hH.!2 /; A; 2i: We must prove that the transitive collapse, MX , of X is A-iterable. For each i < ! let Ni be the image of NXi under the transitive collapse of X and let ji W MXi ! Ni the image of jXi . Thus for all i < !, (2.1) .MXi ; ji ; Ni / 2 NiC1 , (2.2) Ni †2 NiC1 , (2.3) ji W MXi ! Ni is an iteration map, and further MX D [¹Ni j i < !º: By Theorem 3.19, MX is iterable. Suppose j W MX ! M is a countable iteration. Then for each i < !, j.ji / W MXi ! j.Ni / is an iteration and so for each i < !, j jNi W Ni ! j.Ni / is an iteration. For each i < !, Ni is an iterate of MXi and MXi is A iterable. Hence Ni is A-iterable. Finally M D [¹j.Ni / j i < !º; and for each i < !, Ni 2 NiC1 . Hence M D ¹j.Ni / j i < !º: Therefore j.A \ MX / D [¹j.A \ Ni / j i < !º D [¹A \ j.Ni / j i < !º D A \ M: This verifies that MX is A-iterable.

t u

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Lemma 3.35. Suppose that the nonstationary ideal on !1 is !2 -saturated. Suppose A  R and that B is weakly homogeneously Suslin for each set B which is projective in A. Let M be a transitive set such that M  ZFC , P .!1 /  M , and such that M # exists. Then ¹X  M j X is countable and MX is A-iterableº contains a club in P!1 .M /. Here MX is the transitive collapse of X . Proof. We recall that if N is a countable transitive model of ZFC and if j W N ! N is a countable iteration then j.A \ N / is defined to be the set j.A \ N / D [¹j.Z/ j Z 2 N; and Z  Aº: We first prove the lemma in the case that M D H.!2 /. Set M0 D H.!2 /. Let G be V -generic for the Levy collapse of 2!1 to !. Thus M0 is countable in V ŒG. Let T be a weakly homogeneous tree in V such that A D pŒT . Define AG to be the projection of T in V ŒG. It is a standard fact that AG does not depend on the choice of T . Let T  be a tree in V such that in V ŒG, pŒT  D R n pŒT  : The tree T  exists since T is weakly homogeneous in V and since G is generic for a partial order of size less than the least measurable cardinal. Finally let S be a weakly homogeneous tree in V such that B D pŒS  where B is the set of reals which code A-iterable transitive models of ZFC . Since B is projective in A, the tree S exists. Further we have that in V ŒG, BG is the set of reals which code AG -iterable transitive models of ZFC where BG is the projection of S in V ŒG. We claim that M0 is AG -iterable in V ŒG. Let j0 W M0 ! N0 be a countable iteration of M0 in V ŒG. Then by Lemma 3.8, there corresponds an iteration j W V ! .N; E/  V ŒG such that N0 D j.M0 / and j jM0 D j0 . By Lemma 3.10, .N; E/ is wellfounded and we identify N with its transitive collapse. Note that j.A/ D pŒj.T / \ N and that j0 .A/ D j.A/. It suffices to show that in V ŒG, pŒT  D pŒj.T /. However in V ŒG; (1.1) pŒT   pŒj.T /, (1.2) pŒT    pŒj.T  /, (1.3) pŒT  D R n pŒT  , (1.4) pŒj.T / \ pŒj.T  / D ;.

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3 The nonstationary ideal

Condition (1.4) holds by absoluteness since by elementarity of j it must hold in N . From these conditions it follows immediately that pŒT  D pŒj.T / in V ŒG. Thus M0 is AG -iterable in V ŒG. Let x0 be a real in V ŒG which codes M0 . Thus x0 2 BG and so x0 2 pŒS . In V fix a set Z  P!1 .M0 / such that Z is stationary. Let  be large enough such that T; S 2 V and such that V is admissible. Let X  V be a countable elementary substructure such that T 2 X , S 2 X and such that X \ M0 2 Z. Let NX be the transitive collapse of X and let SN be the image of S under the collapsing map. Finally let GN be N -generic for the Levy collapse of .2!1 /N . Thus by the argument above we have that there exists a real xN 2 N ŒGN  such that xN 2 pŒSN  and xN codes H.!2 /N . Therefore xN 2 pŒS  and so H.!2 /N is A-iterable. However H.!2 /N is the transitive collapse of X \ H.!2 / and X \ M0 2 Z. This proves the lemma in the case that M D H.!2 /. The general case follows using Lemma 3.8 and Lemma 3.14. The point is that if N is a countable transitive iterable model of ZFC in which the nonstationary ideal is saturated and if A is any set of reals then N is A-iterable if and only if H.!2 /N is A-iterable. t u The covering theorems are more easily proved using the following theorem which is a routine generalization of the (correct) results of .Woodin 1983/ from the setting of L.R/ to that of L.A; R/. We shall only need parts (2) and (3). For the sake of completeness we include a sketch of the proof. Theorem 3.36. Suppose A  R, L.A; R/  AD and that G  Coll.!;
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79

It is a standard fact that AD implies **uniformization and that **uniformization implies that every set of reals has the property of Baire. Assume **uniformization and that V D L.A; R/ where A  R. Then **AC holds; i. e. if F W R ! V n ¹;º is a function then there exists a function H WR!V such that ¹x j H.x/ 2 F .x/º is comeager in R. This easily generalizes as follows. Suppose Q is a partial order with a countable dense set and let  2 V Q be a term for a real. For each condition p 2 Q define an ideal I p as follows. Let S be the Stone space of Q. I p D ¹Z  R j ¹G  Q j p 2 G; IG ./ 2 Zº is meager in Sº where if G  Q is a filter then IG is the associated (partial) interpretation map. For a comeager collection of filters G  Q, IG ./ is defined; i. e. for each n 2 ! there exists m 2 ! such that for some q 2 G, q  .n/ D m: p We say Z  R is I -positive if Z … I p . The set Z is I p -large if X 2 I p where X D R n Z. The following facts are easily verified. The ideals I p are countably complete and for all Z  R, Z is I p -positive if and only if there exists q < p such that Z is I q -large. Now assume Q is a partial order with a countable dense set,  is a term for a real, p 2 Q, V D L.A; R/ and **uniformization. Suppose F W R ! V n ¹;º is a function. Then there is a function H WR!V such that ¹x j H.x/ 2 F .x/º is I p -large. Suppose g  Q is L.A; R/-generic and let z 2 L.A; R/Œg be the interpretation of  by g. Let Ug D ¹Z  R j Z 2 L.A; R/; Z is I p -large for some p 2 gº and so Ug is an L.A; R/-ultrafilter on L.A; R/ \ P .R/. Let N D Ult.L.A; R/; Ug /. It is easily verified that N is wellfounded (use DC in L.A; R/) and we identify N with its transitive collapse. Let j W L.A; R/ ! N be the associated generic elementary embedding. It follows that j is the identity on the ordinals, RN D RL.A;R/Œz , and that j .A/ D [¹X V Œz j X 2 V; X is borel and X  Aº where if X 2 V is a borel set then X V Œz denotes its interpretation in V Œz.

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3 The nonstationary ideal

Now suppose that G  Coll.!;
X \ S˛

is comeager in S˛ , (1.2) for all x 2 X , .x; f .x// 2 Z, 1 (1.3) f is …  1 in the codes. It is straightforward to show that AD implies !1 -**uniformization. Assume !1 -**uniformization and that V D L.A; R/ where A  R. Then !1 **AC holds; i. e. if F W H.!1 / ! V n ¹;º

3.1 The nonstationary ideal and ı12

81

is a function then there exists X  H.!1 / and there exists a function H WX !V such that for all g 2 X , H.g/ 2 F .g/ and such that for each ˛ < !1 ,

X \ S˛

is comeager in S˛ . In fact, both assertions (1) and (2) in the statement of theorem follow simply from the assumption that !1 -**uniformization holds in L.A; R/ and we shall prove (1) and (2) from only this weaker assumption. Let S be the following partial order. Conditions are triples .N; g; ˛/ such that (2.1) N  !1 , (2.2) ˛ < !1 , (2.3) g 2 S˛ . Suppose .N1 ; g1 ; ˛1 / 2 S and .N2 ; g2 ; ˛2 / 2 S. Then .N2 ; g2 ; ˛2 / < .N1 ; g1 ; ˛1 / if N1 2 LŒN2 , ˛1 < ˛2 , g1  g2 and g2 \ Q˛1 ;˛2 is LŒN1 -generic. We will need the following consequences of !1 -**AC and **uniformization. Suppose X  H.!1 / and that ˛ < !1 . Suppose that for each ˇ < !1 , X \ S˛;ˇ is comeager in S˛;ˇ . Then there exists N  !1 such that for all ˇ < !1 , if ˛ < ˇ then ¹g  Q˛;ˇ j g is LŒN -genericº  X: By **uniformization, every set of reals has the property of Baire and so for every set N  !1 and for every ˇ < !1 , if ˛ < ˇ then ¹g  Q˛;ˇ j g is LŒN -genericº is comeager in S˛;ˇ . We first prove the following. Suppose .N; g; ˛/ 2 S and D0  S is a set which is dense below .N; g; ˛/. Let D be the set of p 2 S such that for some q 2 D0 , p  q 2 G where G is the term for the generic filter. We claim there exist N   !1 and ˇ < !1 such that ˛ < ˇ and such that .N  ; g  h; ˇ/ 2 D for all h  Q˛;ˇ which are LŒN  -generic. By !1 -**AC, there exists X  H.!1 / and there exists a function F W X ! P .!1 / such that for all ˇ < !1 , if ˛ < ˇ then (3.1) X \ S˛;ˇ is comeager in S˛;ˇ , (3.2) if h 2 S˛;ˇ \ X and if there exists M  !1 such that .M; g  h; ˇ/ 2 D then .F .h/; g  h; ˇ/ 2 D.

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Let T be the set of triples .ˇ; q;  / such that (4.1) ˛ < ˇ < !1 , (4.2) q 2 Q˛;ˇ , (4.3)  < !1 , (4.4) ¹h 2 S˛;ˇ j  2 F .h/º is comeager in the open subset of S˛;ˇ given by q. Let X  be the set of h  Q˛;ˇ such that ˛ < ˇ < !1 and such that h is LŒT generic. Define F  W X  ! P .!1 / by

F  .h/ D ¹ j .ˇ; q;  / 2 T for some q 2 hº

where ˇ < !1 is such that h is LŒT -generic for Q˛;ˇ . Let X  D ¹h 2 X \ X  j F .h/ D F  .h/º: Since every set of reals has the property of Baire, for every ˇ < !1 , if ˛ < ˇ then X  \ S˛;ˇ is comeager in S˛;ˇ . Let S be the set of pairs .ˇ; p/ such that (5.1) ˛ < ˇ < !1 , (5.2) p 2 Q˛;ˇ , (5.3) ¹h 2 S˛;ˇ j .F .h/; g  h; ˇ/ 2 Dº is comeager in the open subset of S˛;ˇ given by p. For each ˇ < !1 such that ˛ < ˇ let Yˇ D ¹h 2 S˛;ˇ j .F .h/; g  h; ˇ/ 2 D $ .ˇ; p/ 2 S for some p 2 hº: Thus for each ˇ, Yˇ is comeager in S˛;ˇ . Let Y D [¹Yˇ j ˛ < ˇ < !1 º: Let N   !1 be a set such that .g; S; T / 2 LŒN   and such that for all ˇ < !1 , if ˛ < ˇ then ¹g  Q˛;ˇ j g is LŒN  -genericº  X  \ Y: Suppose ˇ < !1 and ˛ < ˇ. Suppose h  Q˛;ˇ is LŒN  -generic and that there exists M  !1 such that .M; g  h; ˇ/ 2 D. h is LŒN  -generic and so h 2 X  . Therefore .F .h/; g  h; ˇ/ 2 D and further F .h/ D F  .h/. However T 2 LŒN   and so F .h/ 2 LŒN  Œh. Thus .N  ; g  h; ˇ/  .F .h/; g  h; ˇ/ 2 G and so

.N  ; g  h; ˇ/ 2 D:

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83

Let E D ¹p j .ˇ; p/ 2 S for some ˇ < !1 º: We claim that E is predense in Coll.!;
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where Gp0 D g0  .G \ Q˛0 ;!1 / is the perturbation of G to extend g0 . We prove that FGp0 \ jG .D/ ¤ ; for all D  S such that D is dense. Suppose D  S is dense. We may assume that for all p 2 S, if p  q 2 G for some q 2 D then p 2 D. Since every set of reals has the property of Baire and since !1 -**AC holds, it follows by the remarks above, that there exists ˛ < !1 such that ˛0 < ˛ and there exists N1  !1 such that .N1 ; g0  h; ˛/ 2 D for all h  Q˛0 ;˛ such that h is LŒN1 -generic. Thus .N1 ; g0  G˛0 ;˛ ; ˛/ 2 jG .D/ and .N1 ; g0  G˛0 ;˛ ; ˛/ < jG .p0 /. Clearly G \ Q˛;!1 is LŒN1 -generic. Thus .N1 ; g0  G˛0 ;˛ ; ˛/ 2 jG .D/ \ FGp0 : This proves that FGp0 \ jG .D/ ¤ ; for all D  S such that D is dense. It now follows that FG \ D ¤ ; for all D 2 L.AG ; RG / such that D  jG .S/ and such that D is dense. The point is that any set in L.AG ; RG / is the image of a set in L.AG˛ ; RG˛ / for some ˛ and so genericity follows by relativizing the previous argument, with a suitable choice of p0 , to L.AG˛ ; RG˛ /  V ŒG˛ : Finally we prove (2). It suffices to show that if F S is L.A; R/-generic then !1 -AC holds in L.A; R/ŒF . We work in L.A; R/. Suppose  is a term, .N; g; ˛/ 2 S and .N; g; ˛/   ¤ ;: We prove that there exists N



 !1 and a term such that .N  ; g; ˛/  2 :

Let D be the set of q < .N; g; ˛/ such that q 2 for some term . Therefore D is open and D is dense below .N; g; ˛/. By the claim proved above, there exists N   !1 and there exists ˇ < !1 such that ˛ < ˇ and such that .N  ; g  h; ˇ/ 2 D for all h  Q˛;ˇ which are LŒN  -generic. By **AC there exists a set X which is comeager in S˛;ˇ and a function F W X ! L.A; R/ such that for all h 2 X , F .h/ is a term and .N  ; g  h; ˇ/  F .h/ 2 :

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Let N   !1 be a set such that N  2 LŒN   and such that if h  Q˛;ˇ is LŒN  generic then h 2 X . F defines a term and .N  ; g; ˛/  2 : Now suppose  2 L.A; R/S is a term, .N; g; ˛/ 2 S, and .N; g; ˛/   W !1 ! V n ;: Let hˇ W ˛ < ˇ < !1 i be a sequence of terms such that for all ˇ < !1 , .N; g; ˛/   ./ D ˇ : where ˛ C 1 C  D ˇ. By !1 -**AC and by **uniformization and by the result proved above, there exists X  H.!1 / and two functions, F0 W X ! P .!1 / and F1 W X ! L.A; R/ with the following properties. For all ˇ < !1 , if ˛ < ˇ then X \ S˛;ˇ is comeager in S˛;ˇ and for all h 2 X \ S˛;ˇ , F1 .h/ is a term and .F0 .h/; g  h; ˇ/  F1 .h/ 2 ˇ : As we did above we extract the term defined by F0 . Let T be the set of triples .ˇ; q;  / such that (7.1) ˛ < ˇ < !1 , (7.2) q 2 Q˛;ˇ , (7.3)  < !1 , (7.4) ¹h 2 S˛;ˇ j  2 F0 .h/º is comeager in the open subset of S˛;ˇ given by q. For each ˇ < !1 such that ˛ < ˇ let Yˇ be the set of h 2 X \ S˛;ˇ such that F0 .h/ D ¹ < !1 j .ˇ; q;  / 2 T for some q 2 hº: Thus Yˇ is comeager in S˛;ˇ . Let Y D [¹Yˇ j ˛ < ˇ < !1 º: 

Finally let N  !1 be such that .N; T / 2 LŒN   and such that for all ˇ < !1 , if ˛ < ˇ then ¹h 2 S˛;ˇ j h is LŒN  -genericº  X \ Y: Suppose h  Q˛;ˇ and that h is LŒN  -generic. Therefore h 2 X and so .F0 .h/; g  h; ˇ/  F1 .h/ 2 ˇ : The genericity of h relative to LŒN   also implies that h 2 Yˇ . Therefore F0 .h/ 2 LŒT Œh:

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However T 2 LŒN   and so .N  ; g  h; ˇ/  .F0 .h/; g  h; ˇ/ 2 G where G is the term for the generic filter. Thus for all ˇ < !1 , if ˛ < ˇ and if h 2 S˛;ˇ is LŒN  -generic then .N  ; g  h; ˇ/  F1 .h/ 2 ˇ : The function F1 yields a term 2 L.A; R/S such that .N  ; g; ˛/  “ is a choice function for  ”:

t u

Remark 3.37. (1) As indicated in the proof, one does not need AD for this. For example, (3) follows assuming only that **uniformization holds in L.A; R/. See .Woodin 1983/. (2) The partial order S, defined in the proof of Theorem 3.36, is equivalent to the forcing notion of .Steel and VanWesep 1982/. Assuming AD, ¹.N; g; ˛/ 2 S j N  !º is dense in S and the order on S can be refined to make the partial order !-closed; i. e. S is !-strategically closed. (3) With additional requirements on the inner model, L.A; R/, one in fact gets !1 DC in L.AG ; RG /ŒG. The additional assumption is ADC , it is implied by AD if V D L.R/. !1 -choice is sufficient for our purposes. A brief survey of ADC is given in the first section of Chapter 9. t u As a corollary to Theorem 3.36 we obtain the following theorem. Theorem 3.38. Suppose A  R, L.A; R/  AD and that G  Coll.!;
3.1 The nonstationary ideal and ı12

of subsets of !1 such that

87

5¹S˛ j ˛ < !1 º

contains a club and such that for each ˛ < !1 , there is a club C on which f j.S˛ \ C / D g for some g 2 L.AG ; RG /. (4) Suppose X  ı is a set of size !1 in L.AG ; RG /ŒG. Then there is a set Y  ı in L.AG ; RG / of size !1 such that X  Y . Proof. (1) follows immediately from Theorem 3.36(3) noting that the club filter on !1 is an ultrafilter in L.A; R/ and that the ultrapower Ord!1 =U is wellfounded in L.A; R/ where U is the club filter on !1 . (2) is an elementary consequence of the fact that the partial order, Coll.!;
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Let T D ¹.p; ˛; / j p 2 Coll.!;
kQy0 .g˛ /.!1V / D ˛ :

For each ˛ < !1 let S˛ D ¹ < !1 j f . / D g˛ . /º: It suffices to show that in L.AG ; RG /ŒG, S D 5¹S˛ j ˛ < !1 º contains a closed unbounded subset of !1 . Suppose E  !1 is a stationary subset of !1 in L.AG ; RG /ŒG. Let be a term for E and let S D ¹.p;  / j p  “  2 ”º: S 2 L.A; R/ and so there exists y 2 R such that S 2 LŒy. We may suppose that 1  “ is a stationary subset of !1 ”:

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Fix z 2 R such that S 2 LŒz and such that y 2 LŒz. Thus ¹S; Eº  LŒzŒG: Let SQ D kz .S /: Thus SQ is a term for a subset of ı12 in the forcing language for Coll.!; E \ S ¤ ;: Therefore in L.AG ; RG /ŒG, S contains a closed unbounded subset of !1 .

t u

Lemma 3.39. Suppose A  R, L.A; R/  AD and that for all B 2 P .R/ \ L.A; R/ the set ¹X  hH.!2 /; B; 2i j MX is B-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose G  Coll.!;
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Proof. Suppose G  Coll.!;
B D f 1 ŒCG 

for some continuous function f W RG ! RG with f 2 V ŒG. Thus it suffices to prove that for all C 2 P .R/ \ L.A; R/, the set ® ¯ X  hH.!2 /V ŒG ; CG ; 2i j MX is CG -iterable and X is countable is stationary. Fix C 2 P .R/ \ L.A; R/. Let U RR 1 be a universal † 1 set. For each z 2 R let

Uz D ¹y 2 R j .z; y/ 2 U º: If M is a transitive model of ZFC we let U M D U \ M . By absoluteness U M is defined in M by the same †11 formula which defines U in V . Suppose X  hH.!2 /; C; 2i is a countable elementary substructure in V such that MX is D-iterable and MX is E-iterable where D D ¹z j Uz  C º and E D ¹z j Uz  R n C º: Let Y D X ŒG and let N be the transitive collapse of Y . Therefore N D MX ŒG \ Coll.!;
3.1 The nonstationary ideal and ı12

Suppose in V ŒG,

91

k W N ! N

is a countable iteration of N . Let g D G \ Coll.!;
N  D k.MX /Œk.g/

and k.g/ is k.MX /-generic for Coll.!; !. Suppose that Y  R  R is a prewellordering of length . Then there exists a set X  R and a surjection WX ! such that 1 (1) for each † 1 set Z  X ,

sup¹.t / j t 2 Zº < ;

(2) the set ¹.x; y/ j .x/  .y/º  X  X 1 is † 1 .Y /.

Proof. Suppose that there exists .; X / which satisfies (1). Then by the Moschovakis Coding Lemma there exists .; X / satisfying both (1) and (2). We prove (1). We fix some notation. Suppose A  R. Let A be the least ordinal such that LA .A; R/  ZF n Powerset and let A D P .R/ \ LA .A; R/:

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It is useful to note that A  B if and only if A  B , (by Wadge). Let   P .! ! / be such that (1.1) for each A 2 ,

A  ;

(1.2) ordertype¹ A j A 2 º D . These conditions uniquely specify . Let ı D sup¹ A j A 2 º: By (1.2), cof.ı / D cof./ and so cof.ı / > !. We shall assume the basic facts concerning Wadge reducibility in the context of AD, see the discussion after Definition 9.25. One such fact is that there exists a set B  ! ! of minimum Wadge rank such that B … . Fix B0 and let  D ¹B   ! ! j B  is a continuous preimage of B0 º: Let

O D ¹! ! n A j A 2 º

be the dual pointclass. By the choice of B0 , O  D  \ ; this is the second basic fact we require. In fact we shall use a slightly stronger form of this. Let L.! ! ; ! ! / denote the set of continuous functions f W !! ! !! such that for all x 2 ! ! , for all y 2 ! ! , and for all k 2 !, if xjk D yjk then f .x/jk D f .y/jk. It follows from the determinacy of the relevant Wadge games, the closure properties of , and the definition of , that: (2.1) Suppose B  ! ! . Then ¹B; ! ! n Bº  ¹f 1 ŒB0  j f 2 L.! ! ; ! ! /º if and only if B 2 . By the results of .Steel 1981/: (3.1)  is closed under finite unions or O is closed under finite unions. (3.2) Suppose that  is closed under finite unions. Then for each  †11 set Z  ! ! and for each A 2 , A \ Z 2 :

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It is the latter claim, which requires that cof./ > !; which is the key claim. Without loss of generality we can suppose that  is closed under finite unions. Fix a surjection  W ! ! ! L.! ! ; ! ! / such that the set ¹.x; y; z/ j .x/.y/ D zº is borel; i. e. a reasonable coding of L.! ! ; ! ! /. Fix A0 2  n  and let R be the set of pairs .x0 ; x1 / such that (4.1) ..x0 //1 ŒA0  \ ..x1 //1 ŒA0  D ;, (4.2) ..x0 //1 ŒA0  [ ..x1 //1 ŒA0  D ! ! . Note that by the definition of , for each .x0 ; x1 / 2 R, ..x0 //1 ŒA0  2 : Define  W R ! ı by .x0 ; x1 / D B where B D ..x0 //1 ŒA0 . Thus by (1.2),  defines a prewellordering of R with length  and ı D sup¹.x0 ; x1 / j .x0 ; x1 / 2 Rº: Finally we show that if ZR is

1 † 1

then sup¹.x0 ; x1 / j .x0 ; x1 / 2 Zº < ı :

This is where we use (3.2). Since the range of  has ordertype , this boundedness property will suffice to prove (1). Let Z  D ¹.x0 ; x1 ; y; z0 ; z1 / j .x0 ; x1 / 2 Z; .x0 /.y/ D z0 ; and .x1 /.y/ D z1 º: 1 Thus Z  is † 1 . Let

A D ¹.x0 ; x1 ; y; z0 ; z1 / j .x0 ; x1 ; y; z0 ; z1 / 2 Z  and z0 2 A0 º D Z  \ ¹.x0 ; x1 ; y; z0 ; z1 / j z0 2 A0 º: Then because  is closed under intersections with  †11 sets (and closed under continuous preimages), A 2 :

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But ! ! n A D ¹.x0 ; x1 ; y; z0 ; z1 / j .x0 ; x1 ; y; z0 ; z1 / … Z or z0 … A0 º D ¹.x0 ; x1 ; y; z0 ; z1 / j .x0 ; x1 ; y; z0 ; z1 / … Z or z1 2 A0 º D .! ! n Z  / [ ¹.x0 ; x1 ; y; z0 ; z1 / j z1 2 A0 º and so since  is closed under finite unions (and contains all  …11 sets), ! ! n A 2 :

Therefore

A 2  \ O D :

But for each .x0 ; x1 / 2 Z, ..x0 //1 ŒA0  is a continuous preimage of A and so B  A < ı where B D ..x0 //

1

ŒA0 . Therefore

sup¹.x0 ; x1 / j .x0 ; x1 / 2 Zº  A < ı ; t u

and so Z is bounded. We begin with a technical lemma. Lemma 3.41. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) for all A 2 M \ P .R/, the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose ı < ‚M , S  !1 is stationary and f W S ! ı. Suppose that g W !1 ! ı is a function such that g 2 M and such that f .˛/  g.˛/ for all ˛ 2 S . Then there exists a sequence h.T ; g / W  < !1 i such that S D 5¹T j  < !1 º and such that for all  < !1 , (1) T is stationary, (2) g 2 M ,

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(3) either f jT D gjT ; or for all ˛ 2 T ,

f .˛/  g .˛/ < g.˛/:

ı 12 D !2 and so we can assume that !2 < ı. Proof. Fix ı < ‚M . By Theorem 3.19,  Fix f W S ! ı such that S  !1 and such that S is stationary. For notational reasons we assume that the range of f is bounded in ı. Fix a set A  R such that A 2 M and such that A codes a prewellordering of length ı. Suppose G  Coll.!;
f .˛/  gi .˛/ < g.˛/:

By Theorem 3.38(3) the lemma follows from this claim. We work in V ŒG. We may suppose without loss of generality that for all ˛ 2 S  , f .˛/ < g.˛/: We first divide S  into three parts. Let S0 D ¹˛ 2 S  j g.˛/ is a successor ordinalº; let and let

S1 D ¹˛ 2 S  j cof.g.˛// D !º; S2 D ¹˛ 2 S  j cof.g.˛// > !º:

Clearly we may suppose that S  D S0 , S  D S1 or S  D S1 ; for if the claim holds for each of S0 ; S1 , and S2 then it trivially holds for S  . If S  D S0 then the claim is trivial.

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We next suppose that S  D S1 . Note that for ordinals less than ı, L.AG ; RG /ŒG correctly computes the cofinality if the ordinal has countable cofinality in V ŒG. Since !1 -choice holds in L.AG ; RG /ŒG, there exists a sequence hhˇi˛ W i < !i W ˛ 2 S1 i 2 L.AG ; RG /ŒG such that for each ˛ 2 S1 , hˇi˛ W i < !i is an increasing cofinal sequence in g.˛/. For each i < ! define a function gi W !1 ! ı by gi .˛/ D ˇi˛ : For each i < ! let Ti D ¹˛ 2 S1 j f .˛/  gi .˛/ < g.˛/º: Clearly S D [¹Ti j i < !º: This proves the claim holds for .g; f; S1 /. We finish by proving that the claim holds for the triple .g; f; S2 /. Let Y  ı be a set in L.AG ; RG / of cardinality !1 in L.AG ; RG / such that the range of g is a subset of Y . Y exists by Theorem 3.38 (3). Fix in L.AG ; RG / a function h W !1 ! Y which is onto. Let U  RG  R2G be a universal set for the relations 1 2 which are † 1 .AG /. Let P  RG be the set of pairs .x; y/ such that: (2.1) x codes a countable ordinal ˛; (2.2) Uy is a prewellordering y of length h.˛/ with the property that if Z  field.y / and Z is

1 † 1

then Z is bounded relative to y .

By Theorem 3.40, dom.P / is exactly the set of x 2 R such that x codes a countable ordinal. The key point is that !1 -choice holds in L.AG ; RG /ŒG and so we can find a sequence h.x˛ ; y˛ / W ˛ < !1 i 2 L.AG ; RG /ŒG of elements of P such that for each ˛ < !1 , x˛ codes a countable ordinal  such that h. / D g.˛/. Choose in V ŒG an !1 sequence of reals hz˛ W ˛ < !1 i such that for each ˛ < !1 , z˛ 2 field.y˛ / and such that for each ˛ 2 S2 , f .˛/ is the rank of z˛ relative to y˛ . Let S D h.x˛ ; y˛ / W ˛ < !1 i and let T D hz˛ W ˛ < !1 i. Choose a countable elementary substructure X  H.!2 /V ŒG containing the sequences S and T and such that MX is P -iterable where MX is the transitive collapse of X . Let SX and TX be the images of S and T under the collapsing

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3 The nonstationary ideal

map. Thus SX D Sj!1MX and similarly for TX . By Corollary 3.13, there is an iteration j W MX ! N of length !1 such that j.SX / D S and j.TX / D T : Fix ˛ 2 !1 . Let Z˛ be the set of all z 2 RG such that there is an iteration k W MX ! N of length ˛ C 1 such that k.SX /j.˛ C 1/ D Sj.˛ C 1/ and z D k.TX /.˛/. Thus Z˛ 1 is a † 1 set and z˛ 2 Z˛ . Further since MX is P -iterable we have Z˛  field.y˛ /. Thus this set is bounded. The definition of Z˛ is uniform in Sj.˛ C 1/ and hence hZ˛ W ˛ < !1 i 2 L.AG ; RG /: Therefore there is a function g  2 L.AG ; RG /ŒG such that for all ˛ 2 S2 , f .˛/  g  .˛/ < g.˛/. This proves that the claim holds for t u the triple .f; g; S2 /. Lemma 3.41 yields the following theorem as an easy corollary. Theorem 3.42. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) for all A 2 M \ P .R/, the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose ı < ‚M and that f W !1 ! ı. Then there exists a sequence h.S˛ ; g˛ / W ˛ < !1 i such that !1 D 5¹S˛ j ˛ < !1 º and such that for all ˛ < !1 , (1) S˛ is stationary, (2) g˛ 2 M , (3) f jS˛ D g˛ jS˛ . Proof. By Lemma 3.41 there exists a sequence hFi W i < !i of functions such that for each i < !:

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(1.1) dom.Fi /  P .!1 / n INS and jdom.Fi /j D !1 ; (1.2) if ¹S; T º  dom.Fi / and if S ¤ T then S \ T 2 INS I (1.3) dom.Fi / is predense in .P .!1 / n INS ; /; (1.4) for each S 2 dom.Fi /, Fi .S / 2 M and either

Fi .S / W !1 ! ı; Fi .S /jS D f jS;

or for all ˛ 2 S , f .˛/ < Fi .S /.˛/I (1.5) for each T 2 dom.FiC1 / there exists S 2 dom.Fi / such that T  S ; (1.6) suppose that S 2 dom.Fi /, T 2 dom.FiC1 / and that T ¨ S , then for each ˛ 2 T, FiC1 .T /.˛/ < Fi .S /.˛/I (1.7) for each S 2 dom.Fi / if

Fi .S /jS ¤ f jS

then there exists T 2 dom.FiC1 / such that T ¨ S . Let A be the set of S  !1 such that for some i < !, S 2 dom.Fj / for all j > i . By (1.2) and (1.6), for each S 2 A, f jS D gjS for some g 2 M . Let be closed unbounded filter as computed in M . Since M  AD C DC;

is an ultrafilter in M and the ultrapower ¹g W !1 ! ı j g 2 M º= is wellfounded. This in conjunction with (1.6) yields the following. Suppose that hSi W i < !i is an infinite sequence such that for all i < j < !, Sj  Si and Si 2 dom.Fi /. Then there exists i0 < ! such that for all i > i0 , Si D Si0 : By (1.4) and (1.6), Sj 2 A for all j i0 . Therefore by (1.3), for each T 2 P .!1 / n INS there exists S 2 A such that S \ T … INS :

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Thus A is predense in .P .!1 / n INS ; /. Finally A  [¹dom.Fi / j i < !º t u

and so jAj  !1 . We obtain as an immediate corollary the first covering theorem.

Theorem 3.43. Suppose that the nonstationary ideal on !1 is !2 -saturated. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) every set A 2 M \ P .R/ is weakly homogeneously Suslin in V . Suppose ı < ‚M , S  !1 is stationary and f W S ! ı. Then there exists g 2 M such that ¹˛ 2 S j f .˛/ D g.˛/º is stationary. Proof. By Lemma 3.35, for each A 2 P .R/ \ M , the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº contains a set closed and unbounded in P!1 .H.!2 //. Therefore the theorem follows from Theorem 3.42. t u Corollary 3.44. Suppose that the nonstationary ideal on !1 is !2 -saturated. Suppose that ¯ ® !3  sup ‚M where M ranges over transitive inner models such that (i) R  M , (ii) Ord  M , (iii) M  ZF C DC C AD, (iv) every set A 2 M \ P .R/ is weakly homogeneously Suslin in V . Suppose G  P .!1 / n INS is V -generic and that j WV !M is the induced generic elementary embedding. Then j j˛ 2 V for every ordinal ˛. Proof. By the last theorem j j!3 2 V . It follows on general grounds that j jOrd is a definable class in V . u t

3.1 The nonstationary ideal and ı12

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The second covering theorem is stronger. Again we prove a preliminary version. Theorem 3.45. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) for all A 2 M \ P .R/, the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose ı < ‚M , X  ı and jX j D !1 . Then there exists Y 2 M such that M  “jY j D !1 ” and such that X  Y . Proof. Fix ı < ‚M and let A 2 M be a prewellordering of the reals of length ı. Fix a set X  ı of cardinality !1 . As in the proof of the first covering theorem suppose that G  Coll.!;
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This is easily done. Let N D L .AG ; RG /ŒG. Thus N  !1 -choice C !1 -Replacement: By Theorem 3.42, has cofinality !2 in V ŒG, and so V ŒG  ZFC , Choose Z  V ŒG such that (3.1) Z is countable, (3.2) AG 2 Z, X 2 Z and ı 2 Z, (3.3) the transitive collapse MZ of Z is iterable. Define two sequences hX˛ W ˛ < !1 i and hZ˛ W ˛ < !1 i by induction on ˛ such that: (4.1) X0 D N \ Z and Z0 D Z; (4.2) Xˇ D [¹X˛ j ˛ < ˇº and Zˇ D [¹Z˛ j ˛ < ˇº for all limit ordinals ˇ < !1 ; (4.3) X˛C1 D ¹f .X˛ \ !1 / j f 2 X˛ º; (4.4) Z˛C1 D ¹f .Z˛ \ !1 / j f 2 Z˛ º. Define a sequence hX˛ W ˛ < !1 i by X˛ D Z˛ \ N . Thus for all ˛ < ˇ < !1 : (5.1) Z˛  Zˇ  V ŒG ; (5.2) X˛  Xˇ  N ; (5.3) X˛  X˛ . It is because !1 -choice and !1 -replacement hold in N that hX˛ W ˛ < !1 i is an elementary chain. The key claim is that for all ˛ < !1 , X˛ \ D X˛ \ and so for all ˛ < !1 , X˛ \ D Z˛ \ . This will follow from the first covering theorem. Once we prove this claim the theorem follows. This is because !1  [¹Z˛ j ˛ < !1 º and so since X 2 Z0 , X  [¹Z˛ \ j ˛ < !1 º  [¹X˛ j ˛ < !1 º: Further X0 2 L.AG ; RG /ŒG and so hX˛ W ˛ < !1 i 2 L.AG ; RG /ŒG: Thus Y D [¹X˛ \ ı j ˛ < !1 º is the desired cover of X . To finish we must prove that for all ˛ < !1 , X˛ \ D X˛ \ . This follows by induction provided we can prove the following:

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Claim: Suppose X   X   N , Z   V ŒG and that X  D Z  \N . Suppose that X  \ D X  \ , Z  is countable and that ; AG 2 X  . Then for each function f 2 Z  where f W !1 ! there exists g 2 X  such that f .Z  \ !1 / D g.Z  \ !1 /: We prove this claim. Fix f W !1 ! . By Theorem 3.42 we have in V ŒG that there is a sequence h.S˛ ; g˛ / W ˛ < !1 i such that: (6.1) hS˛ W ˛ < !1 i is a sequence of pairwise disjoint stationary sets; (6.2) 5¹S˛ j ˛ < !1 º contains a club in !1 ; (6.3) g˛ W !1 ! and g˛ 2 L.AG ; RG /; (6.4) g˛ jS˛ D f jS˛ . Since L .AG ; RG /  L‚ .AG ; RG / and since has cofinality !2 in L.AG ; RG /, ¹g W !1 ! j g 2 L.AG ; RG /º  L .AG ; RG /: Thus for each ˛ < !1 , g˛ 2 N . Since Z   V ŒG , we can suppose that h.S˛ ; g˛ / W ˛ < !1 i 2 Z  : It follows that

f .Z  \ !1 / D g  .Z  \ !1 /

for some function g  W !1 ! with g  2 Z  \ L .AG ; RG /: Let j W‚!‚ be the ultrapower embedding computed in L.AG ; RG / using the club measure on !1 . Let  W ‚!1 \ L‚ .AG ; RG / ! ‚ be the map that assigns to each function the ordinal it represents. By the Moschovakis Coding Lemma j j W ! : 

Let  D .g / be the ordinal represented by g  . Thus since g  2 Z  ,  2 X  and so  2 X  . But X   N and so since  is definable there exists g 2 X  \ L .AG ; RG / such that .g/ D  . Therefore g D g  on a club and so g.Z  \ !1 / D g  .Z  \ !1 / D f .Z  \ !1 /: This proves the claim.

t u

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3 The nonstationary ideal

There is another formulation of Theorem 3.45. Recall P!1 .X / denotes the set of all countable subsets of X . Theorem 3.46. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) for all A 2 M \ P .R/, the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose ı < ‚M and that f W !1 ! ı: Then there exists a function g W !1 ! P!1 .ı/ such that g 2 M and such that for all ˛ < !1 , f .˛/ 2 g.˛/. Proof. Let X D ¹f .˛/ j ˛ < !1 º: By Theorem 3.45, there exists a set Y  ı such that X  Y and such that jY jM D !1 : By Theorem 3.19, .!2 /M D !2 and so we may reduce to the case that ı D !1 . Let C D ¹˛ < !1 j f Œ˛  ˛º: The set C is closed and unbounded in !1 . By Theorem 3.19, there exists a closed, cofinal, set D  C such that for some x 2 R, D 2 LŒx: Therefore D 2 M . Define g W !1 ! P!1 .!1 / by g.˛/ D min.D n ˇ/; where ˇ D ˛ C 1. Thus g is as required.

t u

The second covering theorem is an immediate corollary of Theorem 3.45. Theorem 3.47. Suppose that the nonstationary ideal on !1 is !2 -saturated. Suppose that M is a transitive inner model such that M  ZF C DC C AD;

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and such that (i) R  M , (ii) Ord  M , (iii) every set A 2 M \ P .R/ is weakly homogeneously Suslin in V . Suppose ı < ‚M , X  ı and jX j D !1 . Then there exists Y 2 M such that M  “jY j D !1 ” and such that X  Y . Proof. By Lemma 3.35, for each A 2 P .R/ \ M , the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº contains a set closed and unbounded in P!1 .H.!2 //. Therefore the theorem follows from Theorem 3.45. t u Corollary 3.48. Assume the nonstationary ideal on !1 is !2 -saturated and that there exist ! many Woodin cardinals with a measurable cardinal above them all. Let ‚ D ‚L.R/ . (1) Suppose that X is a bounded subset of ‚ of cardinality !1 . Then there exists a set Y 2 L.R/ of cardinality !1 in L.R/ such that X  Y . (2) Suppose G  P .!1 / n INS is V -generic and that j W V ! M is the induced generic elementary embedding. Let k W ‚ ! ‚ be the map derived from the ultrapower ‚!1 =U computed in L.R/ where U is the club measure on !1 . Then j j‚ D k: Proof. From the large cardinal hypothesis, AD holds in L.R/ and further every set of reals which is in L.R/ is weakly homogeneously Suslin. The corollary follows by the covering theorems. t u We end this section with the following theorem which in the special case of L.R/ approximates the converse of Theorem 3.46. Theorem 3.49. Assume ADL.R/ . Suppose that for all ı < ‚L.R/ , if f W !1 ! ı then there exists a function g W !1 ! P!1 .ı/ such that g 2 L.R/ and such that for all ˛ < !1 , f .˛/ 2 g.˛/. Let  be the least ordinal such that L .R/ †1 L.R/: Then for each set A  R such that A 2 L .R/ there exists a countable elementary substructure X  H.!2 / such that hX; A \ X; 2i  hH.!2 /; A; 2i and such that MX is A-iterable where MX is the transitive collapse of X .

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Proof. By the definition of ,  is a regular cardinal in L.R/ and  < ‚L.R/ . Therefore since  > !2 , cof./ > !1 , and so L .H.!2 //  ZFC : Further by the Moschovakis Coding Lemma, for each ˛ < , P .˛/ \ L.R/ 2 L .R/: Let X  L .H.!2 // be a countable elementary substructure and let MX be the transitive collapse of X . We prove that MX is A-iterable for each A  R such that A 2 X \ L .R/: Fix A. Thus for some t 2 R, A is definable in L .R/ from t . The set A is 21 .t / in L.R/ and so by the Martin–Steel theorem, Theorem 2.3, there exist < , and trees T0 ; T1 on !  such that A D pŒT0 ; such that, R n A D pŒT1 ; and such that .T0 ; T1 / is †1 -definable in L.R/ from .t; R/. Since L .R/ †1 L.R/; it follows that L .R/ \ .HOD t /L.R/ D .HOD t /L .R/ ; and so .T0 ; T1 / 2 .HOD t /L .R/ : Let j W .HOD t /L.R/ ! N t be the elementary embedding computed in L.R/ where N t D .HOD!t 1 /L.R/ = and where is the club filter on !1 . Since DC holds in L.R/, this ultrapower is wellfounded and we identify it with its transitive collapse. It follows that j  .HOD t /L.R/ . Since L .R/ †1 L.R/ and since cof./ > !1 , j./ D . The structure   .HOD t /L .R/ ; j j.HOD t /L .R/ is naturally iterable and the iterates are wellfounded. The notion of iteration is the conventional (non-generic) one.

3.1 The nonstationary ideal and ı12

107

Let .N; k/ be the image of ..HOD t /L .R/ ; j j.HOD t /L .R/ / under the transitive collapse of X . Thus N and k are definable subsets of MX . Let T0X be the image of T0 under the transitive collapse of X and let T1X be the image of T1 . Suppose j W .N; k/ ! .N  ; k  / is a countable iteration. Then it follows that there exists an elementary embedding  W N  ! .HOD t /L .R/ such that .j.T0X // D T0 and such that .j.T1X // D T1 : Thus N  is wellfounded, pŒ.j.T0X //  pŒT0  and pŒ.j.T1X //  pŒT1 : We now come to the key points. By the Moschovakis Coding Lemma, if h W !1 !  and h 2 L.R/ then h 2 L .R/. Thus the hypothesis of the theorem holds in L .H.!2 //. Suppose that jO W MX ! MX is an iteration of MX . Then, abusing notation slightly, jOjN W .N; k/ ! .jO.N /; jO.k// is an iteration of .N; k/ and so MX is wellfounded. Let B D R n A D pŒT1 . Thus jO.A \ MX /  pŒjO.T0X /  pŒT0  and

jO.B \ MX /  pŒjO.T1X /  pŒT1 :

Therefore

jO.A \ MX / D A \ MX :

This verifies that MX is A-iterable.

t u

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3 The nonstationary ideal

3.2

The nonstationary ideal and CH

We still do not know if CH implies that the nonstationary ideal on !1 is not saturated. In light of the results in the previous section this seems likely. Shelah, Shelah .1986/, has proved that assuming CH the nonstationary ideal is not !1 -dense. We prove a generalization of this theorem. It is a standard fact, which is easily verified, that the boolean algebra, P .!1 /=INS , is !2 -complete; i. e. if X  P .!1 /=INS is a subset of cardinality at most @1 then _X exists in P .!1 /=INS . Theorem 3.50. Suppose that the quotient algebra P .!1 /=INS is !1 -generated .equivalently !-generated/ as an !2 -complete boolean algebra. Then 2@0 D 2@1 :

t u

We shall actually prove the following strengthening of Theorem 3.50. We fix some notation. Suppose A  !1 . For each  < !2 such that !1   , let bA 2 P .!1 /=INS be defined as follows. Fix a bijection  W !1 ! : Let S D ¹ < !1 j ordertype.Œ/ 2 Aº: Set to be the element of P .!1 /=INS defined by S . It is easily checked that bA is unambiguously defined. We let BA denote the !2 -complete subalgebra of P .!1 /=INS generated by ® A ¯ b j !1   < !2 : bA

Suppose Z  P .!1 /=INS is of cardinality @1 . Then there exists a set A  !1 such that Z  BA : Thus Theorem 3.50 is an immediate corollary of the next theorm. Theorem 3.51. Suppose that for some set A  !1 BA D P .!1 /=INS Then

2@0 D 2@1 :

3.2 The nonstationary ideal and CH

109

Proof. The key point is the following. Suppose Y  hH.!2 /; 2i is a countable elementary substructure such that A 2 Y . Let N be the transitive collapse of Y and suppose that j W N ! N is a countable iteration such that N  is transitive. Then we claim that j is uniquely determined by j.AN / where AN D A \ !1N . To see this let hNˇ ; G˛ ; j˛;ˇ j ˛ < ˇ   i be the iteration giving j . We first prove that G0 is uniquely determined by j.AN / \ N . This follows from the definitions noting that the property of A, BA D P .!1 /=INS is a first order property of A in H.!2 /. Therefore since Y  hH.!2 /; 2i it follows that N  Ba D P .!1 /=INS where a D AN . For each  2 N \ Ord with  !1N , let .ba /N be as computed in N . Strictly speaking .ba /N is not an element of N , instead it is a definable subset of N . G0 is an N -generic filter and so it follows since N  Ba D P .!1 /=INS that G0 is uniquely determined by ¯ ®  2 N j G0 \ .ba /N ¤ ; : Finally

¯ ®  2 N j G0 \ .ba /N ¤ ; D .j.a/ \ N / n !1N :

This verifies that G0 is uniquely determined by j.AN /\N . It follows by induction that j is uniquely determined by j.AN /. Fix B  !1 and fix a countable elementary substructure X  H.!2 / with A 2 X and B 2 X . Let hX W  < !1 i be the sequence of countable elementary substructures of H.!2 / generated by X as follows. (1.1) X0 D X . (1.2) For all  < !1 , X C1 D X ŒX \ !1  D ¹f .X \ !1 / j f 2 X º:

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(1.3) For all  < !1 , if  is a limit ordinal then X D [¹X j  < º: Let hM X W  < !1 i be the sequence of countable transitive sets where for each  < !1 , M X is the transitive collapse of X . Let X!1 D [¹X j  < !1 º and let M!1 be the transitive collapse of X!1 . For each  <   !1 let j; W M ! M be the elementary embedding given by the image of the inclusion map X  X under the collapsing map. For each  < !1 , .!1 /M is the critical point of j ; C1 and M C1 is the restricted ultrapower of M by G where G is the M -ultrafilter on .!1 /M given by j ; C1 . By Lemma 3.12, hM ; G ; j; W  <   !1 i is an iteration of M0 . For each   !1 let A be the image of A under the collapsing map. Therefore A D A \ .!1 /M and for each  < !1 , j ; C1 .A / D A C1 : Similarly for each   !1 let B be the image of B under the collapsing map. Thus by Corollary 3.13, j0;!1 .B0 / D B. For all  < !1 , j ; C1 .A / D A C1 . By the claim proved above, for all  < !1 , G is uniquely determined by j ; C1 .A /. But for each  < !1 , j ; C1 .A / D A C1 : Therefore the iteration hM ; G ; j; W  <   !1 i is uniquely determined by M0 and A. Finally j0;!1 .B0 / D B. This induces a map t u from H.!1 / onto P .!1 /. Remark 3.52. One can also prove Theorem 3.51 using a form of ˘, weak diamond, due to Devlin and Shelah, Devlin and Shelah .1978/. This weakened form of diamond holds whenever 2@0 ¤ 2@1 . t u Suppose that the nonstationary ideal on !1 is !2 -saturated. Then for each A  !1 there exists A  !1 such that A is definable in LŒA  and such that the quotient algebra .P .!1 /=INS /=BA is atomless.

3.2 The nonstationary ideal and CH

111

Thus if the nonstationary ideal on !1 is saturated and CH holds then P .!1 /=INS decomposes as B  T where T is a Suslin tree in V B . We now define two weak forms of ˘. We shall see that if ˘ holds in a transitive inner model which correctly computes !2 then these forms of ˘ hold in V . To motivate the definitions we recall the following equivalents of ˘, stating a theorem of Kunen. Theorem 3.53 (Kunen). The following are equivalent. (1) ˘. (2) There exists a sequence hS˛ j ˛ < !1 i of countable sets such that for each A  !1 the set ¹˛ j A \ ˛ 2 S˛ º is stationary in !1 . (3) There exists a sequence hS˛ j ˛ < !1 i of countable sets such that for each A  !1 the set ¹˛ ! j A \ ˛ 2 S˛ º is nonempty. (4) There exists a sequence hS˛ j ˛ < !1 i of countable sets such that for each countable X  P .!1 / the set ¹˛ ! j A \ ˛ 2 S˛ for all A 2 X º is nonempty. Proof. .2/ is commonly referred to as weak ˘. That .3/ is also equivalent to ˘ is perhaps at first glance surprising. We prove that .3/ is equivalent to .2/. Let hS˛ j ˛ < !1 i be a sequence witnessing .3/. For each ˛ < !1 let T˛ D P .˛/ \ L .hSˇ j ˇ < ˛ C !i/ where  < !1 is the least ordinal such that L .hSˇ j ˇ < ˛ C !i/  ZF n Powerset: We claim that hT˛ j ˛ < !1 i witnesses .2/. To verify this fix A  !1 and fix a closed unbounded set C  !1 . We may suppose that C contains only limit ordinals. It suffices to prove that for some ˇ 2 C , A \ ˇ 2 Tˇ . Let B0 D ¹2 ˛ j ˛ 2 Aº: For each  2 C [ ¹0º, let x  ! be a set which codes A \  where  is the least element of C above . Let B1 D ¹ C 2k C 1 j  2 C and k 2 x º: Let B D B0 [ B1 . Since hS˛ W ˛ < !1 i witnesses .3/, there exists an infinite ordinal ˛ such that B \ ˛ 2 S˛ : If ˛ 2 C then set ˇ D ˛. Thus ˇ is as required since S˛  T˛ . If ˛ … C let  be the largest element of C below ˛. Let  D 0 if C \ ˛ D ;. Let  be the least element of C above ˛. There are two cases. If  C !  ˛ then A \  2 T  since x D ¹k < ! j . C 2k C 1/ 2 B \ ˛º: If ˛ <  C ! then  ¤ 0. Therefore  2 C and since ˛ <  C !, A \  2 T . t u In either case A \ ˇ 2 Tˇ for some ˇ 2 C .

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3 The nonstationary ideal

Our route toward a weakening of ˘ starts with .4/ which is reminiscent of ˘C . Definition 3.54. Suppose hS˛ j ˛ < !1 i is a sequence of countable sets. Suppose X  P .!1 / is countable. Then X is guessed by hS˛ j ˛ < !1 i if the set ¹˛ j A \ ˛ 2 S˛ for all A 2 X º t u

is unbounded in !1 .

Q There exists a sequence hAˇ j ˇ < !2 i of distinct subsets of !1 Definition 3.55. ˘: and there exists a sequence hS˛ j ˛ < !1 i of countable sets such that ¹ˇX j X  P .!1 / is countable and hS˛ j ˛ < !1 i guesses X º is stationary in !2 . Here ˇX D sup¹ C 1 j A 2 X º.

t u

We weaken (possibly) still further in the following definition. QQ There exists a sequence Definition 3.56. ˘: hAˇ j ˇ < !2 i of distinct subsets of !1 and a sequence hS˛ j ˛ < !1 i of countable sets such that for a stationary set of countable sets X  !2 , there exists ˛ < !1 such that X \ !1  ˛ and such that ¹ˇ j ˇ 2 X \ !2 and Aˇ \ ˛ 2 S˛ º is t u cofinal in X \ !2 . QQ the sequence Remark 3.57. (1) Suppose that 2@1 D @2 . Then in the definition of ˘, hAˇ j ˇ < !2 i can be taken to be any enumeration of P .!1 /. (2) If there is a Kurepa tree on !1 then ˘Q holds. We shall show in Section 6.2.5 that the existence of a weak Kurepa tree is consistent with the nonstationary ideal on !1 is !1 -dense. Therefore ˘QQ is not implied by the existence of a weak Kurepa tree. Recall that a tree T  ¹0; 1º
3.2 The nonstationary ideal and CH

113

Let hAˇ W ˇ < !2 i be a sequence of distinct subsets of !1 with hAˇ W ˇ < !2 i 2 M: The key point is that the set M \ P!1 .!2 / is stationary in P!1 .!2 /. To verify this, let F W !2
F Œ
t u

Theorem 3.59. Assume that the nonstationary ideal on !1 is !2 -saturated. Then ˘QQ fails. QQ Proof. Suppose hS˛ W ˛ < !1 i and hAˇ W ˇ < !2 i together witness ˘. Therefore there exists a countable elementary substructure X  H.!3 / such that (1.1) hS˛ W ˛ < !1 i 2 X , (1.2) hAˇ W ˇ < !2 i 2 X , (1.3) for some ˛ < !1 , X \ !1 < ˛ and ¹ˇ j ˇ 2 X and Aˇ \ ˛ 2 S˛ º is cofinal in X \ !2 .

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3 The nonstationary ideal

Fix ˛ satisfying (1.3). Let hX W  < !1 i be the elementary chain where X0 D X and for all  < !1 , (2.1) XC1 D ¹f .X \ !1 / j f 2 X º, (2.2) if  is a limit ordinal, X D [¹X j  <  º: Fix  < !1 such that !1 \ X  ˛ < !1 \ XC1 : Note that for all  < !1 , X \ !2 is cofinal in X \ !2 . Therefore ¹ˇ j ˇ 2 X and Aˇ \ ˛ 2 S˛ º is cofinal in X \ !2 . Thus by replacing X by X if necessary we may assume that  D 0; i. e. that !1 \ X  ˛ < !1 \ Y where Y D ¹f .X \ !1 / j f 2 X º: Let NX be the transitive collapse of X , let NY be the transitive collapse of Y and let j W NX ! NY be the induced elementary embedding (the image of the inclusion map). However the nonstationary ideal on !1 is !2 -saturated and P .!1 /  H.!3 /. Therefore by Lemma 3.12, NY is a generic ultrapower of NX and j is the induced embedding. Transferring to V (or equivalently, working in NX ) there exists a stationary set S  !1 and ordinal ˛0 such that !1  ˛0 < !2 and such that if G  P .!1 / n INS is V -generic with S 2 G then ¹ < !2 j j.A / \ ˛0 2 S˛0 º is cofinal in !2 where

j W V ! N  V ŒG

is the induced embedding and hS W  < !2 i D j.hS˛ W ˛ < !1 i/: However for all  < !2 , j.A / \ !1 D A , and so for all 1 < 2 < !2 , j.A 1 / \ ˛0 ¤ j.A 2 / \ ˛0 : This is a contradiction since S˛0 is countable in V ŒG and !2V D !1V ŒG .

t u

As an immediate corollary to Theorem 3.58 and Theorem 3.59 we obtain the following. Corollary 3.60. Assume that the nonstationary ideal on !1 is !2 -saturated. Then ˘ u t fails in any transitive inner model which correctly computes !2 .

3.2 The nonstationary ideal and CH

115

Related to the question of CH is the following question: Question. Can there exist countable transitive models M; M  such that M  ZFC C “The nonstationary ideal on !1 is saturated”; 

M is an iterate of M and such that M 2 M  ?

t u

Remark 3.61. (1) For this question the fragment of ZFC is important. The answer should be the same for all reasonably strong fragments. But note the answer is yes for ZFC for trivial reasons. (2) It is straightforward to show that the answer is no if the model M is iterable or if M  “P .!1 /=INS is countably generated”. (3) Suppose the nonstationary ideal on !1 is saturated and CH holds. Suppose there exists an inaccessible cardinal. Then the answer is yes. t u One could ask this question for any iteration of generic embeddings. Suppose V is the inner model for one Woodin cardinal. Suppose G  Coll.!1 ; .!1 /N0 and such that ı is a Silver indiscernible of LŒx for all x 2 [¹R \ Nk j k 2 !º. From this iterability follows by an argument essentially identical to that given in the proof of Theorem 3.16. There it is proved that assuming  ı 12 D !2 and that the nonstationary ideal is saturated then if X  H.!2 / is a countable elementary substructure, the transitive collapse of X is iterable. t u Remark 4.18. (1) It is important to note that the assumptions of Lemma 4.17 do not actually imply that any iterations exist; the only implication is that if iterates exist, they are wellfounded. It is easy to construct sequences which satisfy the conditions of Lemma 4.17 and for which no (nontrivial) iterations exist. Lemma 4.19 isolates a condition sufficient to prove the existence of nontrivial iterations. (2) The conditions (i) and (ii) of the hypothesis of Lemma 4.17 are equivalent to the assertions: a) if C 2 Nk is closed and unbounded in !1N0 then there exists x 2 NkC1 such that ¹˛ < !1N0 j L˛ Œx is admissibleº  C: b) V!C1 \ NkC1 †2 V!C1 .

t u

Lemma 4.19. Suppose that hNk W k < !i is a sequence of countable transitive sets such that for all k < !, Nk 2 NkC1 , Nk  ZFC ; and Nk \ .INS /NkC1 D Nk \ .INS /NkC2 : Suppose that k 2 ! and that a 2 .P .!1 //Nk n .INS /NkC1 : Then there exists G  [¹.P .!1 //Ni j i < !º such that a 2 G and such that for all i < !, G \ Ni is a uniform Ni -normal ultrafilter. Proof. Fix

a 2 .P .!1 //Nk n .INS /NkC1 ; by replacing hNi W i < !i with hNiCk W i < !i, we may suppose that a 2 N0 . Let hfi W i < !i enumerate all functions f W !1N0 ! !1N0 such that f 2 [¹Nj j j < !º and such that for all ˛ < !1N0 , f .˛/ < 1 C ˛. (Thus f .˛/ < ˛ for all ˛ > !.)

4.1 Iterable structures

127

We may suppose that fi 2 Ni for all i < !. Construct a sequence hai W i < !i such that a0  a and such that for all i < !, (1.1) ai  !1N0 , (1.2) ai is cofinal in !1N0 , (1.3) ai 2 Ni n .INS /Ni C1 , (1.4) fi jai is constant, (1.5) aiC1  ai . The sequence is easily constructed by induction on i . Suppose ai is given. By (1.2) it follows that ai … .INS /Nj for all j i . This is the key point. Thus ai is a stationary subset of !1N0 in NiC2 and so since fiC1 is regressive there exists ˇ < !1N0 such that a D ¹ 2 ai j fiC1 ./ D ˇº … .INS /Ni C2 : since fiC1 2 NiC1 . Therefore a satisfies the requirements for

However a 2 NiC1 aiC1 . Let hai W i < !i be a sequence satisfying (1.1)–(1.4) and let

G D ¹b  !1N0 j b 2 [¹Nj j j 2 !º and ai  b for some i < !º: It follows that for each j < !, G \ Nj is a uniform Nj -normal ultrafilter. (1.2) guarantees uniformity and (1.4) guarantees normality. t u Lemma 4.17 yields the following corollary. Corollary 4.20. Suppose h.Nk ; Jk / W k < !i is a countable sequence such that for each k, Nk is a countable transitive model of ZFC and such that for all k: (i) Jk 2 Nk and Nk  “Jk is a set of normal uniform ideals on !1 ”I (ii) Nk 2 NkC1 and jNk jNkC1 D !1N0 ; (iii) for each I 2 Jk there exists I  2 JkC1 such that, (1) I  \ Nk D I , N

(2) for each A 2 Nk such that A  P .!1 k / \ Nk n I if A is predense in .P .!1 / n I /Nk , then A is predense in .P .!1 / n I  /NkC1 ;

128

4 The Pmax -extension

(iv) .Nk ; Jk / is iterable; (v) if C 2 Nk is closed and unbounded in !1N0 then there exists D 2 NkC1 such that D  C , D is closed and unbounded in C and such that D 2 LŒx for some x 2 R \ NkC1 . Then h.Nk ; Jk / W k < !i is iterable. Proof. Any iteration of h.Nk ; Jk / W k < !i naturally defines an iteration of hNk W k < !i: By Lemma 4.17, the iterates of hNk W k < !i are wellfounded.

t u

Remark 4.21. The previous lemma is also true if condition (iv) is replaced by the condition that for all x 2 R \ .[¹Nk j k 2 !º/; x # 2 [¹Nk j k 2 !º.

t u

We continue our discussion of iterable structures with Lemma 4.22 which is a boundedness lemma for iterations of sequences of structures. Lemma 4.22 which will be used to guarantee that the conditions of Lemma 4.17 are satisfied, is proved by an argument identical to that for Lemma 4.7. Lemma 4.22 (ZFC ). Assume that for all x 2 R, x # exists. Suppose hNk W k < !i is an iterable sequence and that j W hNk W k < !i ! hNk W k < !i is an iteration of length !1 . Let x 2 R code hNk W k < !i. Then (1) for all k < !

rank.Nk / <  ı 12 ;

(2) if C 2 [¹Nk j k < !º is closed and unbounded in !1 then there exists D 2 LŒx such that D  C and such that D is closed and unbounded in t u !1 . Definition 4.15 suggests the following generalization of Definition 3.5. Definition 4.23. Suppose that M is a countable model of ZFC . A sequence hMˇ ; G˛ ; j˛;ˇ j ˛ < ˇ <  i is a semi-iteration of M if the following hold.

4.1 Iterable structures

129

(1) M0 D M . (2) j˛;ˇ W M˛ ! Mˇ is a commuting family of elementary embeddings. (3) For each ˇ C1 <  , Gˇ is an Mˇ -normal ultrafilter, Mˇ C1 is the Mˇ -ultrapower of Mˇ by Gˇ and jˇ;ˇ C1 W Mˇ ! Mˇ C1 is the induced elementary embedding. (4) For each ˇ <  if ˇ is a limit ordinal then Mˇ is the direct limit of ¹M˛ j ˛ < ˇº and for all ˛ < ˇ, j˛;ˇ is the induced elementary embedding. A model N is a semi-iterate of M if it occurs in an semi-iteration of M . The model M is strongly iterable if every semi-iterate of M is wellfounded. t u Clearly if M  “INS is saturated” then every semi-iteration of M is an iteration of M . We recall the following notation. Suppose A  R. Then  †11 .A/ is the set of all BR such that B can be defined from real parameters by a †1 formula in the structure hV!C1 ; A; 2i: if both B and R n B are  †11 .A/. A set B  R is Let  ı 11 .A/ be the supremum of the lengths of the prewellorderings of R that are 1  1 .A/. 1  1 .A/

Lemma 4.24. Suppose that A  R and that there exists X  H.!2 / such that hX; A \ X; 2i  hH.!2 /; A; 2i and such that the transitive collapse of X is A-iterable. Suppose that M is a transitive set, H.!2 /  M , M  ZFC ; and that M \ Ord <  ı 11 .A/: Then the set of ¹Y  M j Y is countable and MY is strongly iterableº contains a club in P!1 .M /. Here MY is the transitive collapse of Y . Proof. Let  D rank.M / and let  WR! be a surjection such that 1 ¹.x; y/ j .x/  .y/º 2  1 .A/:

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4 The Pmax -extension

Let B D ¹.x; y/ j .x/  .y/º: Let N be a transitive set such that N  ZFC and such that ¹M; ; Aº [ H.!2 /  N . Let Y  N be a countable elementary substructure such that ¹; M; Aº  Y and let NY be the transitive collapse of Y . Let Y be the image of  under the transitive collapse and let Y be the image of . Let X D Y \ M and let MX be the transitive collapse of X . Suppose j W .MX ; 2/ ! .M  ; E  / is an elementary embedding given by a countable semi-iteration. Since H.!2 /MX D H.!2 /NY ; j lifts to define a semi-iteration k W .NY ; 2/ ! .N  ; E  /: We identify the standard part of N  with its transitive collapse. Thus kjH.!2 /MX W H.!2 /MX ! k.H.!2 /MX / is a countable iteration. By Theorem 3.34, H.!2 /MX is A-iterable. Therefore k.A \ NY / D A \ N  1 and so since B is  1 .A/ in parameters from NY , k.B \ NY / D B \ N  :

By elementarity, it follows that k.Y / W R \ N  ! k.Y / is a surjection and that B \ N  D ¹.x; y/ j k.Y /.x/  k.Y /.y/º: Therefore k.Y / is an ordinal and so k.MX / is wellfounded. Thus j.MX / is wellfounded since j.MX / elementarily embeds into k.MX /. Therefore MX is strongly-iterable.

t u

Definition 4.25. The nonstationary ideal on !1 is semi-saturated if for all generic extensions, V ŒG, of V , if U 2 V ŒG is a V -normal ultrafilter on !1V , then Ult.V; U / is wellfounded. t u

4.1 Iterable structures

131

Lemma 4.26. Suppose INS is not semi-saturated and that G  Coll.!; P .!1 // is V -generic. Then there exists U 2 V ŒG such that U is a V -normal ultrafilter on !1V and such that Ult.V; U / is not wellfounded. Proof. Suppose INS is not semi-saturated in V . Then there exists a V -normal ultrafilter U0 such that U0 is set generic over V and such that Ult.V; U0 / is not wellfounded. Let  be an ordinal such that ¹f W !1V !  j f 2 V º=U0 is not wellfounded. We work in V ŒG. Let hbi W i < !i be an enumeration of .P .!1 //V and let hgi W i < !i be an enumeration of all functions g W !1V ! !1V such that g 2 V and such that for all ˛ < !1V , g.˛/ < 1 C ˛. Let T be the set of finite sequences h.ai ; fi / W i  ni such that for all i < n, (1.1) ai 2 .P .!1 //V n ¹;º, and aiC1  ai , (1.2) ai  bi or ai \ bi D ;, (1.3) fi W !1V !  , fi 2 V and for all ˇ 2 aiC1 , fiC1 .ˇ/ < fi .ˇ/; (1.4) gi jai is constant. T is a tree ordered by extension. Any infinite branch of T yields a V -normal ultrafilter, U , such that ¹f W !1V !  j f 2 V º=U is not wellfounded. Conversely if U is a V -normal ultrafilter such that ¹f W !1V !  j f 2 V º=U is not wellfounded, then U defines an infinite branch of T . Therefore U0 defines an infinite branch of T and so T is not wellfounded. By absoluteness, T must have an infinite branch in V ŒG. t u Clearly if INS is !2 -saturated then INS is semi-saturated. Lemma 4.27. Suppose that INS is semi-saturated and that U  P .!1 / is a uniform, V -normal ultrafilter which set generic over V . Let j W V ! M  V ŒU  be the associated generic elementary embedding. Then j.!1 / D !2 .

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4 The Pmax -extension

Proof. For each ˛ < !2 let

˛ W !1 ! ˛

be a surjection and define f˛ W !1 ! !1 by f˛ .ˇ/ D ordertype.˛ Œˇ/. Suppose that U  P .!1 / is a uniform, V -normal ultrafilter which set generic over V . Let j W V ! M  V ŒU  be the associated generic elementary embedding. Then for each ˛, j.f˛ /.!1V / D ˛I i. e. the function f˛ necessarily represents ˛ (it is a canonical function for ˛). We begin by noting the following. Suppose that I0  P .!1 / is a normal uniform ideal and that h W !1 ! !1 is a function such that for each ˛ < !2 , ¹ˇ < !1 j f˛ .ˇ/ < h.ˇ/º … I0 : Then there is a normal, uniform, ideal I0  P .!1 / such that I0  I0 and such that for each ˛ < !2 , ¹ˇ < !1 j h.ˇ/  f˛ .ˇ/º 2 I0 I simply define I0 to be the ideal generated by I0 [ ¹¹ˇ < !1 j h.ˇ/  f˛ .ˇ/º j ˛ < !2 º: It is straightforward to verify that this is a normal ideal and that it is proper. The point is that for all ˛1  ˛2 < !2 , ¹ˇ < !1 j f˛2 .ˇ/  h.ˇ/º n ¹ˇ < !1 j f˛1 .ˇ/  h.ˇ/º 2 INS : Assume toward a contradiction that the lemma fails. Then it follows that there exists a function h W !1 ! !1 and a normal, uniform, ideal I on !1 such that if U  P .!1 / is a V -normal ultrafilter which is set generic over V such that U \ I D ;, then j.h/.!1V / D !2V where j W V ! M  V ŒU  be the associated generic elementary embedding. Otherwise one can easily construct a V -normal ultrafilter U  which is set generic over V and such that Ult.V; U  / is not wellfounded. Clearly we can suppose that for all ˇ < !1 , h.ˇ/ is a nonzero limit ordinal. For each ˇ < !1 let hˇk W k < !i be an increasing cofinal sequence in h.ˇ/. For each k < ! define hk W !1 ! !1 by hk .ˇ/ D ˇk .

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For each k < ! there must exist ˛k < !2 such that ¹ˇ < !1 j f˛k .ˇ/  hk .ˇ/º 2 I: Otherwise for some k0 < !1 and for each ˛ < !2 , ¹ˇ < !1 j f˛ .ˇ/ < hk0 .ˇ/º … I: In this case it follows, by the remarks above, that there is a normal ideal I  such that I  I  and such that if U   P .!1 / is a V -normal ultrafilter, set generic over V , with U \ I  D ;, then j  .hk0 /.!1V / !2V where j  is the associated generic elementary embedding. This contradicts the choice of h and I . Let ˛! D sup¹˛k j k < !º. Thus ¹ˇ j f˛! .ˇ/ < h.ˇ/º 2 I since for all ˇ < !1 , h.ˇ/ D sup¹hk .ˇ/ j k < !º: This again contradicts the choice of h and I .

t u

Corollary 4.28. Suppose that INS is semi-saturated and that f W !1 ! !1 . Then there exists ˛ < !2 such that the following holds. Let  W !1 ! ˛ be a surjection. The set ¹ˇ < !1 j f .ˇ/ < ordertype.Œˇ/º contains a closed, unbounded, subset of !1 . Proof. As in the proof of Lemma 4.27, for each ˛ < !2 let ˛ W !1 ! ˛ be a surjection and define f˛ W !1 ! !1 by f .ˇ/ D ordertype.˛ Œˇ/. Assume toward a contradiction that for each ˛ < !2 , ¹ˇ < !1 j f˛ .ˇ/  f .ˇ/º … INS : Then, arguing as in the proof of Lemma 4.27, there is a normal, uniform, ideal I  P .!1 / such that for each ˛ < !2 , ¹ˇ < !1 j f .ˇ/  f˛ .ˇ/º 2 I: Suppose that U  P .!1 / is a V -normal ultrafilter such that U is set generic over V and such that U \ I D ;. Let j W V ! M  V ŒU  be the associated generic elementary embedding. Then !2V  j.f /.!1V / which contradicts Lemma 4.27. t u

4 The Pmax -extension

134

We will encounter situations in which the nonstationary ideal on !1 is semisaturated and not saturated cf. Definition 6.11 and Theorem 6.13. Nevertheless the assertion that INS is semi-saturated has many of the consequences proved in Section 3.1 for the assertion that INS is saturated. For example it is routine to modify the proofs in Section 3.1 to obtain the following variations of Lemma 3.14 and Theorem 3.17, together with the subsequent generalization of Theorem 3.47. Clearly, if the nonstationary ideal is semi-saturated in V then it is semi-saturated in L.P .!1 //. Theorem 4.29. Suppose that the nonstationary ideal on !1 is semi-saturated and that P .!1 /# exists. Suppose that X  H.!2 / is a countable elementary substructure. Then the transitive collapse of X is iterable. Proof. Clearly for all x 2 R, x # exists. Let Y  L.P .!1 // be a countable elementary substructure containing infinitely many Silver indiscernibles of L.P .!1 //. Let X D Y \ H.!2 /, let N be the transitive collapse of Y and let M be the transitive collapse of X . Thus M D .H.!2 //N and N D L˛ .M / where ˛ D N \ Ord. Since Y contains infinitely many indiscernibles of L.P .!1 //, L˛ .M /  L.M /: Finally INS is semi-saturated and so L.P .!1 //  “INS is semi-saturated”: Therefore N  “INS is semi-saturated” and so L.M /  “INS is semi-saturated”: We claim that M is iterable. Suppose M  is an iterate of M occurring in an iteration of length ˛. Let  < !1 be such that ˛ <  and such that L .M /  L.M /:

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135

By an absoluteness argument analogous to the proof of Lemma 3.10, any semiiterate of L .M / occurring in a semi-iteration of L .M / of length less than  is wellfounded. The iteration of M of length ˛ witnessing M  is an iterate of M induces a semiiteration of L .M / of length ˛ producing a semi-iterate of L .M / into which M  can be embedded. Therefore M  is wellfounded and so M is iterable. Thus there exists a countable elementary substructure X  H.!2 / whose transitive collapse is iterable. Thus by Theorem 3.19, if X  H.!2 / is any countable elementary substructure, the transitive collapse of X is iterable. t u Theorem 4.30. Suppose that the nonstationary ideal on !1 is semi-saturated and that ı 12 D !2 . P .!1 /# exists. Then  Proof. By Theorem 3.19, the theorem is an immediate corollary of Theorem 4.29. u t The proof of Lemma 3.35 can similarly be adapted to prove the corresponding generalization of Lemma 3.35. Lemma 4.31. Suppose that the nonstationary ideal on !1 is semi-saturated. Suppose A  R and that B is weakly homogeneously Suslin for each set B which is projective in A. Let M be a transitive set such that M  ZFC , P .!1 /  M , and such that M # exists. Then ¹X  M j X is countable and MX is A-iterableº t u contains a club in P!1 .M /. Here MX is the transitive collapse of X . Finally we obtain the generalization of the second covering theorem, Theorem 3.47, to the case when INS is simply assumed to be semi-saturated. Theorem 4.32. Suppose that the nonstationary ideal on !1 is semi-saturated. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) every set A 2 M \ P .R/ is weakly homogeneously Suslin in V . Suppose ı < ‚M , X  ı and jX j D !1 . Then there exists Y 2 M such that M  “jY j D !1 ” and such that X  Y . Proof. This is an immediate corollary of Lemma 4.31, applied to the set, H.!2 /, and Theorem 3.45. t u

136

4 The Pmax -extension

4.2

The partial order Pmax

We now define the partial order Pmax . Definition 4.33. Let Pmax be the set of pairs h.M; I /; ai such that: (1) M is a countable transitive model of ZFC C MA!1 ; (2) I 2 M and M  “I is a normal uniform ideal on !1 ”; (3) .M; I / is iterable; (4) a  !1M ; (5) a 2 M and M  “!1 D !1LŒaŒx for some real x” . Define a partial order on Pmax as follows: h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i if M0 2 M1 ; M0 is countable in M1 and there exists an iteration j W .M0 ; I0 / ! .M0 ; I0 / such that: (1) j.a0 / D a1 ; (2) M0 2 M1 and j 2 M1 ; (3) I1 \ M0 D I0 .

t u

Remark 4.34. (1) Given the results of Section 3.1 it would be more natural to define Pmax as the set of pairs .M; a/ where M is an iterable model in which the nonstationary ideal is saturated. Assuming  12 -Determinacy this yields an equivalent forcing notion. More precisely assuming  12 -Determinacy, the set of conditions h.M; I /; ai 2 Pmax such that I is a saturated ideal in M and such that I is the nonstationary ideal in M, is dense in Pmax . (2) We shall prove that the nonstationary ideal is saturated in L.R/Pmax and that ZFC holds there. Thus Pmax is in some sense converting the existence of models with precipitous ideals (which are relatively easy to find) into the existence of models in which the nonstationary ideal on !1 is saturated. This is an aspect we shall exploit when we modify Pmax to show the relative consistency that the t u nonstationary ideal on !1 is !1 -dense. There are equivalent versions of Pmax that do not require that the models which appear in the conditions be models of MA!1 , this is a degree of freedom which is essential for the variations that we shall define. In Chapter 5 we shall give three other (T)  0 , Pmax and Pmax . The first of these will involve presentations of Pmax , denoted by Pmax

4.2 The partial order Pmax

137

using the generic elementary embeddings associated to the stationary tower in place of embeddings associated to ideals on !1 . The second will be closer to Pmax , however the stationary tower will be used to generate the necessary conditions and so certain aspects of the analysis will differ. In fact there are strong arguments to support the  0 is actually the best presentation of Pmax . The third, Pmax , is claim that in the end, Pmax (T)  a combination of Pmax and Pmax . In defining two of the variations of Pmax , we shall use these alternate formulations as a template, see Definition 6.54 and Definition 8.30. The following lemma indicates the utility of working with models of MA. We state it in a more general form than is strictly necessary for the analysis of Pmax . Lemma 4.35. Suppose M is a countable transitive model of ZFC C MA!1 . Suppose a 2 M, a  !1M ; and

M  “!1 D !1L.a;x/ for some x 2 R”:

Suppose j1 W M ! M1 and j2 W M ! M2 are semi-iterations of M such that (i) M1 is transitive, (ii) M2 is transitive, (iii) j1 .a/ D j2 .a/, (iv) j1 .!1M / D j2 .!1M /. Then M1 D M2 and j1 D j2 . Proof. This is a relatively standard fact. The key point, which we prove below, is that since both j1 .a/ D j2 .a/ and

j1 .!1M / D j.!2M /;

it follows that j1 .b/ D j2 .b/ for each set b 2 M such that b  !1M . From this it follows easily by induction that at every stage the generic filters are the same and so j1 D j 2 . Let hs˛ W ˛ < !1M i be the sequence of almost disjoint subsets of ! where for each  < !1M , s is the first subset of ! constructed in L.a; x/ which is almost disjoint from sˇ for each ˇ <  . Thus hs˛ W ˛ < !1M i 2 L.a; x/

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4 The Pmax -extension

and this sequence is definable from a and x. Since j1 ..a; !1M // D j2 ..a; !1M // it follows that

j1 .hs˛ W ˛ < !1M i/ D j2 .hs˛ W ˛ < !1M i/:

Let

ht˛ W ˛ < i D j1 .hs˛ W ˛ < !1M i/

where D j1 .!1M / D j2 .!1M /. Suppose that b 2 M and b  !1M . Since M  MA!1 it follows that there exists t 2 M such that t almost disjoint codes b relative to hs˛ W ˛ < !1M i; i. e. b D ¹˛ j t \ s˛ is infiniteº. Therefore j1 .b/ D ¹˛ < j t \ t˛ is infiniteº D j2 .b/: Therefore for each b 2 M such that b  !1M , j1 .b/ D j2 .b/. The lemma follows. t u The next two lemmas are key to proving many of the properties of the partial order Pmax . Because we wish to apply them within the models occurring in conditions we work in ZFC . Lemma 4.36 (ZFC ). Suppose .M; I / is a countable transitive iterable model where I 2 M is a normal uniform ideal on !1M and M  ZFC . Suppose J is a normal uniform ideal on !1 . Then there exists an iteration j W .M; I / ! .M ; I  / such that: (1) j.!1M / D !1 ; (2) J \ M D I  . Proof. Fix a sequence hAk;˛ W k < !; ˛ < !1 i of J -positive sets which are pairwise disjoint. The ideal J is normal hence each Ak;˛ is stationary in !1 . We suppose that Ak;˛ \ .˛ C 1/ D ;. Fix a function f W !  !1M ! P .!1M / \ M n I such that (1.1) f is onto, (1.2) for all k < !, f jk  !1M 2 M, (1.3) for all A 2 M if A has cardinality !1M in M and if A  P .!1M / n I then A  ran.f jk  !1M / for some k < !.

4.2 The partial order Pmax

139

The function f is simply used to anticipate subsets of !1 in the final model. Suppose j  W .M; I / ! .M ; I  / is an iteration. Then we define j  .f / D [¹j  .f jk  !1M / j k < !º 

and it is easily verified that the range of j  .f / is P .!1M /\M nI  . This follows from (1.3). We construct an iteration of M of length !1 using the function f to provide a book-keeping device for all of the subsets of !1 which belong to the final model and do not belong to the image of I in the final model. More precisely construct an iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i such that for each ˛ < !1 , if !1M˛ 2 Ak; then j0;˛ .f /.k; / 2 G˛ . The set C D ¹j0;˛ .!1M / j ˛ < !1 º is a club in !1 . Thus for each B  !1 such that B 2 M!1 and B … j0;!1 .I / there exists k < !;  < !1 such that .C n  C 1/ \ Ak;  B \ Ak; : Further if B  !1 , B 2 M!1 and B 2 j0;!1 .I / then B \ C D ;. Thus J \ M!1 D I!1 .

t u

Lemma 4.37 is the analog of Lemma 4.36 for iterable sequences. The proof is a straightforward modification of the proof of Lemma 4.36. Lemma 4.37 (ZFC ). Suppose h.Nk ; Jk / W k < !i is an iterable sequence such that Nk  ZFC for each k < !. Suppose J is a normal uniform ideal on !1 . Then there exists an iteration j W h.Nk ; Jk / W k < !i ! h.Nk ; Jk / W k < !i such that: (1) j.!1N0 / D !1 ; (2) J \ Nk D Jk for each k < !.

t u

We analyze the conditions in Pmax in a variety of circumstances. The partial order Pmax is nontrivial under fairly mild assumptions. Lemma 4.38. Assume that for every real x, x  exists. Then for each x 2 R the set of h.M; I /; ai 2 Pmax such that x 2 M is dense in Pmax . Proof. Suppose x 2 R and h.M0 ; I0 /; a0 i 2 Pmax . Let y 2 R code the pair .x; h.M0 ; I0 /; a0 i/ so that x 2 LŒy, h.M0 ; I0 /; a0 i 2 LŒy and h.M0 ; I0 /; a0 i is countable in LŒy. y  exists and so there is a transitive inner model N and countable ordinals ı < such that y 2 N , N contains the ordinals, N  ZFC C GCH;

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is inaccessible in N , and such that ı is a measurable cardinal in N . Let Q 2 N be a ı-cc poset in N such that; (1.1) N Q  MA C :CH, (1.2) N Q  ı D !1 , (1.3) Q has cardinality < in N . Let J 2 N be an ideal dual to a normal measure on ı in N . Let G  Q be N -generic and let JG be the ideal generated by J in N ŒG. Thus JG is a normal uniform ideal on ı in N ŒG. By .Jech and Mitchell 1983/ JG is a precipitous ideal in N ŒG. Thus by Lemma 4.5, any iteration of .N ŒG; JG / is wellfounded and so by Lemma 4.4, .N ŒG; JG / is iterable. Let j W .M0 ; I0 / ! .M0 ; I0 / be an iteration of .M0 ; I0 / such that j 2 N ŒG and such that I0 D JG \ M0 . Let b D j.a0 /. Thus h.N ŒG; JG /; bi 2 Pmax : Finally x 2 N ŒG and h.N ŒG; JG /; bi < h.M0 ; I0 /; a0 i.

t u

Remark 4.39. Assuming that for every real x, x  exists, it follows that the set of conditions h.M; I /; ai 2 Pmax for which M  ZFC is dense in Pmax . Thus in the definition of Pmax the fragment of ZFC used is not really relevant provided it is strong enough. t u For the analysis of Pmax we need a much stronger existence theorem for conditions. Lemma 4.40. Assume AD holds in L.R/. Suppose that X  R and that X 2 L.R/. Then there is a condition h.M; I /; ai 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; I / is X -iterable, and further the set of such conditions is dense in Pmax . Proof. We work in L.R/. Suppose that for some X  R with X 2 L.R/ no such condition h.M; I /; ai 2 Pmax exists. Then by standard reflection arguments in L.R/ we may assume that X is  21 definable in L.R/. By the Martin-Steel theorem, Theorem 2.3, in L.R/ the pointclass 21 definable in L.R/ †21 has the scale property. Thus any set X  R  R which is  is Suslin in L.R/ and so can be uniformized by a function which is  21 definable in

4.2 The partial order Pmax

141

L.R/. Let F W R ! R be a function such that if N is a transitive model of ZF closed under F then hH.!1 /N ; X \ N; 2i  hH.!1 /; X; 2i: Let Y  R be the set of reals which code elements of F  X . Since X is  21 it follows 2 2  that F may be chosen such that F is  1 in which case Y is  1 . Let T; T be trees  such that Y D pŒT  and R n Y D pŒT . Note that if N is any transitive model of ZF with T 2 N then N is closed under F . Since AD holds, there exists a transitive inner model N of ZFC, containing the ordinals such that T 2 N , T  2 N and such that  is a measurable cardinal in N for some countable ordinal  . !1 is strongly inaccessible in N and so by passing to a generic extension of N if necessary we can require that the GCH holds in N at  . Let Q 2 N be a  -cc poset in N such that; (1.1) N Q  MA C :CH, (1.2) N Q   D !1 , (1.3) jQj D  C in N . Let G  Q be N -generic. Let I 2 N be a normal ideal on  which is dual to a normal measure on  . Let IG be the normal ideal generated by I in N ŒG. Thus in N ŒG, IG is a precipitous ideal on !1N ŒG . Let ı < !1 be an inaccessible cardinal in N ŒG. Thus by Lemma 4.4 and Lemma 4.5, it follows that .Nı ŒG; IG / is iterable. Since T 2 N ŒG it follows that hH.!1 /N ŒG ; X \ N ŒG; 2i  hH.!1 /; X; 2i: We claim that .Nı ŒG; IG / is X -iterable. Suppose j W Nı ŒG ! M is an iteration of .Nı ŒG; IG /. Then by Lemma 4.4, there corresponds an iteration j  W N ŒG ! M  of .N ŒG; IG / and an elementary embedding k W M ! j  .Nı ŒG/ such that k ı j D j  jNı ŒG. (In fact in our situation M D j  .Nı ŒG/ and k is the identity.) Let YN ŒG D pŒT  \ N ŒG. Thus j  .YN ŒG / D pŒj  .T / \ M  : However (2.1) pŒT   pŒj  .T /, (2.2) pŒT    pŒj  .T  /.

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4 The Pmax -extension

Further by absoluteness pŒj  .T / \ pŒj  .T  / D ; and so pŒT  D pŒj  .T / and pŒT   D pŒj  .T  /. Thus j  .YN ŒG / D Y \ M  and so j.YN ŒG / D Y \ M . Therefore j.X \ N ŒG/ D X \ M: This proves that .Nı ŒG; IG / is X -iterable. Let a 2 Nı ŒG be such that Nı ŒG  a  !1

and

!1 D !1LŒa :

h.Nı ŒG; IG /; ai is the desired condition. The density of these conditions follows abstractly. Let h.M; I /; ai 2 Pmax . Let z 2 R code h.M; I /; ai. Choose a condition h.N ; J /; bi 2 Pmax such that; (3.1) Y \ N 2 N , (3.2) hH.!1 /N ; Y \ N i  hH.!1 /; Y i, (3.3) .N ; J / is Y -iterable, where Y is the set of reals which code elements of X  ¹zº. By Lemma 4.36, there exists an iteration j W .M; I / ! .M ; I  / such that j 2 N and I  D J \ M . Let a D j.a/. Thus h.N ; J /; a i 2 Pmax and h.N ; J /; a i < h.M; I /; ai. h.N ; J /; a i is the required condition. t u The entire analysis of Pmax that we give can be carried out abstractly just assuming the following:  For each set X  R with X 2 L.R/, there is a condition h.M; I /; ai 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; I / is X -iterable. This in turn is equivalent to:  For each set X  R with X 2 L.R/, there exists M 2 H.!1 / such that (1) M is transitive, (2) M  ZFC , (3) X \ M 2 M, (4) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (5) .M; I / is X -iterable.

4.2 The partial order Pmax

143

This includes the proof that the nonstationary ideal on !1 is saturated in L.R/Pmax . However we shall see in Chapter 5 that this assumption implies ADL.R/ . This property for a set of reals, X , is really a regularity property which can be established from a variety of different assumptions. For example, it can be established quite easily from just the assumption that every set of reals which is projective in X is weakly homogeneously Suslin. Theorem 4.41. Suppose X  R and that every set of reals which is projective in X is weakly homogeneously Suslin. Then there is a condition h.M; I /; ai 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; I / is X -iterable. Proof. Note that since there are nontrivial weakly homogeneously Suslin sets there must exist a measurable cardinal. Let ı be the least measurable cardinal and let I be a normal uniform ideal on ı such that I is maximal; i. e. the dual filter is a normal measure. By collapsing 2ı to ı C if necessary we can assume that 2ı D ı C . The generic collapse of 2ı to ı C preserves the hypothesis of the theorem and it adds no new reals to V . X is weakly homogeneously Suslin and so there exists a weakly homogeneous tree S such that X D pŒS . The tree S is necessarily ı-weakly homogeneous. Let S  be a weakly homogeneous tree such that pŒS   D R n X . Again S  is necessarily ı-weakly homogeneous and so if G  P is V -generic where P is a partial order of size less than ı then in V ŒG, pŒS  D R n pŒS  . Let Y be the set of reals which code elements of the first order diagram of hH.!1 /; X; 2i: Y is weakly homogeneously Suslin since it is a countable union of weakly homogeneously Suslin sets. Similarly R n Y is also weakly homogeneous Suslin since it too is the countable union of weakly homogeneously Suslin sets. Therefore there exist weakly homogeneous trees T and T  such that pŒT  D Y 

and such that pŒT  D R n Y . The trees T and T  are each necessarily ı-weakly homogeneous. Thus if G  P is V -generic where P is a partial order of size less than ı, then in V ŒG, pŒT  D R n pŒT  . A key point is that ı is measurable and so this also holds if P is a partial order which is ı-cc.

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4 The Pmax -extension

Let > ı C be a regular cardinal such that ¹S; T; S  ; T  º  H. /. Thus H. / is admissible and if Q 2 H. / is any partial order of cardinality at most ı C then H. /ŒG  ZFC and H. /ŒG is admissible, whenever G  Q is V -generic. Let Z  H. / be a countable elementary substructure such that ¹S; T; S  ; T  ; I º  Z.  ; IN be the images of Let N be the transitive collapse of Z and let ıN ; SN ; SN  ı; S; S ; I under the collapsing map. Let Q 2 N be a ıN -cc poset in N such that; (1.1) N Q  MA C :CH, (1.2) N Q  ıN D !1 , C . (1.3) N  jQj D ıN

Let G  Q be N -generic and let J be the normal ideal in N ŒG generated by IN . Note that pŒSN  \ N ŒG 2 N ŒG since N ŒG is admissible. Suppose that j W .N ŒG; J / ! .N  ŒG  ; J  / is an iteration of countable length. Then it follows that j W .N; IN / ! .N  ; j.IN // is an iteration. But IN 2 N is the ideal dual to a normal measure in N on ıN and so this is an iteration in the usual sense. Let  W N ! Z be the inverse of the collapsing map. Thus by standard arguments there exists Z   H. / such that Z  Z  , N  is the transitive collapse of Z  and   ı j jN D  where   W N  ! Z  is the inverse of the collapsing map.  /  pŒS  . Hence Thus pŒj.SN /  pŒS . Similarly pŒj.SN   pŒj.SN / \ N ŒG  D X \ N  ŒG   and so j.X \ N ŒG/ D X \ N  ŒG  . This proves that X \ N ŒG 2 N ŒG and that .N ŒG; J / is X -iterable. It remains to show that hH.!1 /N ŒG ; X \ N ŒG; 2i  hH.!1 /; X; 2i: A key point is the following. Suppose G  P is V -generic where P is a partial order of size less than ı. Then in V ŒG, pŒT  codes the diagram of hH.!1 /; pŒS ; 2i. Again ı is measurable and so if G  P is V -generic where P is a partial order which is ı-cc, then in V ŒG, pŒT  codes the diagram of hH.!1 /; pŒS ; 2i. By elementarity and the remarks above it follows that pŒTN  \ N ŒG codes the diagram of hH.!1 /N ŒG ; N ŒG \ pŒSN ; 2i. Thus Y \ N ŒG codes the diagram of hH.!1 /N ŒG ; N ŒG \ X; 2i and so t u hH.!1 /N ŒG ; X \ N ŒG; 2i  hH.!1 /; X; 2i:

4.2 The partial order Pmax

145

Remark 4.42. The requirement hH.!1 /M ; X \ Mi  hH.!1 /; X i is important in the analysis of the Pmax -extension. It is also more difficult to achieve. For example if there is a measurable cardinal and if X  R is universally Baire then there exists .M; I / which is X -iterable. The proof is identical to that of Theorem 4.41. We do not know if from these assumptions one can find an X -iterable structure .M; I / for which hH.!1 /M ; X \ Mi  hH.!1 /; X i even if one adds the assumption that every set of reals which is projective in X is universally Baire. The notion that a set of reals is universally Baire is defined in .Feng, Magidor, and Woodin 1992/. It has a simple reformulation in terms of Suslin representations which is all that is relevant here: If X is universally Baire then for any partial order P there exist trees T; T  such that X D pŒT  and such that in V P , pŒT  D R n pŒT  . Universally Baire sets are briefly discussed in Section 10.3. t u As a corollary to Lemma 4.37 we easily establish that under suitable hypotheses, the partial order Pmax is !-closed and homogeneous. Lemma 4.43. Assume Pmax ¤ ; and that for each x 2 R the set of h.M; I /; ai 2 Pmax such that x 2 M is dense in Pmax . Then Pmax is !-closed and homogeneous. Proof. We first prove that Pmax is !-closed. Suppose that hpk W k < !i is a descending sequence of conditions in Pmax and that for each k < !, pk D h.Mk ; Ik /; ak i: Let b D [¹ak j k < !º. For each k < ! there is a unique iteration jk W .Mk ; Ik / ! .Nk ; Jk / such that jk .ak / D b. We summarize the properties of the sequence h.Nk ; Jk / W k < !i: (1.1) Nk  ZFC ; (1.2) Jk 2 Nk and Nk  “Jk is a normal uniform ideal on !1 ”I (1.3) .Nk ; Jk / is iterable; (1.4) Nk 2 NkC1 ; (1.5) jNk j D !1 in NkC1 ;

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4 The Pmax -extension

(1.6) 5A is of measure 1 for JkC1 whenever A 2 Nk , N

A  P .!1 k / \ Nk n Jk ; and A is dense; (1.7) JkC1 \ Nk D Jk ; (1.8) if C 2 Nk is closed and unbounded in !1N0 then there exists D 2 NkC1 such that D  C , D is closed and unbounded in C and such that D 2 LŒx for some x 2 R \ NkC1 . These properties are straightforward to verify, (1.6) follows from Lemma 4.10 and (1.8) follows from Lemma 4.6. By Corollary 4.20, the sequence h.Nk ; Jk / W k < !i is iterable. Let z be a real which codes h.Nk ; Jk / W k < !i. Thus there is a condition h.M; I /; ai 2 Pmax such that z 2 M. By Lemma 4.37, there is an iteration j W h.Nk ; Jk / W k < !i ! h.Nk ; Jk / W k < !i such that: (2.1) j 2 M; (2.2) j.!1N0 / D !1M ; (2.3) I \ Nk D Jk for each k < !. Let a D j.b/. Thus h.M; I /; a i 2 Pmax and h.M; I /; a i < h.Mk ; Ik /; ak i for all k < !. This shows that Pmax is !-closed. We finish by showing that Pmax is homogeneous. Suppose h.M0 ; I0 /; a0 i and h.M1 ; I1 /; a1 i are conditions in Pmax . Let z be a real which codes the pair of these conditions. Suppose h.M; I /; ai is a condition in Pmax such that z 2 M. Thus there are iterations j0 W .M0 ; I0 / ! .M0 ; I0 / and j1 W .M1 ; I1 / ! .M1 ; I1 / such that: (3.1) j0 2 M and j1 2 M; (3.2) j0 .!1M0 / D !1M D j1 .!1M1 /; (3.3) I \ M0 D I0 and I \ M1 D I1 .

4.2 The partial order Pmax

147

Let a0 D j0 .a0 / and let a1 D j1 .a1 /. The key point is the following. Suppose that h.N ; J /; bi 2 Pmax and h.N ; J /; bi < h.M; I /; ai. Let j W .M; I / ! .M ; I  / be the unique iteration such that j.a/ D b. Then h.N ; J /; j.a0 /i < h.M0 ; I0 /; a0 i and

h.N ; J /; j.a1 /i < h.M1 ; I1 /; a1 i:

Thus the conditions below h.M; I /; ai have canonical interpretations as conditions below h.M0 ; I0 /; a0 i and as conditions below h.M1 ; I1 /; a1 i. These interpretations are unique given j0 and j1 . Now suppose that G  Pmax is L.R/-generic. Then by genericity there exists a condition h.M; I /; ai 2 G such that z 2 M where z is a real coding both the conditions h.M0 ; I0 /; a0 i and h.M1 ; I1 /; a1 i. From the arguments above it follows that we can define generics G0  Pmax and G1  Pmax such that h.M0 ; I0 /; a0 i 2 G0 , h.M1 ; I1 /; a1 i 2 G1 and such that L.R/ŒG0  D L.R/ŒG1  D L.R/ŒG: This shows that Pmax is homogeneous.

t u

Using the iteration lemmas we prove two more lemmas which we shall use to complete our initial analysis of Pmax . We begin with a definition that establishes some key notation. Definition 4.44. A filter G  Pmax is semi-generic if for all ˛ < !1 there exists a condition h.M; I /; ai 2 G such that ˛ < !1M . Suppose G  Pmax is semi-generic. Define AG  !1 by AG D [¹a j h.M; I /; ai 2 Gº: For each h.M; I /; ai 2 G let jG W .M; I / ! .M ; I  / be the embedding from the iteration which sends a to AG . Let P .!1 /G D [¹P .!1 / \ M j h.M; I /; ai 2 Gº and let

IG D [¹I  j h.M; I /; ai 2 Gº:

t u

Remark 4.45. (1) Suppose G  Pmax is a semi-generic filter. Then Pmax is somewhat nontrivial. Strictly speaking, a filter G  Pmax may be, for example, L.R/generic and not be semi-generic. We shall never consider filters in Pmax without assumptions which guarantee that Pmax is nontrivial.

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(2) The iteration jG is uniquely specified by G and the condition h.M; I /; ai 2 G. It is not in general uniquely specified by simply G and .M; I /. A more accurate notation would denote jG by jp;G where p D h.M; I /; ai. However we shall use the potentially ambiguous notation jG , letting the context arbitrate any ambiguities. t u Lemma 4.46 isolates the combinatorial fact which will be used to prove that !1 DC holds in L.R/Pmax . This lemma will be applied within models occurring in Pmax conditions and so the lemma is proved assuming only ZFC . Lemma 4.46 (ZFC ). Assume Pmax ¤ ; and that for each x 2 R, the set of h.M; I /; ai 2 Pmax such that x 2 M is dense in Pmax . Suppose J is a normal uniform ideal on !1 and that Y  H.!1 / is a .nonempty/ set of pairs .p; f / such that: (i) p 2 Pmax ; (ii) for some ˛ < !1 , f 2 ¹0; 1º˛ . Suppose that for all p 2 Pmax , .p; ;/ 2 Y , and suppose that Y satisfies the following closure conditions. (iii) Suppose .p; f / 2 Y and q < p. Then .q; f / 2 Y . (iv) Suppose .p; f / 2 Y and ˛ < dom.f /. Then .p; f j˛/ 2 Y . (v) Suppose .p; f / 2 Y and ˛ < !1 . Then there exists .q; g/ 2 Y such that q < p, f  g and such that ˛ < dom.g/. (vi) Suppose p 2 Pmax , ˛ < !1 , ˛ is a limit ordinal and f W ˛ ! ¹0; 1º: Then either .p; f / 2 Y or .p; f jˇ/ … Y for some ˇ < ˛. Then for each q0 2 Pmax there is a semi-generic filter G  Pmax and a function f W !1 ! ¹0; 1º such that q0 2 G, IG D J \ P .!1 /G and such that for all ˛ < !1 , .p; f jˇ/ 2 Y for some p 2 G and for some ˇ > ˛.

4.2 The partial order Pmax

149

Proof. Let h.p˛ ; f˛ / W ˛ < !1 i be a sequence such that for all ˛ < ˇ < !1 (1.1) p0 < q0 , (1.2) .p˛ ; f˛ / 2 Y , (1.3) pˇ < p˛ , (1.4) f˛  fˇ , (1.5) ˛  dom.f˛ /, (1.6) J \ M˛ D I˛ , where .M˛ ; I˛ / is defined as follows. Let h.M˛ ; I˛ /; a˛ i D p˛ . Let a D [¹a˛ j ˛ < !1 º: Then for each ˛ there exists a unique iteration j˛ W .M˛ ; I˛ / ! .M˛ ; I˛ / such that j˛ .a˛ / D a . This sequence is easily constructed using the properties of Y and the proof of Lemma 4.36. Let G be the filter generated by ¹p˛ j ˛ < !1 º and let f D [¹f˛ j ˛ < !1 º: Thus G is a semi-generic filter and .G; f / has the desired properties.

t u

The next lemma is simply the formulation of Lemma 4.10 for the special case we are presently interested in. This is the case for structures of the form .M; I /; i. e. when only one ideal is designated. Lemma 4.47 (ZFC ). Suppose .M; I / is a countable transitive model where I 2M is a normal uniform ideal on !1M and M  ZFC . Suppose that j W .M; I / ! .M ; I  / 

is a wellfounded iteration of length !1 and that A  P .!1 /M n I  is a maximal antichain with A 2 M  . Let hA˛ W ˛ < !1 i be an enumeration of A in V . Then 5¹A˛ j ˛ < !1 º contains a club in !1 .

t u

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4 The Pmax -extension

Lemma 4.48 (ZFC ). Assume Pmax ¤ ; and that for each x 2 R, the set of h.M; I /; ai 2 Pmax such that x 2 M is dense in Pmax . Suppose J is a normal uniform ideal on !1 and that Y  H.!1 / is a .nonempty/ set of pairs .p; b/ such that: (i) p 2 Pmax ; (ii) b  !1M , b 2 M, and b … I ; where p D h.M; I /; ai. (iii) Suppose .h.M0 ; I0 /; a0 i; b0 / 2 Y and h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i. Then .h.M1 ; I1 /; a1 i; b1 / 2 Y where b1 is the image of b0 under the iteration of .M0 ; I0 / which sends a0 to a1 . (iv) Suppose h.M0 ; I0 /; a0 i 2 Pmax , b0 2 M0 , b0  !1M0 and b0 … I0 . Then there exists .h.M1 ; I1 /; a1 i; b1 / 2 Y such that h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i and such that b1  j.b0 / where j is the embedding given by the iteration of .M0 ; I0 / which sends a0 to a1 . Then for each p0 2 Pmax there exists a semi-generic filter G  Pmax such that p0 2 G, J \ P .!1 /G D IG ; jP .!1 /G j D !1 ; and such that

!1 n 5A 2 J

where A is the set of j.b/ such that .h.M; I /; ai; b/ 2 Y , h.M; I /; ai 2 G, and j W .M; I / ! .M ; I  / is the embedding given by the iteration of .M; I / which sends a to AG . Proof. Let S  ¹˛ < !1 j ˛ is a limit ordinalº and fix a partition hS˛ W ˛ < !1 i of S into disjoint sets such that S D 5¹S˛ j ˛ < !1 º and such that S˛ … J for each ˛ < !1 . For any uniform normal ideal such a partition exists. We construct a sequence h.q˛ ; b˛ / W ˛ < !1 i of elements of Y such that for all ˛ < ˇ < !1 , qˇ < q˛ < p0 and such that; (1.1) for each ˛ < !1 there is a club C  !1 such that S˛ \ C  j˛ .b˛ /, (1.2) for each ˛ < !1 and for each d 2 P .!1M˛ / \ M˛ with d … I˛ there exists ˇ < !1 such that ˛ < ˇ and bˇ  j˛;ˇ .d /,

4.2 The partial order Pmax

151

where for each ˛ < !1 , h.M˛ ; I˛ /; a˛ i D q˛ , j˛ W .M˛ ; I˛ / ! .M˛ ; I˛ / is the embedding given by the iteration which sends a˛ to [¹aˇ j ˇ < !1 º and where for all ˛ < ˇ < !1 j˛;ˇ W .M˛ ; I˛ / ! .M˛ˇ ; I˛ˇ / is the embedding from the iteration which sends a˛ to aˇ . We construct the sequence h.q˛ ; b˛ / W ˛ < !1 i and at the same time a sequence hd˛ W ˛ < !1 i by induction on ˛ where for each ˛ < !1 , d˛ 2 P .!1M˛ / \ M˛ n I˛ : Suppose h.q˛ ; b˛ / W ˛ <  i and hd˛ W ˛ <  i have been constructed. If  D ˇ C 1 then choose .h.M; I /; ai; b/ 2 Y such that h.M; I /; ai < qˇ and such that b  j.dˇ / where j W .Mˇ ; Iˇ / ! .MO ˇ ; IOˇ / is the iteration such that j.aˇ / D a. By (iv), .h.M; I /; ai; b/ 2 Y exists. Let .q ; b / D .h.M; I /; ai; b/ and let d 2 P .!1M / \ M n I . Now suppose that  is a limit ordinal and let hk W k < !i be an increasing cofinal sequence of ordinals less than  . For each k < ! let .Nk ; Jk / be the iterate of .Mk ; Ik / defined by the iteration which sends ak to [¹aˇ j ˇ <  º. Thus h.Nk ; Jk / W k < !i satisfies the conditions for Corollary 4.20 and so it is an iterable sequence. This is just as in the proof that Pmax is !-closed. Thus  2 5¹Sˇ j ˇ <  º. Let ˇ <  be such that  2 Sˇ . Let h.Nk ; Jk / W k < !i be the generic ultrapower of h.Nk ; Jk / W k < !i by a [¹Nk j k < !º-generic ultrafilter which contains j.bˇ / where j is the embedding from the iteration of .Mˇ ; Iˇ / which sends aˇ to [¹aˇ j ˇ <  º. Let a be the image of [¹aˇ j ˇ <  º under this iteration. Let x be a real which codes h.Nk ; Jk / W k < !i and choose h.M; I /; ai 2 Pmax such that x 2 M. The condition exists since we have assumed that for every real t , t  exists. h.Nk ; Jk / W k < !i is an iterable sequence and so by Lemma 4.37, there exists an iteration j W h.Nk ; Jk / W k < !i ! h.Nk ; Jk / W k < !i 

in M such that j.!1N / D !1M and such that for all k < !, I \ Nk D Jk . Let a D j.a /. Thus h.M; I /; a i 2 Pmax and for all ˛ < 

h.M; I /; a i < q˛ < p0 :

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4 The Pmax -extension

Thus by property (iii) of Y there exists .q; b/ 2 Y such that q < h.M; I /; a i. Let .q ; b / D .q; b/ and let d 2 P .!1M / \ M n I . This completes the construction of the sequences. Notice that we have complete freedom in the choice of d at each stage  . Let G  Pmax be the filter generated by ¹q˛ j ˛ < !1 º. We may assume by a routine book-keeping argument that ¹j˛ .d˛ / j ˛ < !1 º D [¹M˛ \ P .!1 / j ˛ < !1 º n IG D P .!1 /G n IG : We claim that G is the desired semi-generic filter. G is generated by a subset of size !1 and so it follows that jAG j D !1 . All that needs to be verified is that 5AG is of measure 1 relative to J and that IG D J \ P .!1 /G . For each ˛ < !1 there is a club C˛  !1 such that C˛ \ S˛  j˛ .b˛ /. Further by definition j˛ .b˛ / 2 AG and so since S D 5¹S˛ j ˛ < !1 º it follows that there is a club C  !1 such that S \ C  5AG , take C D 4¹C˛ j ˛ < !1 º. However S is of measure 1 relative to J and J is a uniform normal ideal. Hence C \ S is of measure 1 relative to J . By the choice of hd˛ W ˛ < !1 i it follows that ¹j˛ .d˛ / j ˛ < !1 º D P .!1 /G n IG :  .b˛C1 /  j˛ .d˛ /. Therefore every set in However for each ˛ < !1 , j˛C1 P .!1 /G n IG is positive relative to J . Further every set in IG is nonstationary and so IG D J \ P .!1 /G :

t u

The lemma follows.

Suppose G  Pmax is L.R/-generic. We assume also that for all reals x, x  exists so that Pmax is nontrivial. Thus the filter G is semi-generic and so we have defined AG  !1 , P .!1 /G  P .!1 /, and IG  P .!1 /G . The next theorem gives the basic analysis of Pmax . Theorem 4.49. Suppose that for each set X  R with X 2 L.R/, there is a condition h.M; I /; ai 2 Pmax such that (i) X \ M 2 M, (ii) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (iii) .M; I / is X -iterable. Suppose G  Pmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal saturated ideal; (3) IG is the nonstationary ideal.

4.2 The partial order Pmax

153

Proof. We claim that for each set X  R with X 2 L.R/ the set of such conditions in Pmax which satisfy (i)–(iii) is dense in Pmax . The point here is that given X and a condition h.M0 ; I0 /; a0 i 2 Pmax define a new set X   R as follows. Fix a real z which codes h.M0 ; I0 /; a0 i and define X  to be the set of reals which code a pair .z; t / where t 2 X . We assume X is nonempty. Thus X  2 L.R/ and so there is a condition h.M; I /; ai 2 Pmax such that (1.1) X  \ M 2 M, (1.2) hH.!1 /M ; X  \ Mi  hH.!1 /; X  i, (1.3) .M; I / is X  -iterable. By (1.2) it follows that z 2 M. Thus by Lemma 4.36, there is an iteration j W .M0 ; I0 / ! .M0 ; I0 / in M such that I  D I \ M0 . Thus h.M; I /; j.a0 /i 2 Pmax and h.M; I /; j.a0 /i < h.M0 ; I0 /; a0 i: h.M; I /; j.a0 /i is the desired condition. Therefore by Lemma 4.43, Pmax is !-closed and homogeneous. We first prove that if G  Pmax is L.R/-generic then !1 -DC holds in L.R/ŒG. Since Pmax is !-closed it follows that DC holds in L.R/ŒG. Every set in L.R/ŒG is definable from an ordinal, a real and G. Therefore to establish that !1 -DC holds in L.R/ŒG it suffices to show that if T  ¹0; 1º
154

4 The Pmax -extension

We may assume that M contains a real coding h.M0 ; I0 /; a0 i by the remarks above. The following closure properties of Y can be expressed as first order statements in the structure hH.!1 /; X; 2i. (4.1) Suppose .p; f / 2 Y and q < p. Then .q; f / 2 Y . (4.2) Suppose .p; f / 2 Y and ˛ < dom.f /. Then .p; f j˛/ 2 Y . (4.3) Suppose .p; f / 2 Y and ˛ < !1 . Then there exists .q; g/ 2 Y such that q < p, f  g and such that ˛ < dom.g/. (4.4) Suppose p 2 Pmax , ˛ < !1 , ˛ is a limit ordinal and f W ˛ ! ¹0; 1º: Then either .p; f / 2 Y or .p; f jˇ/ … Y for some ˇ < ˛. Since it follows that

hH.!1 /M ; X \ Mi  hH.!1 /; X i hH.!1 /M ; Y \ H.!1 /M ; 2i  hH.!1 /; Y; 2i:

Further from this it follows that for all x 2M\R there exists h.N ; J /; bi 2 M \ Pmax such that x 2 N and N is countable in M. We can now apply Lemma 4.46 in M to obtain .g; f / 2 M such that the following hold in M. (5.1) f W !1 ! ¹0; 1º. (5.2) g  Pmax and g is a semi-generic filter. (5.3) h.M0 ; I0 /; a0 i 2 g. (5.4) Ig D I \ P .!1 /g . (5.5) For all ˛ < !1 ,

.p; f jˇ/ 2 Y

for some p 2 g and for some ˇ > ˛. Let ag  !1M be the set determined by g. Thus for all p 2 g, h.M; I /; ag i < p: Now suppose that G  Pmax is L.R/-generic and that h.M; I /; ag i 2 G. There exists a unique iteration j W .M; I / ! .M ; I  / such that j.ag / D AG . Let F D j.f /. We claim that for all ˛ < !1 , F j˛ 2 :

4.2 The partial order Pmax

155

To see this fix ˇ < !1 . Choose h.N ; J /; bi 2 G such that h.N ; J /; bi < h.M; I /; ag i and such that ˇ < !1N . Hence there is a unique iteration k W .M; I / ! .M ; I  / such that k.ag / D b. This iteration is an initial segment of the iteration which defines j , k.f /  F , and ˇ < dom.k.f //.  .M; I / is X -iterable and so it follows that for all ˛ < !1M , .p; k.f /jˇ/ 2 Y for some p 2 k.g/ and for some ˇ > ˛. Finally for all p 2 k.g/, h.N ; J /; bi < p; and so k.g/  G. Therefore we have that for all ˛ < !1N , k.f /j˛ 2  . Thus k.f /jˇ 2  and so F jˇ 2  . This proves that !1 -DC holds in L.R/Pmax . In fact we have proved something stronger:  Suppose that G  Pmax is L.R/-generic. Suppose that T 2 L.R/ŒG, T is an !-closed subtree of ¹0; 1º
P .!1 / D [¹P .!1 /M j h.M; I /; ai 2 Gº since given B  !1 with B 2 L.R/ŒG let T be the subtree of ¹0; 1º
4 The Pmax -extension

156 where

j0 W .M0 ; I0 / ! .M0 ; I0 / and j1 W .M1 ; I1 / ! .M1 ; I1 / are the unique iterations such that j0 .a0 / D AG and j1 .a1 / D AG . Finally we prove that IG is a saturated ideal in L.R/ŒG. For this we prove the following holds in L.R/ŒG.  Suppose A  P .!1 / n INS is dense. Then there exists A  A such that A has cardinality !1 and such that 5A contains a club in !1 . We work in L.R/. Fix a term  2 L.R/Pmax and fix a condition p0 2 Pmax . We assume that 1    P .!1 / n INS and 1

is dense:

Let Y  H.!1 / be the set of all pairs .h.M; I /; ai; b/ such that, (7.1) h.M; I /; ai 2 Pmax , (7.2) b 2 M and b  !1M , (7.3) h.M; I /; ai  b  2 , where if G  Pmax is L.R/-generic and h.M; I /; ai 2 G then b  is the image of b under the iteration of .M; I / which sends a to AG . Observe that because IG is the nonstationary ideal it follows that if .h.M; I /; ai; b/ 2 Y then necessarily b … I . The following properties of Y are easily verified. (8.1) Suppose .h.M0 ; I0 /; a0 i; b0 / 2 Y and h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i. Then .h.M1 ; I1 /; a1 i; b1 / 2 Y where b1 is the image of b0 under the iteration of .M0 ; I0 / which sends a0 to a1 . (8.2) Suppose h.M0 ; I0 /; a0 i 2 Pmax , b0 2 M0 , b0  !1M0 and b0 … I0 . Then there exists .h.M1 ; I1 /; a1 i; b1 / 2 Y such that h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i and such that b1  j.b0 / where j is the embedding given by the iteration of .M0 ; I0 / which sends a0 to a1 .

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157

The second of properties, (8.2), follows from the fact that if G  Pmax is L.R/generic then in L.R/ŒG 

P .!1 / D [¹P .!1 /M j h.M; I /; ai 2 Gº which we have just proved. We introduce some additional notation. Suppose G  Pmax is a semi-generic filter. Let AG be the set of subsets of !1 given by evaluating  using G and Y , AG D ¹j.b/ j h.M; I /; ai 2 G and .h.M; I /; ai; b/ 2 Y º; where as above j W .M; I / ! .M ; I  / is the embedding from the iteration of .M; I / which sends a to AG . Let X  R be the set of reals which code elements of Y . Let h.M1 ; I1 /; a1 i 2 Pmax be a condition such that h.M1 ; I1 /; a1 i < p0 and such that: (9.1) X \ M1 2 M1 ; (9.2) hH.!1 /M1 ; X \ M1 i  hH.!1 /; X i; (9.3) .M1 ; I1 / is X -iterable. We shall obtain a condition in Pmax by modifying a1 in the condition h.M1 ; I1 /; a1 i. Let Y M1 be the set of elements of Y coded by a real in X \ M1 . Thus Y M1 2 M1 and in M1 has the properties (8.1) and (8.2) stated above for Y . Therefore we may apply Lemma 4.48 within M1 to obtain a semi-generic filter G1 such that p0 2 G1 and such that I1 \ P .!1 /G1 D IG1 and such that

!1 n 5AG1 2 I1

where AG1 is the set of subsets of !1 given by evaluating  using G1 and using Y \ H.!1 /M : Let

a10 D A

where A is AG as computed in M1 relative to the filter G1 . By absoluteness .Pmax /M1 D Pmax \ H.!1 /M1 : Further and so

a10 D [¹b j h.N ; J /; bi 2 G1 º h.M1 ; I1 /; a10 i < h.N ; J /; bi

for all h.N ; J /; bi 2 G1 . Note that since p0 2 G1 , h.M1 ; I1 /; a10 i < p0 :

158

4 The Pmax -extension

Now suppose that G  Pmax is L.R/-generic and that h.M1 ; I1 /; a10 i 2 G. Let j W .M1 ; I1 / ! .M1 ; I1 / be the embedding from the iteration which sends a10 to AG . The key point is that since .M1 ; I1 / is X -iterable it follows that j.Y M1 / D Y \ M1 : Further suppose h.M; I /; ai < h.M1 ; I1 /; a10 i and let k W .M1 ; I1 / ! .M1 ; I1 / be the countable iteration of .M1 ; I1 / which sends a10 to a. By the properties of G1 in M1 it follows that h.M; I /; ai < p for all p 2 k.G1 /. From these facts it follows that j.AG1 /  AG : However 5AG1 is of measure 1 in M1 relative to I1 . Therefore 5j.AG1 / is of measure 1 relative to IG and so it contains a club in !1 since IG is the nonstationary ideal in L.R/ŒG. Finally AG1 is of cardinality !1 in M1 and so j.AG1 / has cardinality !1 in L.R/ŒG. Thus j.AG1 / is the desired subset of AG . This proves that IG is a saturated ideal in L.R/ŒG and this completes the proof of the theorem. t u Combining Lemma 4.40 and Theorem 4.49 we obtain as an immediate corollary the following theorem. Theorem 4.50. Assume AD holds in L.R/. Suppose G  Pmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal saturated ideal; (3) IG is the nonstationary ideal.

t u

We continue with our analysis of L.R/Pmax and prove that the conclusion of Corollary 3.48 holds in L.R/Pmax . This theorem can also be proved abstractly by using Corollary 3.48 together with the absoluteness theorem, Theorem 4.64. But a proof along these lines requires stronger hypotheses. Remark 4.51. (1) It is possible to prove that INS is saturated in L.R/Pmax using Lemma 4.52 instead of Lemma 4.48, see the proof of Theorem 10.54. (2) There are Pmax -variations, P 2 L.R/, for which INS is not saturated in L.R/P . However Lemma 4.52 will generalize to these models, yielding the semi-satut u ration of INS in these models, see Section 6.1.

4.2 The partial order Pmax

159

Lemma 4.52. Assume ADL.R/ and suppose that G  Pmax is L.R/-generic. Then in L.R/ŒG, for every set A 2 P .R/ \ L.R/ the set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Proof. Suppose G  Pmax is L.R/-generic. From the basic analysis of Pmax summarized in Theorem 4.49 it follows that H.!2 /L.R/ŒG D H.!2 /L.R/ ŒAG : We work in L.R/ŒG. Fix A  R with A 2 L.R/. Fix a countable elementary substructure X  hH.!2 /; A; G; 2i: Let hXi W i < !i be an increasing sequence of countable elementary substructures of X such that X D [¹Xk j k < !º and such that for each k 2 !, Xk 2 XkC1 . Therefore for each k < !, there exists h.M; I /; ai 2 G \ XkC1 satisfying (1.1) Xk \ P .!1 /  M , (1.2) A \ M 2 M, (1.3) .M; I / is A-iterable, where M is the iterate of M given by the iteration of .M; I / which sends a to AG . Let MX be the transitive collapse of X . We claim that MX is A-iterable. Given this the lemma follows. For each k < ! let h.Mk ; Ik /; ak i 2 G \ XkC1 be a condition satisfying the requirements (1.1), (1.2) and (1.3). For each k < ! let k h.Mk˛ ; Ik˛ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i k be the iteration of .Mk ; Ik / such that j0;! .ak / D AG . Thus for each k < !, 1 k h.Mk˛ ; Ik˛ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1X i 2 MX

and further

MX D [¹Mk j k < !º;

where  D X \ !1 . Suppose j W MX ! N is given by a countable iteration of MX . Let  D j.!1MX /: For each k < ! let

.Nk ; Jk / D j..Mk ; Ik //

160

4 The Pmax -extension

where  D !1MX D !1X . Therefore for each k < !, .Nk ; Jk / is an iterate of .Mk ; Ik / by an iteration of length  which extends the iteration k h.Mk˛ ; Ik˛ /; G˛ ; j˛;ˇ W ˛ < ˇ  i:

For each k < ! this is the (unique) iteration of .Mk ; Ik / which sends ak to j.AG \ !1MX /. By induction on  , N D [¹Nk j k < !º and so MX is iterable. The argument here is identical to proof that Pmax is !-closed, cf. Lemma 4.43. We finish by analyzing AQ D [¹j.B/ j B  A and B 2 MX º: We must show that AQ D A \ N . Let  D !1MX . Thus MX D [¹Mk j k < !º. For each k < !, .Mk ; Ik / is A-iterable. Therefore k A \ MX D [¹j0; .A \ Mk / j k < !º:

For each k < ! let AQk be the image of A \ Mk under the iteration of .Mk ; Ik / which sends ak to j.AG \ /. This is the iteration which defines Nk . Thus AQ D [¹AQk j k < !º since for all B 2 MX , B  A if and only if B  Mk \ A for some k < ! and since for all k < !, k Mk \ A D j0; .A \ Mk /:

The latter equality holds since .Mk ; Ik / is A-iterable. Finally, using the A-iterability of .Mk ; Ik / once more, it follows that for each k < !, AQk D A \ Nk ; and so AQ D A \ N . t u We obtain as a corollary the following theorem. Theorem 4.53. Assume ADL.R/ and suppose G  Pmax is L.R/-generic. Then in L.R/ŒG the following hold. (1) ı12 D !2 . (2) Suppose S  !1 is stationary and f W S ! Ord: Then there exists g 2 L.R/ such that ¹˛ 2 S j f .˛/ D g.˛/º is stationary.

4.2 The partial order Pmax

161

Proof. (1) is an immediate corollary to Lemma 4.52 and Theorem 3.19. (1) also follows from (2). We prove (2). Let ‚ D ‚L.R/ and suppose f W S ! Ord where S is a stationary subset of !1 . By the chain condition satisfied by Pmax in L.R/, there exists a set X  Ord such that X 2 L.R/, f ŒS   X and such that jX j < ‚ in L.R/. Therefore we may suppose that f WS !

for some < ‚. (2) now follows from Lemma 4.52 and Theorem 3.42.

t u

We shall prove the following theorem in Section 5.1. Theorem 4.54. Assume ADL.R/ . Then L.R/Pmax  ZFC:

t u

Definition 4.55. Suppose that A  !1 . The set A is L.R/-generic for Pmax if there exists a filter G  Pmax which is L.R/-generic and such that A D AG . t u The following lemma shows that the generic for Pmax can be identified with the subset of !1 it creates. Lemma 4.56. Assume ADL.R/ . Suppose that A  !1 is L.R/-generic for Pmax . Define in L.R/ŒA a subset F  Pmax as follows. h.M; I /; ai 2 F if there exists an iteration

j W .M; I / ! .M ; I  /

such that (1) j.a/ D A, (2) I  D INS \ M . Then F is a filter in Pmax , F is L.R/-generic and A D AF . Proof. Fix a filter G  Pmax such that G is L.R/-generic and such that A D AG : Note that for each h.M; I /; ai 2 G, the corresponding iteration j W .M; I / ! .M ; I  /

162

4 The Pmax -extension

such that j.a/ D A can be computed in L.A; .M; I // and so M 2 L.R/ŒA. Therefore by Theorem 4.50 it follows that P .!1 /L.R/ŒG  L.R/ŒA: Thus the set F  Pmax is the same computed in L.R/ŒA or L.R/ŒG. Thus it suffices to show that in L.R/ŒG, F D G: By Theorem 4.50, G  F and so we need only show that F  G, i. e. that the requirement specifying membership in F fails for conditions which do not belong to G. Suppose h.M; I /; ai 2 Pmax and h.M; I /; ai … G. We prove that h.M; I /; ai … F : Let z 2 R code h.M; I /; ai. Therefore there is a condition h.N ; J /; bi 2 G such that z 2 N and such that h.M; I /; ai and h.N ; J /; bi are incompatible. First suppose there is no iteration of .M; I / which sends a to b. If there exists an iteration of .M; I / which sends a to A then it is easily verified that there must be an iteration of .M; I / which sends a to b. Therefore there is no iteration of .M; I / which sends a to A and so h.M; I /; ai … F : Therefore we may assume that there is an iteration k W .M; I / ! .M ; I  / such that k.a/ D b. The iteration k is unique and k 2 N . If I  D M \ J then h.N ; J /; bi < h.M; I /; ai which contradicts the incompatibility of these conditions. Therefore I  ¤ M \ J in particular there must exist B 2 P .!1N / \ M n I  such that B 2 J . Let j  W .N ; J / ! .N  ; J  / be the iteration such that j  .b/ D A. Thus j  .k/ W .M; I / ! .M ; I  / is the iteration of .M; I / which sends a to A. But j  .B/ 2 M n I  and j  .B/ 2 j  .J /. Therefore j  .B/ is nonstationary in L.R/ŒG since j  .J /  IG and IG is the nonstationary ideal. Thus h.M; I /; ai … F ; and this proves F D G.

t u

4.2 The partial order Pmax

163

The proof of Lemma 4.59 requires the following technical lemma. Lemma 4.57 (ZFC ). Suppose D  !1 and hyk W k < !i is a sequence of reals such that for all k < !, (1) yk# is recursive in ykC1 , (2) every subset of !1 which is constructible from yk and D contains or is disjoint from a tail of the indiscernibles of LŒykC1  below !1 . Then D is constructible from a real. Proof. For each k < ! let Ck be the set of indiscernibles of LŒyk  below !1 . First we show that if f W !1 ! !1 is a function in LŒD; yk  then there is a function h 2 LŒykC1  such that f D h on a tail of CkC1 . Fix f . For each ˛ < !1 , let ı˛ be the least element of CkC1 above ˛. Thus f .ˇ/ < ıˇ for all sufficiently large ˇ < !1 . This is because every club in !1 which is in LŒD; yk  contains a tail of CkC1 . Fix ˇ0 < !1 such that f .˛/ < ı˛ for all ˛ > ˇ0 . For each ı 2 CkC1 if ˇ0 < ı then f .ı/ is definable in LŒykC1  from ı, finitely many elements of CkC1 below ı and finitely many of the !n ’s. Working in V we can find a stationary set S  CkC1 , a finite set of ordinals t and a definable Skolem function,  , of LŒykC1  such that if ı 2 S then f .ı/ D  .ı; t /. Thus we have produced a function h W !1 ! !1 such that h 2 LŒykC1  and such that T D ¹˛ < !1 j f .˛/ D h.˛/º is stationary in !1 . Clearly T 2 LŒD; ykC1  and so T must contain a tail of CkC1 since it cannot be disjoint from a tail of CkC1 . Thus h is as desired. Let X  H.!2 / be a countable elementary substructure containing D and ¹yk j k < !º. Let Z D X \ .[¹L!2 ŒD; yk  j k < !º/: Define a †0 -elementary chain

hZ˛ W ˛ < !1 i

as follows by induction on ˛ < !1 . Set Z0 D Z and for ˛ a limit ordinal let Z˛ D [¹Zˇ j ˇ < ˛º: Define Z˛C1 D ¹f .Z˛ \ !1 / j f 2 Z˛ º: It is easily verified by induction on ˛ that for every k < !, Z˛ \ L!2 ŒD; yk   L!2 ŒD; yk :

164

4 The Pmax -extension

We prove by induction on ˛ < !1 that Z˛ \ !1 is an initial segment of !1 . This is clearly preserved at limits and so we may assume this holds for Z˛ and we prove it for Z˛C1 . Note that since Z˛ \ !1 is an ordinal it follows that it is necessarily an indiscernible of L.yk / for each k < !1 . Let ı D Z˛ \ !1 . Suppose  2 Z˛C1 \ !1 . Then  D f .ı/ for some function f W !1 ! !1 with f 2 Z˛ \ LŒD; yk  for some k < !. Fix k. Therefore from the remarks above  D h.ı/ for some function h W !1 ! !1 with h 2 LŒykC1 \Z˛ . Thus  < ı  where ı  is the next indiscernible of LŒykC1 . But every ordinal less than ı  can be generated from finitely many ordinals less ı together with ı and finitely many indiscernibles above !1 for LŒykC1  using definable Skolem functions of LŒykC1 . X contains infinitely many indiscernibles for LŒyi  above !1 for every i < ! and so  D g.ı/ for some g 2 Z˛ . Let Z  D [¹Z˛ j ˛ < !1 º. Thus !1  Z  . The key point is the following. For each ˛ < !1 let M˛ be the transitive collapse of Z˛ and let M  be the transitive collapse of Z  . For each ˛ < ˇ < !1 let j˛;ˇ W M˛ ! Mˇ be the †0 elementary embedding induced by the identity map taking Z˛ into Zˇ and let j  W M0 ! M  be the embedding induced by the identity map taking Z0 into Z  . Let ZFC denote the axioms, ZFC n Powerset. It is useful to note that M˛ is not a model of ZFC , however it is an !-length increasing union of transitive models of ZFC . Let ˛ be the image of !1 under the collapsing map of Z˛ . Then in M˛ the club filter on ˛ is a measure and j  W M0 ! M  is simply the iteration of length !1 of M0 by the club measure on 0 . This follows easily from the fact that M˛C1 is the ultrapower of M˛ by the club measure on ˛ and j˛;˛C1 is the induced embedding. This fact we verify by induction on ˛. It suffices to prove that the critical point of j˛;˛C1 is ˛ ; i. e. that for every ˛ < !1 , Z˛ \ !1 is an initial segment of !1 and this we proved above. This iteration of M0 is a non-generic analog of the iteration of a sequence of structures as defined in Definition 4.15, cf. Remark 4.16. Note that D 2 M  since !1  Z  . Let t be a real which codes M0 . Thus D 2 LŒt . t u Lemma 4.57 has the following corollary. Corollary 4.58. Assume ZF C DC and that for all x 2 R; x # exists. The following are equivalent. (1) Every subset of !1 is constructible from a real. (2) The club filter on !1 is an ultrafilter and every club in !1 contains a club which is constructible from a real. t u

4.2 The partial order Pmax

165

For example assume the nonstationary ideal on !1 is !2 -saturated, there is a measurable cardinal and there is a transitive inner model of ZF C DC containing the reals, containing the ordinals, and in which the club filter on !1 is an ultrafilter. Then in L.R/ every subset of !1 is constructible from a real. Lemma 4.59. Assume that for every real x, x  exists. Suppose h.M; I /; ai 2 Pmax ; d

!1M

and d 2 M.

(i) Let D0 be the set of h.N ; J /; bi 2 Pmax such that a) h.N ; J /; bi < h.M; I /; ai, b) N  “!1 D !1L.d

 ;x/

for some real x”.

(ii) Let D1 be the set of h.N ; J /; bi 2 Pmax such that a) h.N ; J /; bi < h.M; I /; ai, b) N  “d  is constructible from a real” . Then D0 [ D1 is open, dense in Pmax below h.M; I /; ai. Here d  denotes the image of d under the iteration of .M; I / which sends a to b. Proof. Fix a condition p 2 Pmax with p < h.M; I /; ai. There are two cases. First suppose there is a sequence h.pk ; xk / W k < !i such that for all k < !; (1.1) pk 2 Pmax and pkC1 < pk < p, (1.2) xk 2 R \ Mk and xk# is recursive in xkC1 , (1.3) x0 codes p and h.M; I /; ai, M

(1.4) every subset of !1 kC1 which belongs to LŒdkC1 ; xk  either contains or is disM joint from a tail of the indiscernibles of LŒxkC1  below !1 kC1 . where for each k < !, pk D h.Mk ; Ik /; ak i, dk D jk .d / and jk is the elementary embedding from the unique iteration of .M; I / such that jk .a/ D ak . Implicit in (1.4) M M is the fact that if A  !1 k and if A 2 Mk then every subset of !1 k which is in LŒA # belongs to L˛ ŒA where ˛ D Mk \ Ord. This is because A 2 Mk which in turn follows from the iterability of .Mk ; Jk /. We use this frequently. Choose a condition h.N ; J /; bi 2 Pmax such that for all k < !, h.N ; J /; bi < h.Mk ; Ik /; ak i: For each k < ! let

jk W .Mk ; Ik / ! .Mk ; Jk /

166

4 The Pmax -extension

be the unique iteration such that jk .ak / D b. Let d  D jk .dk /. This is unambiguously defined and we may apply Lemma 4.57 in N to obtain that there is a real t 2 N such that d  2 LŒt . The condition h.N ; J /; bi 2 D1 and h.N ; J /; bi < p. The second case is that no such sequence h.pk ; xk / W k < !i exists. Notice that if h.N1 ; J1 /; b1 i < h.N0 ; J0 /; b0 i in Pmax and if

j W .N0 ; J0 / ! .N0 ; J0 /

is the unique iteration such that j.b0 / D b1 then for every D 2 J  a tail of indiscernibles of LŒx below !1N1 is disjoint from D where x is any real in N1 which codes N0 . Therefore since the sequence h.pk ; xk / W k < !i does not exist it follows that there exist a condition h.N0 ; J0 /; b0 i < p; a real x0 2 N0 , and a set D  !1N0 , such that (2.1) D 2 LŒx0 ; d0 , (2.2) both D and !1N0 n D are positive relative to J0 , where d0 D j.d / and j W .M; I / ! .M ; I  / is the unique iteration such that j.a/ D b0 . Fix a condition h.N1 ; J1 /; b1 i < h.N0 ; J0 /; b0 i: By modifying b1 we shall produce a condition in D1 below h.N0 ; J0 /; b0 i. We work in N1 . Fix a real t which codes .N0 ; J0 /. Let C be the set C D ¹ı < !1N1 j Lı Œt   LŒt º: Therefore C is a club in !1N1 and C 2 LŒt . Let XC be the elements of C which are not limit points of C and let  W !1N1 ! XC be the enumeration function of XC . Fix A  !1N1 such that A 2 N1 and !1N1 D !1LŒA . Let A  C be the image of A under . Working in N1 construct an iteration j0 W .N0 ; J0 / ! .N0 ; J0 / of length .!1 /N1 such that; (3.1) J0 = J1 \ N0 , (3.2) j0 .D/ \ XC D A .

4.2 The partial order Pmax

167

The iteration exists because the requirements given by (2.1) and (2.2) do not interfere. One achieves (2.1) by working on C n XC as in the proof of Lemma 4.36 and (2.2) is achieved by working on XC . Let b1 D j0 .b0 /, let d1 D j0 .d0 / and let D  D j0 .D/. Thus h.N1 ; J1 /; b1 i < h.N0 ; J0 /; b0 i and  !1N1 D !1LŒt;D  since A 2 LŒt; D  . However D 2 LŒx; d0  and so D  2 LŒx; d1 . Therefore and so

h.N1 ; J1 /; b1 i

!1N1 D !1LŒx;t;d1  2 D0 .

t u

The next theorem reinforces the analogy between Pmax and Sacks forcing. Theorem 4.60. Assume AD holds in L.R/. Suppose that G  Pmax is a filter which is L.R/-generic. Suppose that A  !1 and that A 2 L.R/ŒG n L.R/. Then A is L.R/-generic for Pmax and L.R/ŒG D L.R/ŒA: Proof. This is immediate, the argument is similar to that for the homogeneity of Pmax together with the analysis provided by Theorem 4.49 and Lemma 4.59. Let G  Pmax be L.R/-generic. Fix A  !1 , A 2 L.R/ŒG n L.R/. By Theorem 4.49 there exists a condition h.M0 ; I0 /; a0 i 2 G  such that for some d 2 M0 , j .d / D A where j  W .M0 ; I0 / ! .M0 ; I0 / is the iteration which sends a to AG . By Lemma 4.59 we may assume that M0  “!1 D !1L.d;x/ for some real x”: Therefore there exists a real x 2 M0 such that !1 D !1LŒA;x : We first show that L.R/ŒG D L.R/ŒA. Since M0  MA!1 it follows, by Lemma 4.35, that there exists a real y 2 M0 with AG 2 LŒA; y: Therefore L.R/ŒA D L.R/ŒAG  D L.R/ŒG. To finish we must prove that A is L.R/-generic for Pmax . Let g  Pmax be the filter generated by ¹h.N ; J /; A \ !1N i j h.N ; J /; bi 2 G and h.N ; J /; bi < h.M0 ; I0 /; a0 iº: It follows that g is L.R/-generic and that A D [¹b j h.N ; J /; bi 2 gºI i. e. that A is the set “AG ” computed from g. t u Therefore A is L.R/-generic for Pmax .

4 The Pmax -extension

168

The next theorem is the key for actually verifying that specific …2 sentences hold in

Pmax

hH.!2 /; 2; INS iL.R/

:

Theorem 4.61. Assume AD holds in L.R/. Suppose language for the structure hH.!2 /; 2; INS i and that

Pmax

hH.!2 /; 2; INS iL.R/

.x/ is a …1 formula in the

 9x .x/:

Then there is a condition h.M0 ; I0 /; a0 i 2 Pmax and a set b0  !1M0 with b0 2 M0 such that for all h.M1 ; I1 /; a1 i 2 Pmax , if h.M1 ; I1 /; a1 i  h.M0 ; I0 /; a0 i; then

hH.!2 /M1 ; 2; I1 i 

where b1 D j.b0 / and

Œb1 

j W .M0 ; I0 / ! .M0 ; I0 /

is the iteration such that j.a0 / D a1 . Proof. Assume V D L.R/ and let G  Pmax be generic. For each h.M; I /; ai 2 G let j W .M; I / ! .M ; I  / be the iteration such that j.a/ D AG . By Theorem 4.50, in L.R/ŒG 

P .!1 / D [¹P .!1 /M j h.M; I /; ai 2 Gº and so



H.!2 /L.R/ŒG D [¹H.!2 /M j h.M; I /; ai 2 Gº: t u

The theorem now follows.

The next theorem is simply a reformulation. This theorem strongly suggests that if AD holds in L.R/ and if G  Pmax is L.R/-generic then in L.R/ŒG one should be able to analyze all subsets of P .!1 / which are definable in the structure hH.!2 /; 2; INS iL.R/ŒG by a …1 formula. Thus while a …2 sentence may fail in L.R/ŒG one can analyze completely the counterexamples.

4.2 The partial order Pmax

169

Theorem 4.62. Assume AD holds in L.R/. Suppose .x/ is a …1 formula in the language for the structure hH.!2 /; 2; INS i: Suppose G  Pmax is L.R/-generic and that hH.!2 /; 2; INS iL.R/ŒG  ŒA where A  !1 and A 2 L.R/ŒG n L.R/. Let G   Pmax be the L.R/-generic filter such that A D AG  . Then there is a condition h.M; I /; ai 2 G  such that for all h.M ; I  /; a i 2 Pmax ;    if h.M ; I /; a i  h.M; I /; ai then  hH.!2 /M ; 2; I  i  Œa : Proof. By Theorem 4.60, A is L.R/-generic for Pmax and so the generic filter G  exists. As in the proof of Theorem 4.61,   H.!2 /L.R/ŒG  D [¹H.!2 /M j h.M; I /; ai 2 G  º; where for each h.M; I /; ai 2 G  let j W .M; I / ! .M ; I  / is the iteration such that j.a/ D AG  D A. t u The next theorem we prove gives the key absoluteness property of L.R/Pmax . Using its proof one can greatly strengthen the previous theorems. To prove this we use the following corollary of Theorem 2.61. This theorem is discussed in Section 2.4. An alternate proof is possible using the stationary tower forcing and the associated generic elementary embedding. The choice is simply a matter of taste, working with Theorem 2.61 is more in the spirit of Pmax . In Chapter 6 we shall consider various generalizations of Pmax and for some of the variations we shall prove the corresponding absoluteness theorems which are analogous to the absoluteness theorems proved here for Pmax . There we will have to use the stationary tower forcing cf. Theorem 6.85. Theorem 4.63. Suppose ı is a Woodin cardinal. Let Q D Coll.!1 ;
171

Let C be a V Œg-generic club in !1 which is disjoint from S . Conditions for C are initial segments and so V is closed under ! sequences in V ŒgŒC . A key point is that V ŒgŒC  J D INS \V V ŒgŒC  where INS is the nonstationary ideal as computed in V ŒgŒC . This follows from the normality of the ideal J in V . V ŒgŒC  is a small generic extension of V and so ı is a Woodin cardinal in V ŒgŒC  and is measurable in V ŒgŒC . Let Q D Coll.!1 ; (4.1) ˛0 < ˛1 . (4.2) Suppose  2 Œt2  and let X D ¹f .jk/ j k < !º: Then a) X \ !1 D ˛0 , b) ordertype.X / D ˛1 . To see this suppose that g  P is V -generic with .s2 ; t2 / 2 G. Then in V Œg, g 2 Œt2 . Let Xg D ¹f .g jk/ j k < !º: Thus Xg 2 Cg . By absoluteness it follows from (4.2(a)) and (4.2(b)) that Xg \ !1 D ˛0 and that ordertype.Xg / D ˛1 . Therefore Xg 2 Zg ŒS0 ; S1  and so Cg \ Zg ŒS0 ; S1  ¤ ;: To find .˛0 ; ˛1 / and .s2 ; t2 / we associate to each pair .0 ; 1 / 2 S0  S1 with 0 < 1 a game, G .0 ; 1 /, as follows: Player I plays to construct a sequence h.i ; ˇiI / W i < !i of pairs such that .i ; ˇiI / 2 !2  1 . Player II plays to construct a sequence h.bi ; ni ; ˇiII / W i < !i of triples .bi ; ni ; ˇiII / 2 t1  !  1 . Let hˇi W i < !i be the sequence such that for all i < !, ˇ2iC1 D ˇiI and ˇ2i D ˇiII . The requirements are as follows: For each i < j < !, (5.1) bi  biC1 and dom.bi / D i , (5.2) if s1  bi then biC1 D bi _ ı for some ı > i , (5.3) f .b2iC1 / < !1 if and only if ˇi < 0 , (5.4) ni is odd and

f .bi / D f .bni /;

(5.5) f .b2iC1 /  f .b2j C1 / if and only if ˇi  ˇj . The first player to violate the requirements loses otherwise Player II wins. Thus the game is determined. The key property of the game is the following. Suppose that h.i ; ˇiI / W i < !i and h.bi ; ni ; ˇiII / W i < !i define an infinite run of the game which satisfies (5.1)–(5.5) (and so represents a win for Player II). Let hˇi W i < !i be the sequence such that for all i < !, ˇ2i D ˇiI and ˇ2iC1 D ˇiII . Suppose that 1 D ¹ˇi j i < !º:

190

5 Applications

Let X D ¹f .bi / j i < !º. Then X \ !1 D 0 and ordertype.X / D 1 : We claim that there must exist .˛0 ; ˛1 / 2 S0  S1 such that ˛0 < ˛1 and such that Player II has a winning strategy in the game G .˛0 ; ˛1 /. The proof requires only that INS is presaturated. Let G0  .P .!1 / n INS ; / be V -generic with S0 2 G and let j0 W V ! M0  V ŒG0  be the associated generic elementary embedding. Let G1  .P .!1 / n INS ; /M0 be M0 -generic with j0 .S1 / 2 G1 and let j1 W M0 ! M1  M0 ŒG1  be the generic elementary embedding given by G1 . Thus j1 ı j0 .!2 / D sup¹j1 ı j0 .˛/ j ˛ < !2 º: Further, since !2V D j0 .!1V /, .!1V ; !2V / 2 j1 ı j0 .S0 /  j1 ı j0 .S1 /: It follows by absoluteness, using the property (3.2) of f , that in M1 , Player II has a winning strategy in the game G .!1V ; !2V /. Therefore in V there must exist a pair .˛0 ; ˛1 / 2 S0  S1 such that ˛0 < ˛1 and such that Player II has a winning strategy in the game G .˛0 ; ˛1 /. Fix such a pair .˛0 ; ˛1 / and let hˇi W i < !i enumerate ˛1 . Let † be a winning strategy for Player II in the game G .˛0 ; ˛1 /. It is straightforward to construct a condition .s2 ; t2 /  .s1 ; t1 / such that if  2 Œt2  is a cofinal branch of t2 then there exists a sequences hi W i < !i and h.bi ; ni / W i < !i such that (6.1) for all i < !, i < !2 , (6.2) h.bi ; ni / W i < !i is the response of † to Player I playing h.i ; ˇi / W i < !i, (6.3)  D [¹bi j i < !º.

5.2 Martin’s Maximum and AC

191

Since ˛1 D ¹ˇi j i < !º it follows that if  is a cofinal branch of t2 then ¹f .ji / j i < !º \ !1 D ˛0 and that ordertype.¹f .ji / j i < !º/ D ˛1 : Thus .s2 ; t2 / is as required. This proves the claim that if G  P is V -generic then in V ŒG, the set ZG ŒS0 ; S1  is stationary in P!1 .!2V /. For each set A  !1 and for each ordinal ˛ > !1 let QŒA; ˛ be the partial order defined as follows. Condition are partial functions p W ˛
192

5 Applications

such that: (9.1)  D sup¹.i / j i < !º. (9.2) ¹.i / j i < !º   . (9.3) C \  is cofinal in  . (9.4) Suppose X 2 P!1 . / and that F ŒX
t u

From Lemma 5.8 and the results of .Foreman, Magidor, and Shelah 1988/ we obtain the following corollary. Theorem 5.9. Assume Martin’s Maximum. Then H.!2 /  AC : Proof. The relevant result of .Foreman, Magidor, and Shelah 1988/ is the following. Assume Martin’s Maximum. Suppose hTi W i < !i are pairwise disjoint subsets of !1 and that !1 D [¹Ti j i < !º. Suppose hSi W i < !i are pairwise disjoint stationary subsets of !2 such that for all i < !, Si  C! where C! D ¹˛ < !2 j cof ˛ D !º: Then there exists an ordinal  < !2 and a continuous (strictly) increasing function F W !1 !  with cofinal range such that F ŒTi   Si for each i < !. This together with the previous lemma yields that Martin’s Maximum implies AC . t u Thus: Theorem 5.10. Assume Martin’s Maximum. Then L.P .!1 //  ZFC:

5.3

The sentence

AC

We prove that ./ implies a variant of AC . This sentence implies  L.P .!1 //  AC, and in addition it implies  2@0 D 2@1 D @2 .

t u

5.3 The sentence

AC

193

Further this sentence can be used in place of MA!1 in defining Pmax , an alternate approach which will be useful in defining some of the Pmax variations, cf. Definition 6.91. We will also consider, in Section 7.2, versions of this sentence relativized to a normal ideal on !1 . Definition 5.11. AC : Suppose S  !1 and T  !1 are stationary, co-stationary, sets. Then there exist ˛ < !2 , a bijection  W !1 ! ˛; and a closed unbounded set C  !1 such that ¹ < !1 j ordertype.Œ/ 2 T º \ C D S \ C:

t u

Thus AC asserts that for each pair .S; T / of stationary, co-stationary, subsets of !1 , there exists an ordinal ˛ < !2 such that ŒS INS D ŒŒ˛ 2 j.T / in V B where B D RO.P .!1 /=INS / and

j W V ! .M; E/  V B

is the corresponding generic elementary embedding. This implies (in ZF) that the boolean algebra P .!1 /=INS can be wellordered (in length at most !2 ). Lemma 5.12 (ZF + DC). Assume

AC

holds in

hH.!2 /; 2i: Suppose hS˛ W ˛ < !1 i is a partition of !1 into !1 many stationary sets. Then there is a surjection  W !2 ! P .!1 / which is 1 definable in

hH.!2 /; INS ; 2i

from hS˛ W ˛ < !1 i. Proof. For each set A  !1 let SA D [¹S˛C1 j ˛ 2 Aº and let SA D S0 if A D ;. The key point is that if A  !1 , B  !1 , and if A ¤ B then SA M SB … INS : Define  W !2 ! P .!1 /

194

5 Applications

by .˛/ D A if there is a surjection  W !1 ! ˛ and a closed set C  !1 such that ¹ < !1 j ordertype.Œ/ 2 S0 º \ C D SA \ C: If no such set A exists then .˛/ D ;. Since AC holds,  is a surjection. It is easily verified that  is is 1 definable in hH.!2 /; INS ; 2i from hS˛ W ˛ < !1 i.

t u

The proof that Martin’s Maximum implies AC is actually much simpler then the proof we have given that Martin’s Maximum implies AC . The reason is that our approach to proving AC from Martin’s Maximum was through Lemma 5.8 which established quite a bit more than is necessary. Here we take a more direct approach which only requires a special case of the reflection principle, SRP, an observation due independently to P. Larson. The special case is SRP for subsets of P!1 .!2 /, which can be proved from just Martin’s Maximum.c/. This special case is discussed in Section 9.3. Theorem 5.13. Assume Martin’s Maximum.c/. Then H.!2 / 

AC :

Proof. Fix stationary sets S0  !1 and T0  !1 . Let Z D ¹X 2 P!1 .!2 / j X \ !1 2 S0 if and only if ordertype.X / 2 T0 º: It suffices to prove that for each stationary set S  !1 , the set ZS D ¹X 2 Z j X \ !1 2 S º is stationary in P!1 .!2 /. Let S  !1 be stationary. The claim that ZS is stationary in P!1 .!2 / follows by an absoluteness argument using the fact that INS is !2 -saturated. Fix a function H W !2
5.3 The sentence

AC

195

Let X D j0;2 Œ!2V ; the image of !2V under j0;2 . Thus (2.1) ordertype.X / < j0;2 .!1V /, (2.2) j0;2 .H /ŒX
AC

holds for .S0 ; T0 /.

t u

Larson has also noted that the proof of Theorem 5.13 easily adapts to show that Martin’s Maximum.c/ implies AC . We note that Lemma 5.8 cannot be proved from just Martin’s Maximum.c/. Therefore, for the proof that we have given that Martin’s Maximum implies AC , Martin’s Maximum.c/ does not suffice. Finally Larson has proved versions of Lemma 5.8 showing for example that Martin’s Maximum.c/ implies that for each stationary set S  !1 , SQ is stationary in !2 and that SQ \ ¹˛ < !2 j cof.˛/ D !º ¤ ;: The sentence, boolean algebra

AC ,

implies that for each stationary, co-stationary, set T  !1 , the P .!1 /=INS

is (trivially) generated by the term for j.T /. This fact combined with Theorem 3.51 yields the following lemma as an immediate corollary. Lemma 5.14. Suppose that

AC

holds. Then 2@0 D 2@1 D @2 :

Proof. By Theorem 3.51, 2@0 D 2@1 . By Lemma 5.12, 2@1  @2 .

t u

The next lemma shows that AC serves successfully in place of MA!1 in the definition of Pmax . This lemma is really just a special case of the claim given at the beginning of the proof of Theorem 3.51.

196

5 Applications

Lemma 5.15. Suppose M is a countable transitive set such that M  ZFC C Suppose a 2 M,

AC :

a  !1M ;

and M  “a is a stationary, co-stationary, set in !1 ”: Suppose j1 W M ! M1 and j2 W M ! M2 are semi-iterations of M such that M1 is transitive, M2 is transitive and such that j1 .a/ D j2 .a/: Then M1 D M2 and j1 D j2 . Proof. Fix a and suppose that hMˇ ; G˛ ; j˛;ˇ j ˛ < ˇ   i is a semi-iteration of M such that Mˇ is transitive for all ˇ   . We prove that G0 , M1 and j0;1 W M ! M1 are uniquely specified by j0; .a/ \ !2M . We note that since G0 is an M-normal ultrafilter, G0 D ¹b  !1M j b 2 M and !1M 2 j0;1 .b/º: Therefore since M

AC

it follows that G0 is completely determined by j0;1 .a/ \ !2M : To see this fix b 2 M such that b  !1M . We may suppose that b … .INS /M and that

!1M n b … .INS /M :

Therefore there exist ˛ < !2M , a bijection  W !1M ! ˛ and c  !1M such that (1.1)  2 M, c 2 M, (1.2) c is closed and cofinal in !1M , (1.3) ¹ < !1M j ordertype.Œ/ 2 aº \ c D b \ c.

5.3 The sentence

AC

197

But this implies that b 2 G0 $ ˛ 2 j0;1 .a/ since ˛ is the ordertype of j0;1 ./Œ!1M . Thus G0 is determined by j0;1 .a/ \ !2M and so M1 and j0;1 are uniquely specified by j0;1 .a/ \ !2M . Finally j0; D j1; ı j0;1 and !2M  !1M1 . Therefore j0;1 .a/ \ !2M D j0; .a/ \ !2M and so G0 , M1 and j0;1 are uniquely specified by j0; .a/ \ !2M . By induction on  it follows in a similar fashion that for all  <  , hMˇ ; j˛;ˇ j ˛ < ˇ   C 1i and hG˛ j ˛  i M

are uniquely specified by j0; .a/ \ !2  . The lemma follows noting that since j1 .a/ D j2 .a/ it follows that

j1 .!1M /

D

j2 .!1M /.

t u

The proof that ./ implies AC is simplified by first proving the following technical lemma which isolates the combinatorial essence of the implication. Lemma 5.16 (ZFC ). Suppose that x 2 R codes h.M; I /; ai 2 Pmax and that x # exists. Let C be the set of of the Silver indiscernibles of LŒx below !1 and let C 0 be the limit points of C . Suppose that ¹s; t º  .P .!1 //M n I is such that both !1M0 n s … I and !1M0 n t … I . Suppose J is a normal, uniform, ideal on !1 . Then there exists an iteration j W .M; I / ! .M ; I  / of length !1 such that and such that for all  2 C 0 , if and only if

I  D J \ M  2 j.s/  C 2 j.t /

where  C is the least element of C above  .

198

5 Applications

Proof. We modify the proof of Lemma 4.36 using the notation from that proof. The modification is a minor one. Choose the sequence hAk;˛ W k < !; ˛ < !1 i of J positive, pairwise disjoint, sets such that [¹Ak;˛ j k < ! and ˛ < !1 º  C 0 : Following the proof of Lemma 4.36 construct the iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i of .M; I / to satisfy the additional requirement that for all  2 C 0 ,  2 j0;C1 .s/ if and only if j0;ˇ .t / 2 Gˇ C

where ˇ D  . For each  2 C if h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ   i is any iteration of length  then j0; .!1M / D  and so this additional requirement does not interfere with the original requirements indicated in the proof of Lemma 4.36. Thus j0;!1 W .M; I / ! .M!1 ; I!1 / t u

is as desired. Lemma 5.17. Assume ./ holds. Then

AC

holds.

Proof. Fix a filter G  Pmax such that G is L.R/-generic. Necessarily, P .!1 /  L.R/ŒG: Fix subsets S and T of !1 such that each are both stationary and co-stationary. Therefore there exist h.M0 ; I0 /; a0 i 2 G; s 2 M0 and t 2 M0 such that j.s/ D S and j.t / D T where j W .M0 ; I0 / ! .M0 ; I0 / is the (unique) iteration such that j.a0 / D AG . Thus ¹s; t; !1M0 n s; !1M0 n t º \ I0 D ;: Let x0 code M0 , let C be the set of Silver indiscernibles of LŒx0  below !1 and let C 0 be the set of limit points of C . By Lemma 5.16, since G is generic, we may suppose, by modifying the choice of h.M0 ; I0 /; a0 i if necessary, that for all  2 C 0 ,  2 j.s/

5.4 The stationary tower and Pmax

199

if and only if

 C 2 j.t / where for each  2 C ,  C denotes the least element of C above  . Thus for all  2 C 0 ,  2S if and only if  C 2 T: Let ˛ be the least Silver indiscernible of LŒx0  above !1 and let  W !1 ! ˛ be a bijection. Thus there exists a club D  C 0 such that for all  2 D, ordertype.Œ / D  C : Therefore ¹ < !1 j ordertype.Œ / 2 T º \ D D S \ D: This proves the lemma.

5.4

t u

The stationary tower and Pmax

We sketch a different presentation of Pmax . This leads to different proofs of the absoluteness theorems. This approach will be useful in proving absoluteness theorems for some of the variations of Pmax that we shall define, cf. Theorem 6.85. Another feature of this approach is that it much easier to show that suitable conditions exist. This is because the generic iterations are based on elementary embeddings associated to the stationary tower and not to an ideal on !1 . Thus no forcing arguments are required to produce conditions. Recall from Section 2.3 the following conventions. Suppose ı is a Woodin cardinal. Then Q 209

Lemma 5.28 is really quite general. We state the version for iterable structures .M; I / where I 2 M is an ideal on !1M . This we have already discussed in a different context, see Remark 3.61. This lemma is required for the analysis of any variation of Pmax in which one has dropped all the requirements on the models designed to recover iterations from only the iterates. The proof of Lemma 5.29 is identical to the proof of Lemma 5.28. Lemma 5.29. Suppose M is a countable transitive model of ZFC and that .M; I / is iterable. Suppose j W .M; I / ! .M ; I  / is an iteration of .M; I / of length !1 . Then M … M .

t u

We shall also require a boundedness lemma for iterable structures of the form .M; I/ where I D .I J  D INS \ N  :

218

5 Applications

Fix D  Pmax such that D is open, dense in Pmax and such that D 2 L.R/. Assume toward a contradiction that FA \ D D ;: By Theorem 5.38, there exist h.M0 ; I0 /; X0 i 2 G and a0 2 M0 such that !1M0 D !1LŒa0  and such that j.a0 / D A where j0 W .M0 ; I0 / ! .M0 ; I0 / is the iteration of .M0 ; I0 / given by G. We work in L.R/ and assume that h.M0 ; I0 /; X0 i  FA \ D D ;: Let h.N0 ; J0 /; b0 i 2 Pmax be such that M0 2 .H.!1 //N0 and such that

J0 D .INS /N0 :

Let j1 W .M0 ; I0 / ! .M1 ; I1 / be an iteration such that j1 2 N0 , j.!1M0 / D !1N0 , and such that j1 is full in N0 . Let a1 D j1 .a0 /. Thus !1LŒa1  D !1N0 and so h.N0 ; J0 /; a1 i 2 Pmax . Since D is open, dense in Pmax , there exists h.N1 ; J1 /; b1 i 2 D such that h.N1 ; J1 /; b1 i < h.N0 ; J0 /; a1 i and such that

J1 D .INS /N1 :

(T) Let h.M2 ; I2 /; X2 i 2 Pmax be such that

N1 2 H.!1 /M2 : Let

k0 W .N0 ; J0 / ! .N0 ; J0 /

be the iteration such that k0 .a1 / D b1 . By Lemma 4.36 there exists an iteration k1 W .N1 ; J1 / ! .N2 ; J2 / such that k1 2 M2 and such that .INS /N2 D J2 D N2 \ .INS /M2 : Let a2 D k1 .b1 /. Thus k1 .k0 .j1 // is an iteration k1 .k0 .j1 // W .M0 ; I0 / ! .k1 .k0 .j1 //.M0 /; k1 .k0 .j1 //.I0 //

5.4 The stationary tower and Pmax

219

which is full in M2 . Therefore (T) h.M2 ; I2 /; X i 2 Pmax

and h.M2 ; I2 /; X i < h.M0 ; I0 /; X0 i where X D k1 .k0 .j1 //.X0 / [ ¹.h.M0 ; I0 /; X0 i; k1 .k0 .j1 ///º: By genericity we may assume h.M2 ; I2 /; X i 2 G: Let j2 W .M2 ; I2 / ! .M2 ; I2 / be the iteration given by G. Thus j2 .k1 / W .N1 ; J1 / ! .N1 ; J1 / is an iteration such that 

M

L.R/ŒG .INS /N1 D J1 D N  \ INS 2 D N  \ INS :

Further A D j0 .a0 / D j2 .k1 .k0 .j1 .a0 //// D j2 .k1 .b1 // D j2 .a2 / and j2 .k1 .b1 // D j2 .k1 /.j2 .b1 // D j2 .k1 /.b1 /: Therefore h.N1 ; J1 /; b1 i 2 FA which contradicts the choice of D and A. Therefore .L.P .!1 ///L.R/ŒG is a Pmax generic extension of L.R/. Fix g  Pmax such that g is L.R/-generic and .L.P .!1 ///L.R/ŒG D L.R/Œg: Let P be the following partial order defined in L.R/Œg. (T) P is the set of pairs .h.M; I/; X i; j / such that h.M; I/; X i 2 Pmax and such that O I/ O j W .M; I/ ! .M; is an iteration which is full in L.R/Œg. Suppose .h.M0 ; I0 /; X0 i; j0 / 2 P and that .h.M1 ; I1 /; X1 i; j1 / 2 P . Then .h.M1 ; I1 /; X1 i; j1 / < .h.M0 ; I0 /; X0 i; j0 / if .h.M0 ; I0 /; X0 i; j0 / 2 j1 .X1 /: The two relevant properties of P are the following.

220

5 Applications

(1.1) For each .h.M0 ; I0 /; X0 i; j0 / 2 P and for each B  !1 there exists .h.M1 ; I1 /; X1 i; j1 / 2 P such that .h.M1 ; I1 /; X1 i; j1 / < .h.M0 ; I0 /; X0 i; j0 / and such that B 2 j1 .M1 /. (1.2) For each .h.M0 ; I0 /; X0 i; j0 / 2 P there exist .h.M1 ; I1 /; X1 i; j1 / 2 P , .h.M1 ; I1 /; X1 i; k1 / 2 P , and .h.M2 ; I2 /; X2 i; k2 / 2 P such that a) .h.M2 ; I2 /; X2 i; j2 / < .h.M0 ; I0 /; X0 i; j0 /, b) .h.M2 ; I2 /; X2 i; j2 / < .h.M1 ; I1 /; X1 i; j1 /, c) .h.M2 ; I2 /; X2 i; j2 / < .h.M1 ; I1 /; X1 i; k1 /, d) j1 ¤ k1 . By (1.1), the partial order P is .< !2 /-closed in L.R/Œg and by (1.2), RO.P / has no atoms. The partial order P has cardinality 2@1 in L.R/Œg and so since 2@1 D @2 in L.R/Œg, RO.P / Š RO.!2
L.R/ŒG0  D L.R/ŒgŒh0 : By the definability of forcing it follows that there exists h  P such that h is L.R/Œg-generic and such that L.R/ŒG D L.R/ŒgŒh:

t u

 5.5 Pmax

221

 5.5 Pmax (T) We define a second reformulation of Pmax . This version is quite closely related to Pmax and it involves a reformulation of the sentence AC .

 Definition 5.40. AC : Suppose that hS˛ W ˛ < !1 i and hT˛ W ˛ < !1 i are each sequences of stationary, co-stationary sets. Then there exists a sequence hı˛ W ˛ < !1 i of ordinals less than !2 such that for each ˛ < !1 there exists a bijection

 W !1 ! ı˛ ; and a closed unbounded set C  !1 such that ¹ < !1 j ordertype.Œ/ 2 T˛ º \ C D S˛ \ C:

t u

If M  ZFC then clearly M

AC

M

 AC :

if and only if The reason for introducing iterable sequence and that

 AC

is the following. Suppose that hMi W i < !i is an

[¹Mi j i < !º 

 AC :

Suppose that hMi W i < !i is an iterate of hMi W i < !i. Then [¹Mi j i < !º  This can fail for

 AC :

AC .

 Definition 5.41. Pmax is the set of pairs .hMk W k < !i; a/ such that the following hold.

(1) a 2 M0 , a  !1M0 , and !1M0 D !1LŒa;x for some x 2 R \ M0 . (2) Mk  ZFC . Mk

(3) Mk 2 MkC1 ; !1

MkC1

D !1

.

(4) .INS /MkC1 \ Mk D .INS /MkC2 \ Mk . (5) [¹Mk j k 2 !º 

 AC .

(6) hMk W k < !i is iterable.

222

5 Applications

(7) There exists X 2 M0 such that M1 , a) X  P .!1 /M0 n INS

b) M0  “jX j D !1 ”, M0 . c) for all S; T 2 X , if S ¤ T then S \ T 2 INS  is analogous to Pmax . A condition The ordering on Pmax

.hNk W k < !i; b/ < .hMk W k < !i; a/ if hMk W k < !i 2 N0 , hMk W k < !i is hereditarily countable in N0 and there exists an iteration j W hMk W k < !i ! hMk W k < !i such that: (1) j.a/ D b; (2) hMk W k < !i 2 N0 and j 2 N0 ; 

(3) .INS /MkC1 \ Mk D .INS /N1 \ Mk for all k < !. Remark 5.42.

t u

(1) One can strengthen (4) by requiring that for all k < !, Mk MkC1 D Mk \ INS : INS

In this case (7) necessarily holds.  is equivalent to the partial order Pmax , assuming that for (2) The partial order Pmax  all x 2 R, x exists.  is the better presentation of Pmax . The key difference is that one (3) Arguably Pmax  without using ideals on !1 . This we can directly construct conditions in Pmax shall do in proving Theorem 5.49. t u

The proof of Lemma 5.15 easily adapts to prove the following lemma which is the analog of Lemma 4.35.  . Suppose that Lemma 5.43. Suppose that .hMk W k < !i; a/ 2 Pmax

j1 W hMk W k < !i ! hMk1 W k < !i and j2 W hMk W k < !i ! hMk2 W k < !i are wellfounded iterations such that j1 .a/ D j2 .a/. Then hMk1 W k < !i D hMk2 W k < !i and j1 D j2 .

 5.5 Pmax

223

Proof. Fix x 2 R \ M0 such that !1M0 D !1LŒa;x : It follows that there exists Z  !1M0 such that Z 2 LŒa; x \ M0 M

M

and such that for all k < !, Z … INS kC1 and .!1M0 n Z/ … INS kC1 . Therefore arguing as in the proof of Lemma 5.15, if j1 .Z/ D j2 .Z/ then hMk1 W k < !i D hMk2 W k < !i and j1 D j2 . The sequence hMk1 W k < !i is iterable and so it follows that for all b  !1M0 , if b 2 [¹Mk j k 2 !º then b # 2 [¹Mk j k 2 !º: Therefore .x; a/# 2 [¹Mk j k 2 !º. Thus since j1 .a/ D j2 .a/ it follows that t u j1 .Z/ D j2 .Z/ noting that necessarily j1 .!1M0 / D j2 .!1M0 /.  The basic iteration lemma required for the analysis of Pmax is a modification of Lemma 4.37. The proof is a minor variation of the proof of Lemma 4.36.  . Suppose that Lemma 5.44 (ZFC ). Suppose that .hMk W k < !i; a/ 2 Pmax hS˛ W ˛ < !1 i is a sequence of pairwise disjoint stationary subsets of !1 . Then there is an iteration j W hMk W k < !i ! hMk W k < !i such that for all S  !1 , if M

then S˛ n S 2 INS

S 2 [¹Mk j k < !º n [¹INS k j k < !º for some ˛ < !1 .

t u

As a corollary to Lemma 5.44 we obtain the following iteration lemma. It is for the  proof of this lemma that the requirement (7) in the definition of Pmax is essential. Lemma 5.45. Suppose that  .hMk W k < !i; a/ 2 Pmax ;  .hNk W k < !i; b/ 2 Pmax ;

and that

hMk W k < !i 2 H.!1 /N0 :

Then there is an iteration j W hMk W k < !i ! hMk W k < !i such that j 2 N0 and such that M

N1 INS \ .[¹Mk j k < !º/ D [¹INS k j k < !º:

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5 Applications

Proof. Since

 .hNk W k < !i; b/ 2 Pmax

there exists a sequence hS˛ W ˛ < !1N0 i 2 N0 such that for all ˛ < ˇ < !1N0 , S˛  !1N0 ; N1 S˛ … INS ; N0 . and such that S˛ \ Sˇ 2 INS With this sequence the lemma follows by Lemma 5.44.

t u

 is a routine generalization of Using the iteration lemmas the basic analysis of Pmax the analysis of Pmax provided suitable iterable structures exist.

Definition 5.46. Suppose that hMk W k < !i is an iterable sequence and that A  R. Then the sequence hMk W k < !i is A-iterable if (1) A \ M0 2 [¹Mk j k < !º, (2) for any iteration j W hMk W k < !i ! hMk W k < !i, j.A \ M0 / D A \ M0 :

t u

 . It is (notationally) We prove a very general existence lemma for conditions in Pmax (T) convenient to refer to Pmax in the statements of the two preliminary lemmas that we require; note that the assumption (T) h.M0 ; I0 /; ;i 2 Pmax

simply abbreviates: M0 is a countable transitive model of ZFC, I0 D .I 236

5 Applications

The generalization of Theorem 5.53 and Theorem 5.54 that we seek is the following. Theorem 5.64. The following are equivalent. (1) 2@0 D 2@1 and there exists a countable elementary substructure X  H.!2 / such that the transitive collapse of X is iterable. (2) There exists a semi-generic filter 0 G  Pmax

such that P .!1 /G D P .!1 /. Proof. Let D .2@1 /C . Fix a wellordering,
237

a) For some a0 2 Zi
By Lemma 3.12, the map j specified in (2.2(d)) is an iteration. Let Z0 D [¹Zk j k < !º: 0 which satisfies the It follows, since jZ0 j < 2@0 , that Z0 generates a filter F0  Pmax conditions (1.1)–(1.2). This proves our claim and (2) follows. To finish we assume (2) holds and prove (1). First we must prove that (2) implies that 2@0 D 2@1 . But 0 j  2@0 jGj  jPmax

and for each condition p 2 G, there is a unique iteration  jp;G W hM.p;k/ W k < !i ! hM.p;k/ W k < !i

given by G where p D .hM.p;k/ W k < !i; X.p/ /i. By the definition of P .!1 /G ,  P .!1 /G D [¹P .!1 / \ M.p;0/ j p 2 Gº;

and so by (2), 2@0 D 2@1 . To finish we must prove that (2) implies that there exists a countable elementary substructure X  H.!2 / such that the transitive collapse of X is iterable. By (2), for every x 2 R there exists an iterable sequence, hNk W k < !i; such that x 2 N0 . Thus arguing as in the proof of Theorem 3.19, for every x 2 R, x # exists. Thus the existence of X  H.!2 /, countable and with iterable transitive collapse, is essentially an immediate corollary of Lemma 4.22 and Theorem 3.19. u t

238

5 Applications

5.7

The Axiom

 

We prove that the Pmax -extension can be characterized by a certain kind of generic homogeneity. This property generalizes to L.P .!1 // a well known property which characterizes L.R/ in the case that L.R/ is computed in LŒG where G is L-generic for adding uncountably many Cohen reals to L. This is the symmetric extension of L given by infinitely many Cohen reals. Suppose that L.R/ is a symmetric extension of L for adding infinitely many Cohen reals. Then the following hold. (1) There is an L-generic Cohen real. (2) Let X R be a nonempty set which is ordinal definable in L.R/. Then there exists a term  2 L such that for all L-generic Cohen reals c,  .c/ 2 X; where .c/ is the interpretation of  by the generic filter given by c. It is straightforward to show that the converse is also true: If (1) and (2) hold then L.R/ is a symmetric extension of of L for adding infinitely many Cohen reals. The point is that (1) and (2) imply that for every x 2 R n L, LŒx D LŒc for some c 2 R which is an L-generic Cohen real. Further (1) and (2) also imply that for every x 2 R, there is an LŒx-generic Cohen real. We generalize this to L.P .!1 // in Theorem 5.67. This gives a reformulation of the axiom ./ which seems more suited to the investigation of the consequences of this axiom. As we have indicated above, this property characterizes the Pmax -extension. We fix some notation. Recall that the partial order Coll.!;
t u

5.7 The Axiom

 

239

Remark 5.66. The sequence hS˛g W ˛ < !1 i, defined in Definition 5.65, can be defined using any reasonable sequence h˛ W ˛ < !1 i of terms for pairwise disjoint stationary subsets of !1 . The only important requirement is that h˛ W ˛ < !1V i 2 L:

t u

Theorem 5.67. Assume ./ holds. Suppose X  P .!1 /, X 2 L.P .!1 //; X ¤ ;, and that X is definable in L.P .!1 // from real and ordinal parameters. Then there exist t 2 R and a term   !1  Coll.!;
M

M0 D ¹j.hi /.˛/ j ˛ < !1 0 ; i < !º: Let t 2 R code M0 . Suppose g  Coll.!;
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5 Applications

(1.1) j 2 LŒt Œg, (1.2) j.!1M0 / D !1 , (1.3) for all if S D j.hi /.˛/ then

S 2 P .!1 / \ M0 n I0 ; Sg n S 2 INS

where  D ! ˛ C i (i. e.  is the image of .i; ˛/ under a reasonable bijection of !  !1 with !1 ). The iteration j is easily constructed in LŒt Œg. Let be a term for j . We may suppose that the interpretation of by any LŒt generic filter yields an iteration satisfying (1.1)–(1.3). Let 0 be a term in LŒt  for j.a0 / and let  D ¹.˛; p/ j ˛ < !1 ; p 2 Coll.!;
t u

5.7 The Axiom

 

241

We isolate the conclusion of Theorem 5.67 in defining the following axiom.   Definition 5.68. Axiom  : For all t 2 R, t # exists. Suppose X  P .!1 /, X ¤ ;, and that X is definable from real and ordinal parameters. Then there exist t 2 R and a term   !1  Coll.!;
  

t u one need only consider sets

X  P .!1 / which are definable (without parameters). Lemma 5.69 (For all t 2 R, t # exists). The following are equivalent.   (1)  . (2) Suppose X  P .!1 /, X ¤ ;, and that X is definable by a †2 formula. Then there exist t 2 R and a term   !1  Coll.!;
242

5 Applications

where z D A \ ! and where A D ¹˛ < !1 j ! C ˛ 2 Aº:

  If  fails then X ¤ ;. Assume toward a contradiction that X ¤ ;. X is definable by a †2 formula. Therefore by (2) there exist t 2 R and a term   !1  Coll.!;
Since t exists we may suppose, by replacing t if necessary, that  is definable in LŒt  from t and !1 . Let 0 be the least Silver indiscernible of LŒt  and let g0  Coll.!;  0 / be an LŒt -generic filter. Fix t0 2 LŒt Œg0  \ R such that LŒt Œg0  D LŒt0  and define 0  !1  Coll.!;
5.7 The Axiom

 

243

Thus for all ˛ < !1 , S˛g is stationary. Let A D Ig . /. Therefore A 2 X and so A 2 Xz where as above, z D A \ ! and A D ¹˛ < !1 j ! C ˛ 2 Aº: Since  is definable in LŒt  from t and !1 , it follows that A\! is completely determined by g \ Coll.!; <0 /  g0 : Therefore z 2 LŒt0  and z does not depend on h. Finally by the definition of 0 , Ih .0 / D A and so Ih . / 2 Xz .   Therefore t0 and 0 witness that Xz is not a counterexample to  , which is a contradiction. t u   Many of the consequences of ./ are more easily proved using  . We begin with a straightforward consequence which concerns !1 -borel sets. A set A  R is !1 -borel if it can be generated from the borel sets by closing the borel sets under !1 unions and intersections. Clearly if CH holds then every set of reals is !1 -borel. The following lemma gives a useful characterization of the !1 -borel sets. Lemma 5.70. Suppose A  R. The following are equivalent. (1) A is !1 -borel. (2) There exist S  !1 , ˛ < !2 , and a formula .x0 ; x1 /, such that !1 < ˛ and such that t u A D ¹y 2 R j L˛ ŒS; y  ŒS; yº: Lemma 5.70 can fail if one does not assume AC, the difficulty is that it is possible for a set A  R to be !1 -borel but not effectively !1 -borel. With the latter notion, Lemma 5.70 is true in just ZF. A set A  R is effectively !1 -borel if it has an !1 borel code. The !1 -borel codes are defined by induction as subsets of !1 in the natural fashion generalizing the definition of borel codes as subsets of !. Lemma 5.71 (ZF). Suppose A  R. The following are equivalent. (1) A is effectively !1 -borel. (2) There exist S  !1 , ˛ < !2 , and a formula .x0 ; x1 /, such that !1 < ˛ and such that t u A D ¹y 2 R j L˛ ŒS; y  ŒS; yº:

244

5 Applications

1 It is not difficult to show, assuming AD, that every !1 -borel set is  3 . This is an immediate consequence of the fact that assuming AD, the  13 sets are closed under !1 unions. In fact, Lemma 5.70 can be proved assuming AD, and so, assuming AD, the following are equivalent:

(1) A is !1 -borel, (2) A is effectively !1 -borel, (3) there exist x 2 R, ˛ < !2 , and a formula , such that !1 < ˛ and such that A D ¹y 2 R j L˛ Œx; y  Œx; yº:

t u

  Theorem 5.72. Assume  . Suppose that A  R and that A is definable from ordinal and real parameters. Then the following are equivalent. (1) A is !1 -borel. 1 (2) A is † 3 and

L.R/  A is !1 -borel:

(3) There exist x 2 R, ˛ < !2 , and a formula , such that !1 < ˛ and such that A D ¹y 2 R j L˛ Œx; y  Œx; yº:

t u

Proof. (2) trivially implies (1) and by Lemma 5.70, (3) also implies (1). We assume (1) and prove (3). By Lemma 5.70 there exist S0  !1 , ˛0 < !2 , and a formula 0 .x0 ; x1 /, such that !1 < ˛0 and such that A D ¹y 2 R j L˛0 ŒS0 ; y  0 ŒS0 ; y; !1 º: Clearly we can suppose that ˛0 is less than the least ordinal  such that !1 <  and such that L ŒS0  is admissible. Fix ˛0 and 0 . Let X  P .!1 / be the set of S such that (1.1) A D ¹y 2 R j L˛0 ŒS; y  0 ŒS; yº, (1.2) ˛0 <  where  is the least ordinal above !1 such that L ŒS  is admissible. Thus X ¤ ; and since A is definable from ordinal and real parameters, so is X . By  there exist x 2 R and a term   !1  Coll.!;
5.7 The Axiom

 

245

if g is LŒx-generic and if for each  < !1 , S g is stationary, then Ig ./ 2 X: Let ˛ < !2 be least such that ˛0 < ˛ and such that L˛ .x/  ZFC: Note that for each y 2 R, if g  Coll.!;
such that G is L.R/-generic and such that L.P .!1 // D L.R/ŒG: Thus assuming L.R/  AD; and that 

V D L.P .!1 //;

it follows that ./ and  are equivalent.   In fact we shall also prove that assuming  , the nonstationary ideal has a homogeneity property which can be shown to imply that L.R/  AD: Thus if 

V D L.P .!1 //;

then ./ and  are equivalent.   The proof that  implies that L.P .!1 // is a Pmax -extension of L.R/ requires the following theorem and subsequent lemma. These combine to prove Corollary 5.78. Theorem 5.73 gives more than we need however it does give some interesting consequences of ./.

246

5 Applications

Theorem 5.73. Assume

 

holds. Then:

(1) ı12 D !2 . (2) Every club in !1 contains a club which is constructible from a real. (3) Suppose A  !1 is cofinal. There is a wellordering
 

247

Let p0 and q0 be the LŒt0 -least such conditions relative to the canonical wellordering of LŒt0  given by t0 . Since Ig0 .0 / 2 X0 , ˛0 , p0 and q0 exist. Since 0 is definable in LŒt0  from t0 and !1 , ¹˛0 ; p0 ; q0 º  L 1 Œt0  where 1 is the least indiscernible of LŒt0  above 0 . Let h W  < !1 i be the increasing enumeration of the indiscernibles of LŒt0  below !1 . For each  < !1 let j W LŒt0  ! LŒt0  be the canonical elementary embedding such that j.0 / D  and such that LŒt0  D ¹j .f /.s/ j f 2 LŒt0 ; s 2 Œ 
248

5 Applications

(3.1) ¹p W !  ¹ º !  j p 2 g0 º  g, (3.2) ¹i 2 ! j ˛CiC1 2 Ig .0 /º codes g0 \ Coll.!;
t u

  Remark 5.76. (1) The first 4 consequences of  given in Theorem 5.73 follow from Martin’s Maximum though the proofs seem more involved. (2) We do not know if (5) of Theorem 5.73 can be proved from Martin’s Maximum. This problem seems very likely related to the problem of the relationship of Martin’s Maximum and the axiom ./. Similarly we do not know if Theorem 5.75 can be proved from Martin’s Maximum. The two problems are likely closely related. Note that if B  !1 satisfies the condition (2) of Theorem 5.75 and if t u B # exists then there must exist x 2 LŒB \ R such that x # … LŒB. Lemma 5.77. Assume

 . Suppose that y 2 R and that  M 2 H.!2 /

is definable from y and ordinal parameters. Then M 2 LŒz for some z 2 R.   Proof. This is an immediate consequence of  . Fix y 2 R and suppose that M 2 H.!2 / is ordinal definable from y. Let X be the set of A  !1 such that M 2 LŒA: Thus  is ordinal definable from y (and X ¤ ;).  X By  , there exist z 2 R and   !1  Coll.!;
S˛g

is stationary for each ˛ < !1 then Ig ./ 2 X:

5.7 The Axiom

 

251

Let g1  Coll.!;
t u

Lemma 5.77 has the following corollary. This is also a corollary of Theorem 5.73 and Theorem 3.22, but the proof we give is more direct. Corollary 5.78. Assume

 . Then for all x 2 R, x  exists. 

Proof. Fix x 2 R. By Theorem 5.73(2) and Lemma 5.77, F \ P .!1 / \ HODx is an HODx -ultrafilter where F is the club filter on !1 . Therefore !1 is a measurable cardinal in HODx and F \ P .!1 / \ HODx is a normal measure on !1 . Let N D LŒF ; x: Since !1 is a measurable cardinal in N , jV!1 \ N j D !1 and so by Lemma 5.77 there exists y0 2 R such that V!1 \ N 2 LŒy0 : Since y0# exists, for all A  !1 , if A \ ˛ 2 LŒy0  for all ˛ < !1 then A 2 LŒy0 .

252

5 Applications

Thus P .!1 / \ N  LŒy0  and so jP .!1 / \ N j D !1 : Hence by Lemma 5.77 again there exists y1 2 R such that P .!1 / \ N 2 LŒy1  and such that F \ P .!1 / \ N 2 LŒy1 : Thus N  LŒy1  #

y1#

and so N exists since exists. Therefore x  exists.

t u

By Theorem 4.59 and Corollary 5.78, assuming trivial. Lemma 5.79. Assume

 

, the partial order Pmax is non-

 . Suppose  A 2 P .!1 / n L.R/:

Then A is HODR -generic for Pmax . Proof. For each A 2 P .!1 / n L.R/; let FA be the set of

h.M; I /; ai 2 Pmax

for which there exists an iteration j W .M; I / ! .M ; I  / such that j.a/ D A and such that I  D INS \ M : By Theorem 3.19 and Theorem 5.73(2), there exists a countable elementary substructure X  H.!2 / such that MX is iterable where MX is the transitive collapse of X . Thus by Lemma 4.74 the elements of FA are pairwise compatible. Therefore it suffices to show that FA \ D ¤ ; for each D  Pmax such that D is dense and such that D 2 HODR . Assume toward a contradiction that A0 2 P .!1 / n L.R/;

5.7 The Axiom

 

253

D0  Pmax is dense, D0 2 HODR , and that D0 \ FA0 D ;: By Theorem 5.73(5), there exists t0 2 R such that !1 D .!1 /LŒt0 ;A0  : Let X  P .!1 / n L.R/ be the set of all A 2 P .!1 / n L.R/ such that (1.1) FA \ D0 D ;, (1.2) !1 D .!1 /LŒA;t0  . Since D0 2 HODR ; X is definable with  parameters from R [ Ord. Further A0 2 X and so X ¤ ;. Therefore by  there exist t 2 R and   !1  Coll.!;
254

5 Applications

and let I0 2 N Œg be the ideal generated by ¹a  ı j ı n a 2 º: It follows that .N ŒgŒh; I0 / is iterable. Thus h.N ŒgŒh; I0 /; a0 i 2 Pmax and so, since D0 is dense, there exists h.M; I /; ai 2 D0 such that h.M; I /; ai < h.N ŒgŒh; I0 /; a0 i: Let

j0 W .N ŒgŒh; I0 / ! .N  Œg  Œh ; I0 /

be the iteration such that j0 .a0 / D a. By Lemma 4.36, there exists an iteration j W .M; I / ! .M ; I  / of length !1 such that Let A D j.a/. By elementarity,

I  D INS \ M :

j.g  /  Coll.!;
g  is LŒt -generic and A D Ij.g  / ./: Moreover for each ˛ < ı, .S˛g /N ŒgŒh … I0 : Thus, by the elementarity of j ı j0 , for each ˛ < !1 , S˛j.g Since

/

… I :

I  D INS \ M ; 

it follows that for each ˛ < !1 , S˛j.g / is stationary. Thus implies that A 2 X . However the iteration, j W .M; I / ! .M ; I  / witnesses h.M; I /; ai 2 FA : t u

This is a contradiction.   Corollary 5.80. Assume  . Then MA!1 . Proof. Let A 2 P .!1 / n L.R/ code a pair .P ; D/ 2 H.!2 / such that

5.7 The Axiom

 

255

(1.1) P is a ccc partial order, (1.2) D is a collection of dense subsets of P . By Lemma 5.79, A is L.R/-generic for Pmax . In particular there exists h.M; I /; ai 2 Pmax and an iteration

j W .M; I / ! .M ; I  /

such that

I  D INS \ M

and such that j.a/ D A. By elementarity,

M  MA!1 :

Therefore since A 2 M we have that .P ; D/ 2 M : This implies that there is a filter F  P such that F 2 M and such that F \d ¤; for all d 2 D.

t u

We recall the following notation. Suppose G  Pmax is a semi-generic filter. P .!1 /G D [¹P .!1 / \ M j h.M; I /; ai 2 Gº where for each h.M; I /; ai 2 G, j W .M; I / ! .M ; I  / is the unique iteration such that j.a/ D AG D [¹a j h.M; I /; ai 2 Gº: Of course the filter G is uniquely specified by AG . Theorem 5.81. Assume

 . Suppose  A 2 P .!1 / n L.R/:

Then exists a filter G  Pmax such that G is HODR -generic and such that the following hold. (1) A D AG . (2) P .!1 / D P .!1 /G . (3) HODP .!1 / D HODR ŒG.

256

5 Applications

Proof. Suppose A 2 P .!1 / n L.R/: Fix x 2 R such that

!1 D .!1 /LŒA;x :

By Lemma 5.79 there is a filter G  Pmax such that G is HODR -generic and such that A D AG : Suppose B  !1 . By Corollary 5.80, MA!1 holds and so B 2 L!1 C1 ŒA; x; y for some y 2 R. By genericity there exists, h.M; I /; ai 2 G such that .x; y/ 2 M. Let j W .M; I / ! .M ; I  / is the unique iteration such that A D j.a/. It follows that L!1 C1 ŒA; x; y 2 M and so B 2 M ; i. e.

B 2 P .!1 /G :

Therefore P .!1 / D P .!1 /G and so G satisfies (2). We finish by proving that HODP .!1 / D HODR ŒG: Let P be the predicate defined as follows. .A; ; ˛; b/ 2 P if (1.1) ˛ 2 Ord and ˛ > !2 , (1.2) b 2 R  Ord, (1.3) A 2 P .!1 /, (1.4)  is a formula in the language for set theory, (1.5) V˛  ŒA; b. It is an elementary fact that HODP .!1 / D L.P; P .!1 //: Therefore it suffices to show that for all ı 2 Ord, P \ Vı 2 HODR ŒG: Let Q be the following predicate. .t; ; ; ˛; b/ 2 Q if

5.7 The Axiom

 

257

(2.1) ˛ 2 Ord and ˛ > !2 , (2.2) b 2 R  Ord, (2.3) t 2 R,   !1  Coll.!;
and this proves the theorem.

Using the results of the next section, Section 5.8, the assumption in the Corollary 5.82 that L.R/  AD can be eliminated; i. e. if then ./ and

 

V D L.P .!1 //; are equivalent.

Corollary 5.82. Assume L.R/  AD: Then the following are equivalent. (1) ./. (2) L.P .!1 // 

 

We next prove that of P .!1 /.

.  

t u implies that a perfect set theorem holds for definable subsets

258

5 Applications

  Theorem 5.83. Assume  holds. Suppose X  P .!1 / and that X is definable in L.P .!1 // from real and ordinal parameters. Suppose there exists A 2 X such that A … L.R/. Then there exists a function  W 2
˛ 2 .s/

if and only if s.˛/ D 1I and such that for all F 2 2!1 , [¹.F j˛/ j ˛ < !1 º 2 X: Proof. The proof is quite similar to the proof of Theorem 5.73(5). Let X0 D X n L.R/. Thus X0 is definable from ordinal and real parameters, and X0 ¤ ;.   By  there exist t0 2 R, 0  !1  Coll.!;
5.7 The Axiom

 

259

(1.3) p  “˛0 2 Ig0 .0 /”. (1.4) q  “˛0 … Ig0 .0 /”. Let p0 and q0 be the LŒt0 -least such conditions relative to the canonical wellordering of LŒt0  given by t0 . Since Ig0 .0 / 2 X0 , ˛0 , p0 and q0 exist. Since 0 is definable in LŒt0  from t0 and !1 , ¹˛0 ; p0 ; q0 º  L 1 Œt0  where 1 is the least indiscernible of LŒt0  above 0 . Let h W  < !1 i be the increasing enumeration of the indiscernibles of LŒt0  below !1 . For each  < !1 let j W LŒt0  ! LŒt0  be the canonical elementary embedding such that j.0 / D  and such that LŒt0  D ¹j .f /.s/ j f 2 LŒt0 ; s 2 Œ 
˛ D j .˛0 /;

let p D j .p0 /; and let q D j .q0 /: Suppose that g 2 F t0 and that g \ Coll.!;
 

261

Suppose F 2 2!1 and let g D [¹.F j / j  < !1 º: For each  < !1 , .F j / 2 F t0 and so g is LŒt0 -generic. For each ˛ < !1 , S˛g M S˛g0 2 INS and so for each ˛ < !1 , S˛g is stationary. Therefore Ig .0 / 2 X . Finally Ig .0 / D [¹.F j˛/ j ˛ < !1 º t u

and so  is as desired.

Remark 5.84. Thus subsets of 2!1 which are definable in L.R/Pmax are either in L.R/ t u or contain copies of 2!1 .   The reformulation of ./ as  taken together with the results of Chapter 4 strongly suggests that, assuming ./, one should be able to analyze sets X  P .!1 / which are definable in the structure hH.!2 /; 2; INS i by a …1 formula. We explore the possibilities for classifying specific definable subsets of P .!1 /. For this we assume that the axiom ./ holds and we focus on attempting to classify partitions  W Œ!1 2 ! ¹0;1º for which there is no homogeneous rectangle for 0, of (proper) cardinality @1 . Here we adopt the convention that if Z D A  B  !1  !1 is a rectangle, then Z has proper cardinality @1 if both A and B have cardinality @1 . This is related to the following variation of a question of S. Todorcevic.  Is it consistent that for any partition  W Œ!1 2 ! ¹0;1º; either there is a homogeneous rectangle for  for 0, of (proper) cardinality @1 , or there is no such homogeneous rectangle in any generic extension of V which preserves !1 ?

262

5 Applications

Remark 5.85.

(1) Suppose that  W Œ!1 2 ! ¹0;1º;

and for each ˛ < !1 define B˛ D ¹ˇ < !1 j .˛; ˇ/ D 0º: The partition  has a homogeneous rectangle of (proper) cardinality @1 if and only if there exists a countably complete, uniform, filter F  P .!1 / such that j¹˛ < !1 j B˛ 2 F or !1 n B˛ 2 F ºj D @1 ; similarly the partition  has a homogeneous rectangle for 0 of (proper) cardinality @1 if and only if there exists a countably complete, uniform, filter F  P .!1 / such that j¹˛ < !1 j B˛ 2 F ºj D @1 : Thus one is really attempting to classify the sequences hB˛ W ˛ < !1 i of subsets of !1 for which there exists a uniform countably complete filter on !1 which contains uncountably many of the sets. (2) The problem of whether it consistent for every partition  W Œ!1 2 ! ¹0;1º to have a homogeneous rectangle of (proper) cardinality @1 , has been solved negatively Moore .2006/. t u We fix some more notation. Suppose a 2 H.!1 / and that L.a/  ZFC: Let D jcj

L.a/

where c is the transitive closure of a, Then M3 .a/ D .H.j jC //L.Q3 .a// :

Let b  be a set in L.a/ which codes a. One can show that M3 .a/ is precisely the set of all sets, c, which can be coded by a set z  such that z 2 Q3 .b/. 1 Definition 5.86 ( 2 -Determinacy). Suppose that

 W Œ!1 2 ! ¹0;1º: (1) Suppose that X  !1 . Let E .3/ ŒX  be the set of  < !1 such that there exists Z1  Z2     such that

5.7 The Axiom

 

263

a) Z1 and Z2 each have ordertype , b) .˛; ˇ/ D 0 for all .˛; ˇ/ 2 Z1  Z2 with ˛ < ˇ, c) Q3 .Z1  Z2  .X \ /  jŒ2 / ¤ ;, d) M3 .Z1  Z2  .X \ /  jŒ2 /   D !1 . (2) Suppose that X  !1 and that A D hS˛ W ˛ < !1 i is a sequence of stationary subsets of !1 such that for each ˛ < !1 , S˛ 2 LŒX : Let E .3/ ŒX; A be the set of  < !1 such that there exists Z1  Z2     such that a) Z1 and Z2 each have ordertype , b) .˛; ˇ/ D 0 for all .˛; ˇ/ 2 Z1  Z2 with ˛ < ˇ, c) Q3 .a/ ¤ ;, d) M3 .a/   D !1 , e) for each ˛ < , S˛ \  2 M3 .a/ and S˛ \  is a stationary set within M3 .a/, where a D Z1  Z2  .X \ /  jŒ2 :

t u

Assume there exists a Woodin cardinal with a measurable cardinal above. Then 1 by .Martin and Steel 1989/,  2 -Determinacy holds and so Definition 5.86 applies. If the partition given by  has a homogeneous rectangle for 0, of (proper) cardinality @1 , then necessarily E .3/ ŒX; A contains a club in !1 . Another trivial observation is that if for some .X; A/ the set E .3/ ŒX; A is nonstationary then there exists Y  !1 such that E .3/ ŒY; A D ;: With the notation as above we have the following lemma. Lemma 5.87. Assume there is a Woodin cardinal with a measurable above. Then (1) E .3/ ŒX  contains a closed unbounded set or E .3/ ŒX  is nonstationary, (2) E .3/ ŒX; A contains a closed unbounded set or E .3/ ŒX; A is nonstationary.

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5 Applications

Proof. We prove (1), the proof of (2) is similar. Suppose E .3/ ŒX  is stationary. We show that E .3/ ŒX  contains a closed unbounded set. Let ı be a Woodin cardinal and let S  ı be a set such that Vı 2 LŒS  #

and such that S exists. By the hypothesis of the lemma, ı and S exist. Let N D LŒS : Let Y N be a countable elementary substructure containing infinitely many Silver indiscernibles of N . We prove that Y \ !1 2 E .3/ ŒX : Let NY be the transitive collapse of Y . Let SY be the image of S under the collapsing map and let ıY be the image of ı. Let ˛ D NY \ Ord. Thus NY  “ıY is a Woodin cardinal” and NY D L˛ ŒSY : Since Y contains infinitely many indiscernibles of N , L˛ ŒSY   LŒSY : The key points are that .E .3/ ŒXY /NY D E .3/ ŒX  \ Y \ !1 Y and that ŒXY  is stationary”; NY  “E .3/ Y where XY and Y are the images of X and  under the collapsing map. By elementarity, .E .3/ ŒXY /LŒSY  D E .3/ ŒX  \ Y \ !1 Y and ŒXY  is stationary”: LŒSY   “E .3/ Y Let a D .E .3/ ŒXY /LŒSY  and let  D !1LŒSY  D Y \ !1 . Y Let G  .Q 266

5 Applications

We prove this. By Theorem 5.67, L.P .!1 // 

  : 

Let X be the set of A  !1 for which x and G do not exist satisfying (1.1) and (1.2). By  there exist t 2 R and  2 LŒt  such that (2.1)   !1  Coll.!;
t u

5.7 The Axiom

 

267

Theorem 5.89. Assume ./. Suppose that  W Œ!1 2 ! ¹0;1º; X  !1 and that A D hS˛ W ˛ < !1 i is a sequence of stationary subsets of !1 . Then (1) E .3/ ŒX  contains a closed unbounded set or E .3/ ŒX  is nonstationary, (2) E .3/ ŒX; A contains a closed unbounded set or E .3/ ŒX; A is nonstationary. u t Lemma 5.90 is in essence just Theorem 4.67 adapted to our current context. Lemma 5.90. Assume ./. Suppose  W Œ!1 2 ! ¹0;1º is a partition with no homogeneous rectangle for 0 of .proper/ cardinality @1 . Then there exist X  !1 and a sequence A D hS˛ W ˛ < !1 i of stationary sets such that A 2 LŒX  and such that E .3/ ŒX; A D ;: Proof. Fix a filter G  Pmax such that G is L.R/-generic. Let h.M0 ; I0 /; a0 i 2 G be such that 0 2 M0 and such that j0;!1 .0 / D  where h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i is the unique iteration such that j0;!1 .a0 / D AG and 0 D jŒ!1M0 2 : By Theorem 4.67 we can suppose that for all countable iterations, j W .M0 ; I0 / ! .M; I / if N is a countable, transitive, model of ZFC such that; (1.1) .P .!1 //M  N , (1.2) !1N D !1M , (1.3) Q3 .S /  N , for each S 2 N such that S  !1N , (1.4) if S  !1N , S 2 M and if S … I then S is a stationary set in N ,

268

5 Applications

then N  “ j.0 / has no homogeneous rectangle for 0, of (proper) cardinality @1 ”. Let A D hS˛ W ˛ < !1 i be an enumeration of .P .!1 //M!1 n I!1 : Since I!1 D INS \ M!1 ; for each ˛ < !1 , S˛ is a stationary subset of !1 . Let X  !1 be a set which codes M!1 . Suppose Y  H.!2 / is a countable elementary substructure with h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i 2 Y and let  D Y \ !1 . Then

¹S˛ \  j ˛ < º D .P .!1 //M n I

and X \  codes M . Suppose that N is a countable transitive model of ZFC such that !1N D  and such that X \  2 N . Then .P .!1 //M  N : Therefore by the choice of h.M0 ; I0 /; a0 i,  … E .3/ ŒX; A and so E .3/ ŒX; A D ;.

t u

There is a version of Lemma 5.90 for dealing with the existence of homogeneous sets for partitions  W Œ!1 2 ! ¹0;1º: This requires the obvious adaptation of Definition 5.86. 1 Definition 5.91 ( 2 -Determinacy). Suppose that

 W Œ!1 2 ! ¹0;1º: (1) Suppose that X  !1 . Let F .3/ ŒX  be the set of  < !1 such that there exists Z such that a) Z has ordertype , b) .˛; ˇ/ D 0 for all .˛; ˇ/ 2 Z  Z with ˛ < ˇ, c) Q3 .Z  .X \ /  jŒ2 / ¤ ;, d) M3 .Z  .X \ /  jŒ2 /   D !1 .

5.7 The Axiom

 

269

(2) Suppose that X  !1 and that A D hS˛ W ˛ < !1 i is a sequence of stationary subsets of !1 such that for each ˛ < !1 , S˛ 2 LŒX : Let F .3/ ŒX; A be the set of  < !1 such that there exists Z such that a) b) c) d) e)

Z has ordertype , .˛; ˇ/ D 0 for all .˛; ˇ/ 2 Z  Z with ˛ < ˇ, Q3 .a/ ¤ ;, M3 .a/   D !1 , for each ˛ < , S˛ \  2 M3 .a/ and S˛ \  is a stationary set within M3 .a/,

where a D Z  .X \ /  jŒ2 :

t u

The proof of Lemma 5.90 is easily modified to yield a proof of Lemma 5.92. Assume ./. Suppose  W Œ!1 2 ! ¹0;1º is a partition with no homogeneous set for 0 of cardinality @1 . Then there exist X  !1 and a sequence A D hS˛ W ˛ < !1 i of stationary sets such that A 2 LŒX  and such that F .3/ ŒX; A D ;: u t For the problem of finding homogeneous sets, Lemma 5.92 is essentially the strongest possible result. This is a consequence of the following theorem of Todorcevic. Theorem 5.93 (Todorcevic). Assume ./ and suppose S  !1 is a stationary, costationary, subset of !1 . Then there exists a partition  W Œ!1 2 ! ¹0;1º such that: (1) F O.3/ Œ; D ;; (2) For all X  !1 , F .3/ ŒX  is contains a closed unbounded set; (3) Let A D hS˛ W ˛ < !1 i be any sequence of stationary sets which contains S , then F .3/ ŒX; A is nonstationary; where O W Œ!1 2 ! ¹0;1º is the partition; .˛; O ˇ/ D 0 if and only if .˛; ˇ/ D 1.

t u

270

5 Applications

Remark 5.94. Todorcevic’s theorem is actually stronger, Theorem 5.93 is simply the version relevant to our discussion. Note that Theorem 5.93(1) asserts in effect that  t u cannot have a homogeneous set of cardinality @1 for 1. By combining Theorem 5.67 and Lemma 5.90 we obtain the next theorem. Suppose g  Coll.!;
g   Coll.!;
if g  is LŒt -generic and if 

¹S˛g j ˛ < !1 º  P .!1 / n INS ; then a)  W Œ!1 2 ! ¹0;1º,  b) E .3/  ŒX ; Ag   D ;,

where  D Ig  . / and where X   !1 is such that LŒt Œg   D LŒX  :

5.7 The Axiom

 

271

    Proof. Assuming  , this follows easily from Lemma 5.90. By Theorem 5.67,  u t holds in L.P .!1 //. The key question is the following.  Assume ./. Suppose

 W Œ!1 2 ! ¹0;1º

is a partition with no homogeneous rectangle for 0 of (proper) cardinality @1 . Must there exist a set X  !1 such that E .3/ ŒX  D ;? The point here is the following. By Lemma 5.90, if  W Œ!1 2 ! ¹0;1º is a partition with no homogeneous rectangle for 0 of (proper) cardinality @1 , and if ./ holds, then the nonexistence of the homogeneous rectangle is coupled to the stationarity of certain subsets of !1 . The question we are asking is if this is really possible, perhaps the nonexistence of the homogeneous rectangle can only be coupled to the preservation of !1 as is the case in all of the currently known examples (assuming ./). In contrast to the situation concerning homogeneous rectangles, is that of the existence of homogeneous sets. Todorcevic has proved that given a stationary set S  !1 there exists a partition S W Œ!1 2 ! ¹0;1º such that if V ŒG is a set generic extension of V such that  !1V D !1V ŒG ,  V ŒG  PFA.c/, then in V ŒG: (1) The partition S has no homogeneous set for 1 which is of cardinality !1 ; (2) The partition S has a homogeneous set for 0 of cardinality !1 if and only if the set S is nonstationary. The requirement, V ŒG  PFA.c/; can be weakened substantially. It is a version of this theorem which we state as Theorem 5.93. If the answer to the question stated above is yes, then the hypothesis of the next theorem, Theorem 5.96, can be reduced to the hypothesis of Theorem 5.95 giving a strong version of Lemma 5.90. The key difference in the statement of this theorem is that all LŒt -generic filters  are allowed, the requirement that the sets S˛g each be stationary is not necessary.

272

5 Applications

Theorem 5.96. Assume ./. Suppose  W Œ!1 2 ! ¹0;1º is a partition such that for some X  !1 , E .3/ ŒX  D ;: Then there exist t  !,   L!1  Coll.!;
g   Coll.!;
if g  is LŒt -generic then a)  W Œ!1 2 ! ¹0;1º,  b) E .3/  ŒX  D ;,

where  D Ig  . / and where X   !1 is such that LŒt Œg   D LŒX  : Proof. The theorem is a straightforward consequence of

 

in L.P .!1 //.

t u

The connection with the question of Todorcevic is given in the Theorem 5.97 and Theorem 5.98 below. We state these without giving the proofs for they require some additional machinery which is beyond the scope of this presentation, particularly in the case of Theorem 5.98. The proof of Theorem 5.97 is completely straightforward given that (in the notation of the theorem) .E .3/ ŒX /V D .E .3/ ŒX /V ŒZ which is true by an absoluteness argument. The proof of Theorem 5.98 requires some inner model theory and genericity iterations. Theorem 5.97. Suppose ı is a Woodin cardinal, there is a measurable cardinal above ı, and that  W Œ!1 2 ! ¹0;1º: Suppose Z1  !1 and Z2  !1 are cofinal sets such that

5.7 The Axiom

 

273

(i) .Z1 ; Z2 / is V -generic for a partial order P 2 Vı , (ii) .¹˛; ˇº/ D 0 for all .˛; ˇ/ 2 Z1  Z2 such that ˛ < ˇ, (iii) !1V ŒZ1 ;Z2  D !1V . Then in V , for all X  !1 , E .3/ ŒX  is stationary.

t u

Theorem 5.98. Suppose ı is a Woodin cardinal, there is a measurable cardinal above ı, and that  W Œ!1 2 ! ¹0;1º: Suppose that for all X  !1 , E .3/ ŒX  is stationary. Then for each ˛ < ı there exists a transitive inner model N and a partial order P 2 N such that the following hold. (1) Ord  N . (2) N  ZFC C ı is a Woodin cardinal. (3) V˛ 2 N . (4) Suppose G  P is N -generic. Then !1 D !1N ŒG and there exist cofinal sets Z1  !1 and Z2  !1 such that .Z1 ; Z2 / 2 N ŒG and such that .¹˛; ˇº/ D 0 for all .˛; ˇ/ 2 Z1  Z2 such that ˛ < ˇ.

t u

In Chapter 7 we shall consider Pmax -extensions of inner models other than L.R/; i. e. inner models satisfying stronger determinacy hypotheses. Using these results one can show, for example, that if ZFC C “There are ! 2 many Woodin cardinals ” is consistent then ZFC C

 

C “There are ! 2 many Woodin cardinals ”

is consistent. In particular it is consistent for ./ to hold and for there to exist a Woodin cardinal with a measurable above. Therefore by Theorem 5.97 if ./ implies that for all partitions  W Œ!1 2 ! ¹0;1º either there is a homogeneous rectangle for 0 of (proper) cardinality !1 or there exists a set X  !1 such that E .3/ ŒX  is nonstationary, then the answer to Todorcevic’s question is yes.

274

5 Applications

Remark 5.99. (1) There is no evidence to date that Todorcevic’s question involves large cardinals at all. (2) One can define other versions of E .3/ ŒX . For example define E .1/ ŒX  modifying the definition of E .3/ ŒX  by replacing M3 .a/ by M1 .a/ where for each transitive set a 2 H.!1 /, M1 .a/ D L .a/ where  is the least ordinal such that L .a/ is admissible. E .2/ ŒX  is defined using M2 .a/ D LŒa: (3) Assume ./. Suppose  W Œ!1 2 ! ¹0;1º is a partition such that for some X  !1 , E .3/ ŒX  is nonstationary.  Is E .2/ ŒX  nonstationary for some X  !1 ?  Is E .1/ ŒX  nonstationary for some X  !1 ?

5.8

t u

Homogeneity properties of P .!1 /=INS

Assume ./ holds. We shall show in Section 6.1 that it does not necessarily follow that the nonstationary ideal on !1 is !2 -saturated in V . This suggests that the structure of the quotient algebra P .!1 /=INS is necessarily somewhat complicated. The following lemma, which is well known, shows that assuming MA!1 , if I is a normal, uniform, ideal on !1 which is !2 -saturated then the boolean algebra, P .!1 /=I is rigid. Lemma 5.100 (MA!1 ). Suppose that I0 ; I1 are are normal, uniform, saturated ideals on !1 and that G0  .P .!1 / n I0 ; / is V -generic. Suppose that G1  .P .!1 / n I1 ; / is a V -generic filter such that G1 2 V ŒG0 . Then G0 D G1 .

5.8 Homogeneity properties of P .!1 /=INS

275

Proof. Fix a sequence h ˛ W ˛ < !1 i of (infinite) pairwise almost disjoint subsets of !. For each set A  !1 , let †A be the set of all  ! such that A D ¹˛ < !1 j \ ˛ is infiniteº: Since MA!1 holds, for each A  !1 , †A ¤ ; Let j0 W V ! M0  V ŒG0  be the generic elementary embedding corresponding to the generic ultrapower given by G0 and let j1 W V ! M1  V ŒG1  be the generic elementary embedding corresponding to G1 . Thus RV ŒG0   M0 and G1 2 V ŒG0 . Let  D ! V where 1

h˛ W ˛ < !1M1 i D j1 .h ˛ W ˛ < !1 i/: Thus, by the elementarity of j1 , for all A  !1V , with A 2 V , and for all 2 †A , A 2 G1 if and only if \ is infinite. However  2 M0 since RV ŒG0   M0 and G1 2 V ŒG0 . Let f 2 V be a function such that j0 .f /.!1V / D  ; i. e. a function that represents  . We can suppose that for all ˛ < !1V , f .˛/  ! and that f .˛/ is infinite. Thus for all A 2 P .!1 /V , and for all 2 †A , A 2 G1 if and only if ¹˛ j f .˛/ \ is infiniteº 2 G0 : We work in V . Let Z D ¹[¹ ˛ j ˛ 2 sº j s 2 Œ!1 
a0 \ ˇ

276

5 Applications

We return to V ŒG0 . If B0 2 G0 then ; 2 G1 since a0 2 †; , and so C0 2 G0 . Again we work in V . Define  W C0 ! Œ!1 
¹ˇ < !1 j f .˛/ \ ˇ is infiniteº

is finite and so .˛/ 2 Œ!1 
5.8 Homogeneity properties of P .!1 /=INS

277

Thus there exists an elementary embedding k0 W M1 ! M0 such that j0 D k0 ı j1 . But j1 .!1V / D !2V D j0 .!1V / and so k0 must be the identity. Therefore j0 D j1 and so G0 D G1 .

t u

A natural reformulation of Lemma 5.100 is given in the following corollary. Corollary 5.101 (MA!1 ). Suppose that I is a normal, uniform, saturated ideal on !1 and that G  .P .!1 / n I; / is V -generic. Suppose that U 2 V ŒG is a normal, uniform, V -ultrafilter on !1V . Then U D G: Proof. Let  be a term for U and fix a set S 2 G such that S  “ is a normal, uniform, V -ultrafilter”: Working in V , define J D ¹T  !1 j S  “T … ”º: Thus in V , J is a normal, uniform ideal on !1 . Since .P .!1 / n I; / is !2 -cc, J is a saturated ideal. Thus U  .P .!1 / n J; / and U is V -generic. By Lemma 5.100, U D G:

t u

By Corollary 5.101, if ./ holds then P .!1 /=INS is not homogeneous. In this section we show that if ./ holds then the boolean algebra P .!1 /=INS has a property which approximates homogeneity.  This we define below. A key point is that this property can be proved just assuming  , which is how we shall proceed.

278

5 Applications

Definition 5.102. Suppose I is a normal, uniform, ideal on !1 . The ideal I is quasihomogeneous if the following holds. Suppose that X0  P .!1 / is ordinal definable with parameters from ¹I º [ R. Suppose that there exists A0 2 X0 such that ¹A0 ; !1 n A0 º \ I D ;: Then for all A 2 P .!1 / n I if !1 n A … I there exists B 2 X0 such that A M B 2 I:

t u

Remark 5.103. (1) The condition that an ideal I is quasi-homogeneous is a very strong one, particularly when the ideal is also saturated. We note the following consequence.  Suppose that G  P .!1 / n I is V -generic and let j W V ! M  V ŒG be the associated generic elementary embedding. Suppose that B  Ord and that B is ordinal definable in V . Then for each ordinal ˛, j jL˛ ŒB 2 V: (2) It is easily seen that if I is a normal !1 dense ideal on !1 then the ideal I is not necessarily quasi-homogeneous. Combining the constructions of Qmax and M Qmax which are given in Section 6.2.1 and in Section 6.2.6, respectively, one can construct a partial order Q 2 L.R/ such that if AD holds in L.R/ and if GQ is L.R/-generic then L.R/ŒG  ZFC C “V D L.P .!1 //”; and in L.R/ŒG the following hold. a) INS is !1 dense. b) INS is not quasi-homogeneous.

t u

A key consequence of the existence of a quasi-homogeneous saturated ideal is given in  the following theorem. This seems to be the simplest route to establishing that  implies ADL.R/ , see Remark 5.111. Nevertheless the proof requires the core model induction and so is beyond the scope of this book. Theorem 5.104. Suppose that I is a saturated, normal, ideal on !1 and that I is quasi-homogeneous. Then L.R/  AD: t u   The first step in showing that  implies ./ is to establish that  implies that INS is quasi-homogeneous.

5.8 Homogeneity properties of P .!1 /=INS

Theorem 5.105.

 

279

The nonstationary ideal on !1 is quasi-homogeneous.

Proof. The theorem follows from the following claim which is an immediate corollary of Lemma 4.36. Suppose .M; I / is iterable, b  !1M , b 2 M, b … I and that !1M n b … I . Suppose S  !1 is stationary and co-stationary. Then there exists an iteration j W .M; I / ! .M ; I  / of .M; I / of length !1 such that C \ S D C \ j.b/ for some club C  !1 and such that for all d  !1 , if d 2 M n I  then d is stationary. Suppose that X0  P .!1 / n INS and that X0 is ordinal definable from x where x 2 R. We suppose that X0 is nonempty and that for all A 2 X0 , A is co-stationary. Let Z0 be the set of pairs .t;  / such that (1.1) t 2 R,   !1  Coll.!;
h.M0 ; I0 /; a0 i Pmax AG 2

where is the term for the subset of P .!1 / given by Z0 . Suppose S  !1 is stationary and co-stationary. By the claim above there exists an iteration j0 W .M0 ; I0 / ! .M0 ; I0 / of .M0 ; I0 / of length !1 such that

280

5 Applications

(2.1) C \ S D C \ j0 .a0 / for some club C  !1 , (2.2) I0 D M0 \ INS . Let A1 D j0 .a0 /. Thus A1 is stationary and co-stationary. Again by Theorem 5.81, there is a filter G1  Pmax such that G1 is HODR -generic and such that HODP .!1 / D HODR ŒG1  and such that A1 D AG1 . The embedding, j0 , witnesses that h.M0 ; I0 /; a0 i 2 G1 and so HODR ŒG1   AG1 2 XG1 ; where XG1 is the interpretation of by G1 . However AG1 D A1 and X G1  X0 ; and so A1 2 X0 . Finally by (2.1) and (2.2), S M A1 2 INS ; and this proves the theorem.

t u

Remark 5.106. An immediate corollary of Theorem 5.105 and Theorem 5.67 is that assuming ./, the nonstationary ideal is quasi-homogeneous in L.P .!1 //. This shows that MA!1 is consistent with the existence of a saturated ideal on !1 which is quasi-homogeneous. In Chapter 7, we shall improve this result, replacing t u MA!1 by Martin’s MaximumCC .c/. By Theorem 4.49, the basic analysis of the Pmax extension can be carried out just assuming that for each set B  R such that B 2 L.R/, there exists a condition h.M; I /; ai 2 Pmax such that (1) B \ M 2 M, (2) hH.!1 /M ; B \ Mi  hH.!1 /; Bi, (3) .M; I / is B-iterable.   We now prove that this in fact holds, assuming  . Our goal is to show that assum ing  , the nonstationary ideal is saturated in L.P .!1 //. By Theorem 5.104 it will   follow that  implies ADL.R/ .   We first prove that the conclusion of Lemma 4.52 follows from  .

5.8 Homogeneity properties of P .!1 /=INS

Lemma 5.107. Assume

281

 . Suppose B  R and B 2 HODR . Then the set 

¹X  hH.!2 /; B; 2i j MX is B-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Proof. By Theorem 5.81, there exists a filter G  Pmax such that G is HODR -generic and such that P .!1 / D P .!1 /G : The lemma is a straightforward consequence of this fact. This is more transparent if one reformulates it as follows. Let  H.!2 /G D [¹H.!2 /M j h.M; I /; ai 2 Gº where for each h.M; I /; ai 2 G, j  W .M; I / ! .M ; I  / is the (unique) iteration such that j.a/ D AG . Since P .!1 / D P .!1 /G ; it follows easily that H.!2 / D H.!2 /G : We now fix G. Let F W H.!2 /
282

5 Applications

Since !1  N0 , for each k < !, Nk is transitive. Since H.!2 /G D H.!2 /; there exist sequences hh.Mk ; Ik /; ak i W k < !i; hbk W k < !i; and htk W k < !i such that for all k < !, (2.1) h.Mk ; Ik /; ak i 2 G \ NkC1 \ X , (2.2) h.MkC1 ; IkC1 /; akC1 i < h.Mk ; Ik /; ak i, (2.3) bk 2 Mk , (2.4) for all p 2 Zk \ Pmax , h.MkC1 ; IkC1 /; akC1 i < p; (2.5) jk .bk / D Nk , (2.6) jk .tk / D sk , where Zk is the closure of ¹bk º under F and where jk W .Mk ; Ik / ! .Mk ; Ik / is the iteration such that jk .ak / D AG . For each k < ! let Xk D jk Œbk  D ¹jk .c/ j c 2 bk º: Thus for each k < !, Xk  X and further X D [¹Xk j k < !º: We note that for each k < !, since jk .bk / D Nk , jk .B \ bk / D B \ Nk : For each k < ! and let Dk be the set of h.M; I /; ai < h.Mk ; Ik /; ak i such that

j  .B \ bk / D B \ j  .bk /

and such that for all countable iterations j W .M; I / ! .M ; I  /

5.8 Homogeneity properties of P .!1 /=INS

283

it is the case that j.B \ j  .bk // D B \ j.j  .bk //, where j  W .Mk ; Ik / ! .Mk ; Ik / is the iteration such that j  .ak / D a. We claim that for each k 2 ! there exists q 2 G such that ¹p < q j p 2 Pmax º  Dk : Assume toward a contradiction that this fails for k. Then for all q 2 G there exists p 2 G such that p < q and p … Dk . However G is HODR -generic and so there must exist h.M; I /; ai 2 G and an iteration

j W .M; I / ! .M0 ; I 0 /

such that (3.1) h.M; I /; ai < h.Mk ; Ik /; ak i, (3.2) j.B \ j  .bk // ¤ B \ j.j  .bk // where j  W .Mk ; Ik / ! .Mk ; Ik / is the iteration such that j  .ak / D a, (3.3) h.M0 ; I 0 /; a0 i 2 G where a0 D j.a/. But this contradicts the fact that jk .B \ bk / D B \ Nk . Therefore for each k < ! there exists qk 2 G such that ¹p < qk j p 2 Pmax º  Dk : Note that Dk is definable in the structure hH.!2 /; B; G; 2i from bk . Therefore we can suppose that qk 2 Zk , for such a condition must exist in Zk . This implies that h.MkC1 ; IkC1 /; akC1 i 2 Dk : For each k < n < !, let jk;n W .Mk ; Ik / ! .Mkn ; Ikn / be the iteration such that jk;n .ak / D an and let jk;! W .Mk ; Ik / ! .Mk! ; Ik! / be the iteration such that jk;! .ak / D [¹an j n < !º:

284

5 Applications

Thus for all k < !, ! Mk! 2 MkC1 :

The key points are that MX D [¹Mk! j k < !º D [¹jk;! .bk / j k < !º: and that for each k < !, jk;! .bk / D NkX where NkX is the image of Nk under the collapsing map. These identities are easily verified from the definitions. Finally suppose jO W MX ! MO X is a countable iteration. For each k < !,

! ! ; IkC1 // jO..MkC1

is an iterate of .MkC1 ; IkC1 /. Further for each k < !, h.MkC1 ; IkC1 /; akC1 i 2 Dk : Therefore for each k < !, jO.B \ NkX / D jO.jkC1;! .B \ jk;kC1 .bk /// D B \ jO.jkC1;! .jk;kC1 .bk /// D B \ jO.NkX /: X However for each k < !, NkX is transitive and NkX 2 NkC1 . Therefore

MO X D [¹jO.NkX / j k < !º and so

jO.B \ MX / D B \ MO X : t u

Therefore MX is B-iterable. As a corollary to Lemma 5.107 we obtain that of Pmax . Theorem 5.108. Assume

 

implies the requisite nontriviality

 . Suppose B  R and that B 2 HODR . Then there exists  h.M; I /; ai 2 Pmax

such that (1) B \ M 2 M, (2) hH.!1 /M ; B \ Mi  hH.!1 /; Bi, (3) .M; I / is B-iterable.

5.8 Homogeneity properties of P .!1 /=INS

285

Proof. Fix G  Pmax such that G is HODR -generic and such that P .!1 /G D P .!1 /: Suppose  2 Ord,

L .B; R/ŒG  ZFC ;

and that  < ‚L.B;R/ : By Corollary 5.80, L .B; R/ŒG  MA!1 : Let A  R be such that A 2 HODR and such that  k. Thus jk is uniquely specified by the requirement that jk .ak / D [¹ai j i < !º: For each k < !

pk D h.Mk ; Ik ; Jk /; ak i:

We note that by Corollary 4.20, the sequence h.Mk ; ¹Ik ; Jk º/ W k < !i is iterable (in the sense of Definition 4.8). Lemma 6.7 (ZFC ). Suppose I  J are normal uniform ideals on !1 such that I ¤ J . Suppose hpk W k < !i is a sequence of conditions in 2 Pmax such that for

6.1

2P

max

293

each k < ! pkC1 < pk . Let hpk W k < !i be the associated sequence of 2 Pmax conditions and for each k < !, let h.Mk ; Ik ; Jk /; ak i D pk : Then there is an iteration j W h.Mk ; ¹Ik ; Jk º/ W k < !i ! h.Mk ; ¹Ik ; Jk º/ W k < !i M

such that j.!1 0 / D !1 and such that for all k < !, Ik D I \ Mk and

Jk D J \ Mk :

Proof. By Corollary 4.20 the sequence h.Mk ; ¹Ik ; Jk º/ W k < !i is iterable. The lemma follows by an argument similar to that used to prove Lemma 6.6.

t u

The next lemmas record some of the relevant properties of the partial order 2 Pmax . First we note that the nontriviality of Pmax immediately gives the nontriviality of 2 Pmax . Lemma 6.8. Assume that for each set X  R with X 2 L.R/, there is a condition h.M; I /; ai 2 Pmax such that (i) X \ M 2 M, (ii) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (iii) .M; I / is X -iterable. Then for each set X  R with X 2 L.R/, there is a condition h.M; I; J /; ai 2 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; ¹I; J º/ is X -iterable. Proof. This is immediate by the following observation. Suppose h.M; I /; ai 2 Pmax : !1M

Let S  be such that S 2 M, S … I , and !1M n S … I . Let J 2 M be the ideal generated by I [ ¹S º. Then h.M; I; J /; ai 2 2 Pmax . The point is that any iteration of .M; ¹I; J º/ is an iteration of .M; I /. t u

294

6 Pmax variations

As a corollary we obtain the following lemma. Lemma 6.9. Assume ADL.R/ . Suppose X  R and that X 2 L.R/. Then there is a condition h.M; I; J /; ai 2 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; ¹I; J º/ is X -iterable. Proof. This is immediate by the previous lemma and Lemma 4.40.

t u

Remark 6.10. The analysis of 2 Pmax can be carried out abstractly just assuming: For each set X  R with X 2 L.R/, there is a condition h.M; I; J /; ai 2 2 Pmax such that (1) X \ M 2 M, (2) hH.!1 /M ; X \ Mi  hH.!1 /; X i, (3) .M; ¹I; J º/ is X -iterable.

t u

Suppose that I is a normal ideal on !1 and that S 2 P .!1 / n I . We let I jS denote the normal ideal generated by I [ ¹!1 n S º. We define an operation on normal ideals. Definition 6.11. Suppose that I is a normal ideal on !1 and that I is not saturated. Let t u sat.I / D ¹S 2 P .!1 / j S 2 I or I jS is a saturated idealº: Lemma 6.12. Suppose that I is a normal ideal on !1 and that I is not saturated. Then sat.I / is a normal ideal on !1 . Proof. This lemma is an elementary consequence of the definition of sat.I /.

t u

Theorem 6.13. Suppose that INS is saturated or that sat.INS / is saturated. Then INS is semi-saturated. Proof. Clearly we may suppose that INS is not saturated and so sat.INS / is saturated. Suppose V ŒG is a generic extension of V and that U 2 V ŒG is a V -normal ultrafilter on !1V . If U  .P .!1 / n sat.INS //V

6.1

2P

max

295

then U is V -generic since .sat.INS //V is saturated in V . In this case Ult.V; U / is wellfounded. Therefore we may suppose that U 6 .P .!1 / n sat.INS //V : Thus for some S 2 .P .!1 / n INS /V ; S 2 U and

S 2 .sat.INS //V :

Necessarily .INS jS /V is saturated in V and so U is V -generic. Therefore again we have that Ult.V; U / is wellfounded. This proves the theorem. u t Assume ADL.R/ and suppose G  2 Pmax is L.R/-generic. Then as in the case for Pmax the generic filter G can be used to define a subset of !1 and we denote it by AG . Thus AG D [¹a j h.M; I; J /; ai 2 G for some M; I º: However now the generic filter can also be used to define two ideals which we denote by IG and JG . For each h.M; I; J /; ai 2 G there is an iteration j W .M; ¹I; J º/ ! .M ; ¹I  ; J  º/ such that j.a/ D AG . This iteration is unique because M is a model of MA!1 . Let IG D [¹I  j h.M; I; J /; ai 2 Gº; let and let

JG D [¹J  j h.M; I; J /; ai 2 Gº; 

P .!1 /G D [¹P .!1 /M j h.M; I; J /; ai 2 Gº:

The next lemma gives the basic analysis of 2 Pmax . It shows that IG is the nonstationary ideal, JG is a saturated ideal in L.R/ŒG and JG D sat.IG /. This implies that the ideal IG is presaturated in a very strong sense. Recall that a normal ideal I on !1 is presaturated if for any sequence hAi W i < !i of antichains of P .!1 / n I and for any A 2 P .!1 / n I , there exists B  A such that B … I and such that for each i < !, j¹X 2 Ai j X \ B … I ºj  !1 : Lemma 6.14. Assume ADL.R/ . Then 2 Pmax is !-closed and homogeneous. Suppose G  2 Pmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG:

296

6 Pmax variations

(1) P .!1 /G D P .!1 /; (2) IG is the nonstationary ideal; (3) JG is a normal saturated ideal on !1 ; (4) JG is nowhere the nonstationary ideal; i. e. for all stationary sets S  !1 there exists a stationary set T  S such that T 2 JG ; (5) JG D sat.IG /. Proof. The proof that 2 Pmax is !-closed is immediate by applying Lemma 6.7 within the relevant condition. With the possible exception of (3)–(5) the remaining claims are proved by simply adapting the proofs of the corresponding claims for Pmax , using Lemma 6.6, Lemma 6.8, and Lemma 6.9. (3) is proved following the proof that the nonstationary ideal is saturated in L.R/Pmax . (4) follows by an easy density argument. (5) is also proved by using the proof that INS is saturated in the Pmax -extension. The relevant observation is that one can seal antichains corresponding to IG on sets t u which are IG -positive and in JG . Part (4) of the previous lemma provides another example of how forcing notions like Pmax can be devised to achieve something from very weak approximations. There is a dense set of conditions h.M; I; J /; ai 2 2 Pmax such that ideals I; J differ in a trivial way. J is obtained from I by adding one set. This is how we argued for the nontriviality of 2 Pmax given the nontriviality of Pmax . However in the generic extension the ideal JG is not trivially different from the ideal IG ; it is nowhere equal to IG . One consequence of the next lemma is that if G  2 Pmax is L.R/-generic then L.R/ŒG  AC and so L.R/ŒG  ZFC: One can show directly that L.R/ŒG  AC and we shall do this for the remaining variations of Pmax that we shall define. Lemma 6.15. Assume ADL.R/ . Suppose G  L.R/ŒG: (1) AG is L.R/-generic for Pmax ; (2) P .!1 /  L.R/ŒAG .

2

Pmax is L.R/-generic. Then in

6.1

2P

max

297

Proof. In L.R/ŒG let F be the set of h.M; I /; ai 2 Pmax such that there exists an iteration j W .M; I / ! .M ; I  / such that j.a/ D AG and such that I  D INS \ M : By Lemma 6.14(1), in L.R/ŒG every club in !1 contains a club which is constructible from a real. Therefore by Theorem 3.19, if X  H.!2 / is a countable elementary substructure then the transitive collapse of X is iterable. Thus by Lemma 4.74, the conditions in F are pairwise compatible in Pmax . Therefore we have only to show that F \D ¤; for all D  Pmax such that D is dense and D 2 L.R/. Suppose h.M; I; J /; ai 2 G and that D  Pmax is an open, dense set with D 2 L.R/. Let h.M0 ; I0 /; a0 i 2 Pmax be such that M 2 H.!1 /M0 : Let B0  !1M0 be a set in M0 such that both B0 and !1M0 are I0 -positive. Let J0 2 M0 be the uniform normal ideal on !1M0 defined by I0 [ ¹B0 º. By Lemma 6.6 there exists an iteration j W .M; ¹I; J º/ ! .M ; ¹I  ; J  º/  such that j 2 M0 , I D I0 \ M , and J  D J0 \ M . Thus h.M0 ; I0 /; j.a/i 2 Pmax . Let h.M1 ; I1 /; a1 i 2 D be a condition such that h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i and let j0 W .M0 ; I0 / ! .M0 ; I0 / be the iteration such that j0 .j.a// D a1 . Thus h.M1 ; I1 ; J1 /; a1 i 2 2 Pmax and h.M1 ; I1 ; J1 /; a1 i < h.M0 ; I0 /; a0 i where J1 2 M1 is the (normal) ideal on !1M1 defined by I1 [ ¹j0 .B0 /º. Note that B0 and !1M0 n B0 are I0 -positive, and so j0 .B0 / and !1M1 n j0 .B0 / are I1 -positive. By genericity we may suppose that h.M1 ; I1 ; J1 /; a1 i 2 G: But then h.M1 ; I1 /; a1 i 2 F and so F \D ¤; for all D  Pmax such that D is dense and D 2 L.R/. This proves (1). The second claim of the lemma follows from Lemma 6.14(1). t u

6 Pmax variations

298

The next lemma gives the basic relationship between Pmax and 2 Pmax . Lemma 6.16. Assume ADL.R/ . Suppose G  L.R/ŒG:

2

Pmax is L.R/-generic. Then in

(1) L.P .!1 // is a generic extension of L.R/ for Pmax ; (2) sat.INS / is saturated; (3) There is a filter H0  PNS such that H0 is L.P .!1 //-generic, such that L.P .!1 //ŒH0  D L.R/ŒG and such that IH0 D JG D sat.INS /: Proof. (1) is immediate from Lemma 6.15. (2) follows from Lemma 6.14. Let g  Pmax be the L.R/-generic filter such that AG D Ag . Let PNS be the partial order PNS as computed in L.R/Œg. Conditions are pairs .X; S / such that (1.1) X  P .!1 /, (1.2) jX j  !1 , (1.3) S and !1 n S are stationary. The order is given by .X1 ; S1 /  .X0 ; S0 / if X0  X1 , S0  S1 and if .INS _ S1 / \ X0 D .INS _ S0 / \ X0 where for each S  !1 such that !1 n S is stationary, INS _ S is the normal ideal generated by INS [ ¹S º. Suppose h0  PNS is L.R/Œg-generic. In L.R/ŒgŒh0  define G0  .2 Pmax /L.R/ as follows. h.M; I; J /; ai 2 G0 if h.M; I /; ai 2 g and if for some d 2 M the following two conditions are satisfied. (2.1) J is the ideal generated by I [ ¹d º. (2.2)



..P .!1 //M ; j.d // 2 h0 where j W .M; I / ! .M ; I  / is the iteration such that j.a/ D Ag .

6.1

2P

max

299

We prove that G0 is an L.R/-generic filter for 2 Pmax . Suppose .X0 ; S0 / 2 h0 and that D  2 Pmax is open, dense with D 2 L.R/. Since jX0 j  !1 , there exist h.M0 ; I0 /; a0 i 2 g and .t0 ; b0 / 2 .PNS /M0 M0 such that I0 D INS and such that

j0 ..t0 ; b0 // D .X0 ; S0 / where

j0 W .M0 ; I0 / ! .M0 ; I0 /

is the iteration such that j0 .a0 / D Ag . We work in L.R/. Let h.M1 ; I1 /; a1 i 2 Pmax be such that h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i M1 and such that I1 D INS . Let J0 be the normal ideal in M0 generated by I0 [ ¹b0 º. Let .t1 ; b1 / be the image of .t0 ; b0 / under the iteration of .M0 ; I0 / which sends a0 to a1 . Let J0 be the image of J0 under this iteration. Thus

.t1 ; b1 / 2 .PNS /M1 and b1 … I1 . In M1 , let J1 be the normal ideal on !1M1 generated by I1 [ ¹b1 º. Thus and J1 \ t1 D J0 \ t1 . J1 ¤ I1 and so h.M1 ; I1 ; J1 /; a1 i 2 2 Pmax . Therefore there exists h.M2 ; I2 ; J2 /; a2 i 2 D such that h.M2 ; I2 ; J2 /; a2 i < h.M1 ; I1 ; J1 /; a1 i and such that J2 is the ideal defined by I2 [ ¹d0 º for some d0 2 .P .!1 //M2 : By Lemma 6.6, the set of conditions h.M; I; J /; ai 2 2 Pmax such that J is obtained by adding a single set to I is dense. Thus h.M2 ; I2 ; J2 /; a2 i exists. Let 1 W .M; ¹I1 ; J1 º/ ! .M1 ; ¹I1 ; J1 º/ be the iteration which sends a1 to a2 . An important point is the following. Since J1 is the ideal in M1 generated by I1 [ ¹b1 º,  is also an iteration of .M1 ; I1 /. Thus h.M2 ; I2 /; a2 i < h.M1 ; I1 /; a1 i: By genericity we may assume that h.M2 ; I2 /; a2 i 2 g: Let b2 D .b1 /, let t2 D .t1 / and let d1 D d0 [ b2 . Since J1 D M1 \ J2

6 Pmax variations

300

it follows that b2 2 J2 and

J2 \ t2 D J1 \ t2 :

Therefore .t2 ; d1 /  .t2 ; b2 / in .PNS /M2 . Let j2 W .M2 ; I2 / ! .M2 ; I2 / be the iteration such that j2 .a2 / D Ag and let .X1 ; S1 / D j2 ..t2 ; d1 //. In L.R/Œg, .INS _ S0 / \ X0 D .INS _ S1 / \ X0 and so .X1 ; S1 /  .X0 ; S0 / in PNS . By genericity we may assume that .X1 ; S1 / 2 h0 . This implies that h.M2 ; I2 ; J2 /; a2 i 2 G0 and so G0 \ D ¤ ;. It remains to show that G0 is a filter. Since G0 \ D ¤ ; for each dense set D  2 Pmax , it suffices to show that elements of G0 are pairwise compatible in 2 Pmax . Suppose h.M0 ; I0 ; J0 /; a0 i 2 G0 and h.M1 ; I1 ; J1 /; a1 i 2 G0 : Let d0 2 M0 be such that J0 is the ideal in M0 generated by I0 [ ¹d0 º and similarly let d1 2 M0 be such that J1 is the ideal in M1 generated by I1 [ ¹d1 º. Let j0 W .M0 ; I0 / ! .M0 ; I0 / and

j1 W .M1 ; I1 / ! .M1 ; I1 /

be the iterations such that j0 .a0 / D AG D j1 .a1 /. Since J0 is generated from I0 by adding one set, j0 is also an iteration of .M0 ; ¹I0 ; J0 º/ and similarly j1 is an iteration of .M1 ; ¹I1 ; J1 º/. Let B0 D j0 .d0 / and let B1 D j1 .d1 /. Thus j0 .J0 / D .INS _ B0 / \ M0 and

j1 .J1 / D .INS _ B1 / \ M1 : Let S0 D B0 , S1 D B1 ,

and let

X0 D P .!1 / \ M0 X1 D P .!1 / \ M1 :

Since h.M0 ; I0 ; J0 /; a0 i 2 G0 , it follows .X0 ; S0 / 2 h0 :

6.1

2P

max

301

Similarly .X1 ; S1 / 2 h0 . Let S D S0 [ S1 and let X D X0 [ X1 . Since h0 is a generic filter, .X; S / 2 h0 . Let h.M2 ; I2 /; a2 i 2 g be such that .M0 ; M1 / 2 H.!1 /M2 and such that I2 D .INS /M2 . Thus j0 2 M2 and j1 2 M2 where j2 W .M2 ; I2 / ! .M2 ; I2 / is the iteration such that j2 .a2 / D AG . Let k0 W .M0 ; I0 / ! .MO 0 ; IO0 / and

k1 W .M1 ; I1 / ! .MO 1 ; IO1 /

be the iterations such that k0 .a0 / D a2 D k1 .a1 /. Thus k0 2 M2 and j0 D j2 .k0 /: Similarly j1 D j2 .k1 /. Let b0 D k0 .d0 /, b1 D k1 .d1 /, O

y0 D .P .!1 //M0 and let

O

y1 D .P .!1 //M1 :

Let b D b0 [ b1 and let y D y0 [ y1 . Note that (3.1) j2 ..y0 ; b0 // D .X0 ; S0 /, (3.2) j2 .y1 ; b1 / D .X1 ; S1 /, (3.3) j2 .y; b/ D .X; S /. Now both .X0 ; S0 /  .X; S / and .X1 ; S1 /  .X; S / in PNS . Therefore both M2 . .y; b/  .y0 ; b0 / and .y; b/  .y1 ; b1 / in PNS Let J2 be the normal ideal in M2 generated by I2 [ ¹bº. Thus J2 \ MO 0 D JO0 and J2 \ MO 1 D JO1 where JO0 D k0 .J0 / and JO1 D k1 .J1 /. Finally .MO 0 ; ¹IO0 ; JO0 º/ is an iterate of .M0 ; ¹I0 ; J0 º/ as witnessed by k0 and similarly .MO 1 ; ¹IO1 ; JO1 º/ is an iterate of .M1 ; ¹I1 ; J1 º/ as witnessed by k1 . This is because in M0 , J0 is the normal ideal generated by I0 [ ¹d0 º and because in M1 , I1 is the normal ideal generated by J1 [ ¹d1 º. Thus h.M2 ; I2 ; J2 /; a2 i 2 2 Pmax , h.M2 ; I2 ; J2 /; a2 i < h.M0 ; I0 ; J0 /; a0 i; and h.M2 ; I2 ; J2 /; a2 i < h.M1 ; I1 ; J1 /; a1 i: Therefore h.M0 ; I0 ; J0 /; a0 i and h.M1 ; I1 ; J1 /; a1 i are compatible and so G0 is L.R/-generic.

6 Pmax variations

302

By the genericity of G0 and its definition it follows that L.R/ŒgŒh0  D L.R/ŒG0 : By the homogeneity of 2 Pmax it follows that there exists h  PNS such that h is L.R/Œg-generic and such that L.R/ŒgŒh D L.R/ŒG: Finally since PNS has cardinality !2 in L.R/Œg and since .P .!1 //L.R/Œg D .P .!1 //L.R/ŒG ; PNS is .!1 ; 1/-distributive in L.R/Œg.

t u

As an immediate corollary to Lemma 6.16 we obtain the following theorem. Theorem 6.17. Assume ./ and that V D L.P .!1 //. Then PNS is .!1 ; 1/-distributive. Further, suppose G  PNS is V -generic. Then in V ŒG; (1) IG is a normal saturated ideal, (2) IG D sat.INS /.

t u

There are absoluteness theorems for 2 Pmax analogous to those for Pmax . The proofs are straightforward modifications of those for the Pmax theorems. We prove the absoluteness theorem for 2 Pmax which corresponds to Theorem 4.64. The proof is quite similar to that of Theorem 4.64. Of course in this theorem the ideal I could be simply the nonstationary ideal. Theorem 6.18. Assume ADL.R/ and that there is a Woodin cardinal with a measurable above. Suppose  is a …2 sentence in the language for the structure hH.!2 /; I; J; 2i where I  J are normal uniform ideals on !1 and I ¤ J . Suppose hH.!2 /; I; J; 2i  : Then

2P max

hH.!2 /; INS ; JG ; 2iL.R/

 :

Proof. Let .x; y/ be a †0 formula such that  D 8x9y: .x; y/ (up to logical equivalence). Assume towards a contradiction that hH.!2 /; IG ; JG ; 2i

2P max

 ::

Then by Lemma 6.14, there is a condition h.M0 ; I0 ; J0 /; a0 i 2 2 Pmax and a set b0 2 H.!2 /M0

6.1

such that if

2P

max

303

h.M0 ; I0 ; J0 /; a0 i  h.M0 ; I0 ; J0 /; a0 i

then



hH.!2 /M0 ; 2; I0 ; J0 i  8y Œb0 ;

where b0 D j.b0 / and where j W .M0 ; ¹I0 ; J0 º/ ! .M0 ; ¹I0 ; J0 º/ is the iteration such that j.a0 / D a0 . By Lemma 4.36, there is an iteration j W .M0 ; ¹I0 ; J0 º/ ! .M0 ; ¹I0 ; J0 º/ such that: (1.1) j.!1M0 / D !1 ; (1.2) I0 \ M0 D I0 ; (1.3) J0 \ M0 D J0 . Let B0 D j.b0 /. The sentence  holds in V and so there exists a set D0 2 H.!2 / such that hH.!2 /; 2; I; J i  : ŒB0 ; D0 : Let ı be a Woodin cardinal and be a measurable cardinal above ı. Suppose that G  Coll.!1 ; P .!1 // is V -generic. Let hS˛ W ˛ < !1 i be an enumeration of I in V ŒG, and let hT˛ W ˛ < !1 i be an enumeration of J . Let S D ¹˛ < !1 j ˛ 2 Sˇ for some ˇ < ˛º and let T D ¹˛ < !1 j ˛ 2 Tˇ for some ˇ < ˛º: The key points are the following. We work in V ŒG. First S is co-stationary and .INS _ S /V ŒG \ V D I: This is easily verified by an analysis of terms in V ; since I is a normal ideal in V , for each set A  !1 such that A 2 V and A … I , A \ .!1 n S /

304

6 Pmax variations

is a stationary subset of !1 . This follows from the observation that in V , for each set A 2 P .!1 / n I , ZA  P!1 .H.!2 // is stationary in P!1 .H.!2 // where ZA is the set of countable X  H.!2 / such that X \ !1 2 A and such that for all B 2 X \ I , X \ !1 … B. If B 2 I then B \ .!1 n S / is countable. Similarly T is co-stationary and .INS _ T /V ŒG \ V D J Since I  J , it follows that .INS _ S /V ŒG  .INS _ T /V ŒG ; and since I ¤ J ,

.INS _ S /V ŒG ¤ .INS _ T /V ŒG :

V ŒG is a small generic extension of V and so ı is a Woodin cardinal in V ŒG and is measurable in V ŒG. Let Q D Coll.!1 ; t u

The next two lemmas give useful reformulations of ˘.!1
324

6 Pmax variations

It suffices to prove that there exists a countable elementary substructure, X  H.!2 / such that H ŒX   X and such that f .˛/ is MX -generic. Let N  H.!2 / be an elementary substructure of cardinality @1 such that
!1  N and such that H ŒN   N . Clearly N is transitive. Let hD˛ W ˛ < !1 i enumerate the dense subsets of Coll.!; !1 / which belong to N. Let S D ¹˛ < !1 j f .˛/ \ Dˇ ¤ ; for all ˇ < ˛º:
Since f witnesses ˘.!1
6.2 Variations for obtaining !1 -dense ideals

325

Lemma 6.41. Suppose that for every A  !1 , A# exists and that f W !1 ! H.!1 / is a function such that for all ˛ < !1 , f .˛/ is a filter in Coll.!; ˛/. (1) Suppose f witnesses ˘.!1
t u

Related to the reformulations of ˘.!1
326

6 Pmax variations

Lemma 6.43. Suppose that for every A  !1 , A# exists and that f W !1 ! H.!1 / is a function such that for all ˛ < !1 , f .˛/ is a filter in Coll.!; ˛/. Then following are equivalent. (1) f witnesses ˘.!1
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327

(5) for all ˛ 2 c with ˛ a limit point of c, f .˛/ is a filter in Coll.!; ˛/ and f .˛/ is LŒAh \ ˛  ˛; c \ ˛ generic for Coll.!; ˛/ where Ah D ¹.ˇ;  / j  2 h.ˇ/º: The order on P .f / is given by extension: .h2 ; c2 /  .h1 ; c1 / if h1  h2 , c1  c2 and

c2 \ .˛ C 1/ D c1

where ˛ D [c1 . Suppose  is an ordinal and f W !1 ! H.!1 / is a function. Let P .f;  / denote the countable support iteration where for all ˛ <  , P .f; ˛ C 1/ D P .f; ˛/  P .f / and P .f / is as computed in V P .f;˛/ . We note that P .f;  / is not in general a semiproper partial order. Lemma 6.44. Suppose that for all A  !1 , A# exists, and that f W !1 ! H.!1 / is a function which witnesses ˘.!1
328

6 Pmax variations

Let Y D X \ V and let MY be the transitive collapse of Y . Thus MY 2 L.MX /: Let PY be the image of P .f; / under the collapsing map. Let F W ! ! MY be a surjection such that g is L.MY ; F /-generic where g D f .Y \ !1 /. To see that F exists let z 2 R code the pair .MY ; g/. z # exists and so !1 is inaccessible in LŒz. Therefore there exists a filter G  Coll.!; MY / such that G is LŒz-generic. Let F be the function determined by G. Thus F is a surjection and further g is L.MY ; F /-generic. Let GY be MY -generic for PY with GY 2 L.MY ; F /. Choose GY such that pY 2 GY where pY is the image of p under the collapsing map. Let  W Y ! V be the inverse of the collapsing map. It follows that there is a condition q 2 P .f; / such that q < .p/ for all p 2 GY . The relevant points are that GY 2 L.MY ; F / and that f .Y \ !1 / is L.MY ; F /-generic for Coll.!; MY \ !1 /.

t u

Lemma 6.45. Suppose f W !1 ! H.!1 / is V -generic for Coll.!1 ; H.!1 //. Then f witnesses ˘.!1
t u

Lemma 6.46. Suppose 0 is 1 huge. Suppose V ŒG0 Œf ŒG1  is a generic extension of V such that (i) G0 is V -generic for Coll.!; < 0 /, (ii) f is V ŒG0 -generic for Coll.!1 ; H.!1 // as computed in V ŒG0 , (iii) G1 is V ŒG0 Œf -generic for P .f; 1 / as computed in V ŒG0 Œf . Then in V Œf; G1 , ˘ holds, f witnesses ˘++ .!1
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329

Proof. It is straightforward to show that the sequence h ˛ W ˛ < !1 i witnesses ˘ in V Œf ŒG1  where for each ˛ < !1 , ˛ D f .˛ C 1/: The lemma now follows by a straightforward modification of the proof of Theorem 6.28. t u Lemma 6.47 gives a property of Qmax which is a consequence of the existence of (suitable) large cardinals and yet which cannot be proved simply assuming ADL.R/ . The difficulty is (4). Let ZFC be ZFC together with a finite fragment of ZFC. Lemma 6.47. Suppose there is an huge cardinal. Then for every set A  R with A 2 L.R/ there is a condition h.M; I /; f i 2 Qmax such that (1) A \ M 2 M, (2) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i, (3) .M; I / is A-iterable, (4) ˘ holds in M, (5) f witnesses ˘++ .!1
330

6 Pmax variations

Proof. Let f W !1 ! H.!1 / be a function which witnesses ˘.!1
6.2 Variations for obtaining !1 -dense ideals

331

containing Proof. (1) follows from (2). We prove (2). Suppose G  Qmax is L.R/-generic. We work in L.R/ŒG. (2) is equivalent to the assertion that for each A  !1 there exists an LŒA-generic Cohen real. Suppose that A 2 L.R/ŒG and that A  !1 . We prove that there exists an LŒAgeneric Cohen real. By Theorem 6.30(1), there exists h.M0 ; I0 /; f0 i 2 G and a0 2 M0 such that j0 .a0 / D A where j0 W .M0 ; I0 / ! .M0 ; I0 / is the iteration such that j0 .f0 / D fG . Let h.M1 ; I1 /; f1 i 2 Qmax be such that h.M1 ; I1 /; f1 i < h.M0 ; I0 /; f0 i and let

k0 W .M0 ; I0 / ! .MO 0 ; IO0 /

be the iteration in M1 such that k0 .f0 / D f1 . Let c be a real which is Cohen generic over M1 and let I1.c/ be the normal ideal generated by I1 in M1 Œc. It is easily verified that M1 Œc  ZFC . By Lemma 6.52, it follows that h.M1 Œc; I1.c/ /; f1 i 2 Qmax noting that if k W .M1 Œc; I1.c/ / ! .k.M1 Œc/; k.I1.c/ // is an iteration of .M1 Œc; I1.c/ / then kjM1 W .M1 ; I1 / ! .k.M1 /; k.I1 // is an iteration of .M1 ; I1 /. Therefore .M1 Œc; I1.c/ / is iterable. By genericity we may assume that h.M1 Œc; I1.c/ /; f1 i 2 G: Since c is Cohen over M1 it follows that c is Cohen generic over .LŒk0 .a0 //M1 . Let j1 W .M1 Œc; I1.c/ / ! .M1 Œc; .I1.c/ / / be the iteration such that j1 .f1 / D fG .  Therefore c is Cohen generic over .LŒj1 .k0 .a0 ///M1 . However j1 .k0 .a0 // D j0 .a0 / D A and by condensation, 

.R/LŒA D R \ .LŒA/M1 : Thus c is Cohen generic over LŒA.

t u

6 Pmax variations

334

6.2.2 Qmax We define the variant of Qmax for which the analysis can be carried out assuming just ADL.R/ . The modification is obtained by replacing the model M in a condition with an ! sequence of models. With this we can improve Theorem 6.32 to obtain the consistency of ZFC C “The nonstationary ideal on !1 is !1 dense” from simply the consistency of ZF C AD. This is best possible. The definition of Qmax is motivated by the proof that Qmax is !-closed and the  definition is closely related both to Definition 4.15 and to the definition of Pmax . In fact   there is a dense subset of Qmax which is a suborder of Pmax . Suppose hpk W k < !i is a sequence of conditions in Qmax such that for all k < !, pkC1 < pk . Suppose that for each k < !, pk D h.Mk ; Ik /; fk i: Let

ı D sup¹.!1 /Mk j k < !º

and let f D [¹fk j k < !º. For each k < ! let .Mk ; Ik / be the image of .Mk ; Ik / under the iteration of .Mk ; Ik / which sends fk to f . It follows that iterations of hMk W k < !i in the sense of Definition 4.15, correspond to iterations of h.Mk ; Ik / W k < !i in the sense of Definition 4.8. The relevant point is that for each k < !, 

Ik D .INS /MkC1 \ Mk : Thus if

G  [¹P .ı/ \ Mk j k < !º

is a filter such that G \ Mk is Mk -normal for all k < !, then for all k < !, G \ Mk is Mk -generic. The same point applies to iterates of hMk W k < !i. Therefore hMk W k < !i is iterable. Definition 6.54. Qmax is the set of pairs .hMk W k < !i; f / such that the following hold. (1) f 2 M0 and

f W !1M0 ! M0

is a function such that for all ˛ < !1M0 , f .˛/ is a filter in Coll.!; ˛/. (2) Mk  ZFC .

6.2 Variations for obtaining !1 -dense ideals Mk

(3) Mk 2 MkC1 ; !1 (4) .INS /

MkC1

MkC1

D !1

.

\ Mk D .INS /

MkC2

335

\ Mk .

(5) hMk W k < !i is iterable. (6) For each p 2 Coll.!; !1M0 /, ¹˛ < !1M0 j p 2 f .˛/º … .INS /M1 : (7) Suppose that a  !1M0 , k 2 ! and that a 2 Mk n .INS /MkC1 : Then there exists such that

p 2 Coll.!; !1M0 /

¹˛ < !1M0 j p 2 f .˛/º \ .!1M0 n a/ 2 .INS /MkC1 :

The ordering on Qmax is analogous to Qmax . A condition .hNk W k < !i; g/ < .hMk W k < !i; f / if hMk W k < !i 2 N0 , hMk W k < !i is hereditarily countable in N0 and there exists an iteration j W hMk W k < !i ! hMk W k < !i such that: (1) j.f / D g; (2) hMk W k < !i 2 N0 and j 2 N0 ; 

(3) .INS /MkC1 D .INS /N1 \ Mk for all k < !.

t u

As in the definition of the order on Qmax , clause (3) in the definition of the order on Qmax follows from clauses (1) and (2). The next lemma clarifies the effect of (6) and (7) in Definition 6.54. Lemma 6.55. Suppose that .hMk W k < !i; f / 2 Qmax . (1) Suppose that

j W hMk W k < !i ! hMk W k < !i

is an iteration of length 1. Then j.f /.!1M0 /  Coll.!; !1M0 / and j.f /.!1M0 / is a filter which is generic relative to [¹Mk j k < !º. (2) Suppose that

g  Coll.!; !1M0 /

is a filter which is generic relative to [¹Mk j k < !º. Then there is a unique iteration j W hMk W k < !i ! hMk W k < !i of length 1 such that g D j.f /.!1M0 /.

6 Pmax variations

336

Proof. The lemma is an immediate consequence of the definitions.

t u

Definition 6.56. Suppose .hMk W k < !i; f / 2 Qmax and suppose X  R. Then hMk W k < !i is X -iterable if (1) X \ M0 2 [¹Mk j k < !º, (2) for any iteration j W hMk W k < !i ! hNk W k < !i of hMk W k < !i, j.X \ M0 / D X \ N 0 .

t u

If Qmax is sufficiently nontrivial then Qmax and Qmax are equivalent as forcing notions. More precisely if for every real x there exists a condition h.M; I /; f i 2 Qmax such that x 2 M and such that I D .INS /M then RO.Qmax / Š RO.Qmax /: The proof of this is implicit in what follows. We shall need a slight variant of iterability. Definition 6.57. Let A  R and .M; I; ı/ 2 H.!1 / be such that (i) M is a transitive model of ZFC, (ii) ı is a Woodin cardinal in M and I D .I M Œg  B0 \ MZ Œg 2  WH Z ;

C MZ / . where  D .ıZ C Thus in MZ Œg every set projective in A \ MZ Œg is ıZ -weakly homogeneously Suslin. Let 2 MZ Œg be the least strongly inaccessible cardinal above ıZ . By standard arguments, ıZ is a Woodin cardinal in MZ Œg. Let Y  .MZ Œg/ be an elementary substructure such that A \ MZ Œg 2 Y , Y 2 MZ Œg and Y is countable in MZ Œg. Let N be the transitive collapse of Y and let I be the image of .I t u

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351

Proof. Fix A  R such that A 2 L.R/. By Lemma 6.58 and Lemma 6.67 there exists a countable, A-iterable, structure .N; I/ such that (1.1) N  ZFC C ˘ C ˘++ .!1
Here we use Lemma 6.66 to show that Gˇ exists as required. Let S D ¹ˇ < !1M0 j f .ˇ/ D j0;ˇ C1 .f0 /.ˇ/º: Since .hMk W k < !i; f / 2 Qmax , it follows that !1M0 n S 2 .INS /M1 : Let

j W .N; I/ ! .N  ; I  /

be the limit embedding of the iteration. Let f  D j.f0 /.  Thus !1N D !1M0 and S D ¹ˇ < !1M0 j f .ˇ/ D f  .ˇ/º: Let M0 D N  and for each k > 0 let Mk D Mk .

6 Pmax variations

352

Since N  2 M0 ,

.INS /M1 \ N  D .INS /M2 \ N  ;

and so for all k 2 !, 



.INS /MkC1 \ Mk D .INS /MkC2 \ Mk : Thus

.hMk W k < !i; f  / 2 Qmax t u

and is as required.

With Theorem 6.65 the analysis of Qmax can easily be carried out as in the case of Qmax . We summarize the results of this in the next two theorems. First we prove the main iteration lemmas for conditions in Qmax . Lemma 6.69. Suppose .hMk W k < !i; f / 2 Qmax . Suppose j  W hMk W k < !i ! hMk W k < !i and

j  W hMk W k < !i ! hMk W k < !i

are iterations such that j  .f / D j  .f /. Then hMk W k < !i D hMk W k < !i and j  D j  . Proof. The proof is identical to the proof of Lemma 6.22 which is the corresponding lemma for Qmax . To illustrate we examine an iteration k W hMk W k < !i ! hMO k W k < !i of length 1 so that k corresponds to a weakly generic ultrapower. Let U  [¹.P .!1 //Mk j k < !º be the [¹Mk j k < !º-ultrafilter corresponding to k. Let g D k.f /.!1M0 /. By conditions (6) and (7) in the definition of Qmax , g  Coll.!; !1M0 / and g is [¹Mk j k < !º-generic. Again by the definition of Qmax , a set a belongs to U if and only if there exists p 2 g and k 2 ! such that .!1M0 n a/ \ ¹˛ j p 2 f .˛/º 2 .INS /Mk : Thus the iteration k is completely determined by k.f /.!1M0 /. The lemma follows by induction on the length of iterations.

t u

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Lemma 6.70. Suppose h W !1 ! H.!1 / and that I  P .!1 / is a normal (uniform/ ideal such that for all A  !1 , ¹˛ j h.˛/ is not L.A \ ˛/-generic for Coll.!; ˛/º 2 I: Suppose .hMk W k < !i; f / 2 Qmax . Then there is an iteration j W hMk W k < !i ! hMk W k < !i such that: (1) j.!1M0 / D !1 ; 

(2) for all k < !, I \ Mk D INS \ Mk D .INS /MkC1 \ Mk ; (3) j.f / D h modulo I . Proof. This lemma corresponds to Lemma 6.23. Let hhMkˇ W k < !i; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i be any iteration of hMk W k < !i such that for all ˇ < !1 if j0;ˇ .!1M0 / D ˇ and if h.ˇ/ is an [¹Mkˇ j k < !º-generic filter for Coll.!; ˇ/ then jˇ;ˇ C1 is the corresponding generic elementary embedding. We claim that hhMkˇ W k < !i; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i is as desired. Suppose A  !1 . By assumption ¹˛ < !1 j h.˛/ is not a LŒA \ ˛-generic filter for Coll.!; ˛/º 2 I: Suppose A  !1 and A codes the iteration hhMkˇ W k < !i; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i Then the set of  < !1 such that hhMkˇ W k < !i; G˛ ; j˛;ˇ W ˛ < ˇ  i 2 LŒA \  contains a club in !1 . Further the set of  < !1 such that j0; .!1M0 / D  also contains a club. Let X  !1 be the set of  < !1 such that h./ is an [¹Mkˇ j k < !º-generic filter for Coll.!; / and such that j0; .!1M0 / D . Thus !1 n X 2 I . However by the properties of the iteration, X  ¹ j j0; C1 .f /./ D h./º and so j0;!1 .f / D h modulo I .

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6 Pmax variations

Lemma 6.70 can be reformulated as follows. Lemma 6.71. Suppose that p 2 Qmax , .hNk W k < !i; g/ 2 Qmax and that

p 2 .H.!1 //N0 :

Then there exists h 2 N0 such that (1) .hNk W k < !i; h/ 2 Qmax , (2) .hNk W k < !i; h/ < p, (3) ¹˛ < !1N0 j h.˛/ ¤ g.˛/º 2 .INS /N1 . Proof. Since p 2 .H.!1 //N0 , p 2 .Qmax /N0 . For each a 2 .P .!1 //N0 let a be the set of ˛ < !1N0 such that g.˛/ is LŒa \ ˛-generic for Coll.!; ˛/. Let I 2 N0 be the normal ideal generated by the set ¹a j a 2 .P .!1 //N0 º: The key point is that

I  .INS /N1 ;

which is easily verified since .hNk W k < !i; g/ 2 Qmax . Let .hMk W k < !i; f / D p: Applying Lemma 6.70 within N0 , there exists an iteration j W hMk W k < !i ! hMk W k < !i such that: (1.1) j.!1M0 / D !1N0 ; 

(1.2) for all k < !, I \ Mk D .INS /N0 \ Mk D .INS /MkC1 \ Mk ; (1.3) j.f / D g modulo I . Let h D j.f /. Thus .hNk W k < !i; h/ 2 Qmax and .hNk W k < !i; h/ < p:

t u

As a corollary to Lemma 6.71 and Lemma 6.68, we obtain the set of conditions specified in Lemma 6.68 is dense in Qmax .

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355

Corollary 6.72. Assume AD holds in L.R/. Suppose that A  R, A 2 L.R/; and that p 2

Qmax .

Then there exists .hNk W k < !i; g/ 2 Qmax

such that: (1) A \ N0 2 N0 ; (2) hH.!1 /N0 ; A \ N0 ; 2i  hH.!1 /; A; 2i; (3) hNk W k < !i is A-iterable; (4) ˘ holds in N0 ; (5) g witnesses ˘++ .!1
t u

356

6 Pmax variations

Proof. This is one theorem about Qmax that is actually much simpler than the corresponding theorem about Qmax or Pmax . The !-closure of Qmax is essentially built into its definition. Suppose hpi W i < !i is a strictly decreasing sequence of conditions in Qmax and that for each i < !, pi D .hMki W k < !i; fi /: Let fO D [¹fi j i < !º. For each i < ! let ji W hMki W k < !i ! hMO ki W k < !i be the iteration such that ji .fi / D fO. This iteration exists since hpi W i < !i is a strictly decreasing sequence in Qmax . By Lemma 4.22, hMO kk W k < !i satisfies the hypothesis of Lemma 4.17 and so by Lemma 4.17, hMO kk W k < !i is iterable. Let O0 M Mi ı D !1 0 D sup ¹ !1 0 j k < !º: For q 2 Coll.!; ı/ let Sq D ¹˛ < ı j q 2 fO.˛/º: Then for each q 2 Coll.!; ı/, Sq 2 MO 00 ; and further

Ok

Sq … .INS /Mk

for each k 2 !. Fix k 2 !. Suppose that A  ı and that O kC1

A 2 MO kk n .INS /MkC1 : Then for some q 2 Coll.!; ı/, O kC1

Sq n A 2 .INS /MkC1 : Finally if q1  q2 in Coll.!; ı/ then Sq1  Sq2 . It follows that .hMO kk W k < !i; fO/ 2 Qmax . Further if q 2 Qmax and q < .hMO kk W k < !i; fO/; then q < pi for all i < !. By Corollary 6.72 there exists q 2 Qmax such that q < .hMO kk W k < !i; fO/; and so there exists q 2 Qmax such that q < pi for all i < !.

t u

6.2 Variations for obtaining !1 -dense ideals

357

We adopt the usual notational conventions. Suppose G  Qmax is L.R/-generic. Let fG D [¹f j .hMk W k < !i; f / 2 G for some hMk W k < !iº: For each condition .hMk W k < !i; f / 2 G there is a unique iteration j W hMk W k < !i ! hMk W k < !i such that j.f / D fG . This is the unique iteration such that j.f / D fG . Let  IG D [¹.INS /M1 j .hMk W k < !i; f / 2 Gº and let



P .!1 /G D [¹P .!1 /M0 j .hMk W k < !i; f / 2 Gº:

The basic analysis of Qmax is straightforward, the results are given in the next two theorems. Theorem 6.74. Assume AD holds in L.R/. Suppose G  Qmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal !1 -dense ideal on !1 ; (3) IG is the nonstationary ideal. Proof. The proof is essentially the same as the proof of Theorem 6.30. Here one uses Lemma 6.70 and the proof of Theorem 6.73. t u Theorem 6.75. Assume AD holds in L.R/. Then  L.R/Qmax  AC : Proof. The proof for Qmax naturally generalizes.

t u

Theorem 6.76. Assume ADL.R/ . Let G  Qmax be L.R/-generic and let fG W !1 ! H.!1 / be the function derived from G. Then fG witnesses ˘++ .!1
P .!1 / D [¹P .!1 /M0 j .hMk W k < !i; f / 2 Gº where for each .hMk W k < !i; f / 2 G, j W hMk W k < !i ! hMk W k < !i is the (unique) iteration such that j.f / D fG . The theorem is an immediate corollary of Corollary 6.72.

t u

358

6 Pmax variations

As a corollary to the basic analysis of Qmax we obtain Theorem 6.80 which shows that ADL.R/ implies that Qmax is nontrivial in the sense required for the basic analysis summarized in Theorem 6.30. We require a preliminary lemma. Lemma 6.77. Assume ADL.R/ and suppose G  Qmax is L.R/-generic. Then in L.R/ŒG, for every set A 2 P .R/ \ L.R/ the set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Proof. This is the Qmax version of Lemma 4.52. Suppose G  Qmax is L.R/-generic. From the basic analysis of Qmax summarized in Theorem 6.73 and Theorem 6.74, it follows that H.!2 /L.R/ŒG D H.!2 /L.R/ ŒG: We work in L.R/ŒG. Fix A  R with A 2 L.R/. Fix a stationary set S  P!1 .H.!2 // and fix a countable elementary substructure X  hH.!2 /; A; G; 2i such that X \ H.!2 / 2 S . Let hXi W i < !i be an increasing sequence of countable elementary substructures of X such that X D [¹Xi j i < !º and such that for each i 2 !, Xi 2 XiC1 . Therefore for each i < !, there exists .hMk W k < !i; f / 2 G \ XiC1 satisfying (1.1) Xi \ P .!1 /  j.M0 /, (1.2) A \ M0 2 M0 , (1.3) hMk W k < !i is A-iterable, where

j W hMk W k < !i ! hMk W k < !i

is the iteration such that j.f / D fG . Let MX be the transitive collapse of X . We claim that MX is A-iterable. Given this the lemma follows. For each i < ! let .hMki W k < !i; fi / 2 G \ XiC1 be a condition satisfying the requirements (1.1), (1.2) and (1.3). For each i < ! let ji W hMi W k < !i ! hMO i W k < !i k

k

be the iteration of hMki W k < !i such that ji .fi / D fG j.X \ !1 /.

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Thus for each i < !, ji 2 MX and MX D [¹ji .M0i / j i < !º: Suppose j W MX ! N is an iteration of MX such that j.!1MX / D  and  < !1 . For each i < ! let hNki W k < !i D j.hMO ki W k < !i/: Therefore for each i < !, hNki W k < !i is an iterate of hMki W k < !i and the iteration is the unique iteration which sends fi to j.fG jX \ !1 /. By induction on  N D [¹N0i j i < !º and so MX is iterable. We finish by analyzing C D [¹j.B/ j B  A and B 2 MX º: We must show that C D A \ N . Since MX D [¹ji .M0i / j i < !º it follows that C D [¹j.ji .A \ M0i // j i < !º This is because A \ M0i 2 M0i for each i < !. However for each i < !, hMki W k < !i is A-iterable and so for each i < !, j.ji .A \ M0i // D A \ N0i : This implies that C D A \ N .

t u

As a corollary to Lemma 6.77 and Lemma 4.24 we obtain the following theorem which easily yields a plethora of conditions in Qmax . Theorem 6.78. Assume AD holds in L.R/. Suppose G  Qmax is L.R/-generic. Then in L.R/ŒG the following holds. Suppose  2 Ord, L .R/ŒG  ZFC ; and that L .R/ †1 L.R/: Suppose X  L .R/ŒG is a countable elementary substructure with G 2 X . Let MX be the transitive collapse of Y and let IX D .INS /MX : Then for each A  R such that A 2 X \ L.R/, .MX ; IX / is A-iterable.

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Proof. By an analysis of terms L .R/ŒG †1 L.R/ŒG: We prove that for each ˛ < , if L˛ .R/ŒG  ZFC then the set ¹Y 2 P!1 .L˛ .R/ŒG/ j MY is iterableº contains a club in P!1 .L˛ .R/ŒG/, where MY is the transitive collapse of Y . Assume toward a contradiction that this fails for some ˛ and let ˛0 be the least such ˛. It follows that ˛0 < ‚L.R/ . This contradicts Lemma 4.24 and Lemma 6.77. Now suppose X  L .R/ŒG is a countable elementary substructure with G 2 X . Let Z D ¹˛ 2 X \  j L˛ .R/ŒG  ZFC º: Thus Z is cofinal in X \ . Further since L .R/ŒG †1 L.R/ŒG; for each ˛ 2 Z there exists a function F W L˛ .R/ŒG
MX

is A-iterable. Thus MX is A-iterable.

t u

Another corollary of Lemma 6.77 is the following lemma. Lemma 6.79. Assume ADL.R/ and suppose G  Qmax is L.R/-generic. Then in L.R/ŒG the following hold.

6.2 Variations for obtaining !1 -dense ideals

361

(1) !3 D ‚L.R/ . (2) ı12 D !2 . (3) Suppose S  !1 is stationary and f W S ! Ord: Then there exists g 2 L.R/ such that ¹˛ 2 S j f .˛/ D g.˛/º is stationary. Proof. (2) follows from Theorem 3.16 and Lemma 6.77. By Theorem 6.75, ‚L.R/  !3 in L.R/ŒG since c D !2 in L.R/ŒG. Qmax satisfies the following chain condition in L.R/. Suppose  W Qmax ! ‚ is a function. Then the range of  is bounded in ‚. This is because there is a map of the R onto Qmax . The usual analysis of terms shows that ‚L.R/ is a cardinal in L.R/ŒG. By Theorem 6.74(1), !1L.R/ and !2L.R/ are cardinals in L.R/ŒG. Therefore (1) follows. Similarly for (3) one can reduce to the case that for some ı < ‚, f W !1 ! ı: and so (3) follows from Theorem 3.42, Lemma 6.77, and Theorem 6.74(2).

t u

As an immediate corollary to Theorem 6.78 we obtain that, assuming ADL.R/ , Qmax is suitably nontrivial as required for the analysis of the Qmax -extension. Let ZFC be any finite fragment of ZFC together with ZFC . Theorem 6.80. Assume AD holds in L.R/. Then for each set A  R with A 2 L.R/, there is a condition h.M; I /; f i 2 Qmax such that (1) M  ZFC , (2) I D .INS /M , (3) A \ M 2 M, (4) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i, (5) .M; I / is A-iterable, (6) f witnesses ˘++ .!1
6 Pmax variations

362

Proof. Let G  Qmax be L.R/-generic. We work in L.R/ŒG. Fix  2 Ord such that L .R/ŒG  ZFC ; cof ./ > !1 and such that L .R/ †1 L.R/: Let X  L .R/ŒG be a countable elementary substructure such that ¹A; fG º  X . Let MX be the transitive collapse of X and let fX be the image of fG under the collapsing map. Let IX D .INS /MX which is the image of INS under the collapsing map. By Theorem 6.78, .MX ; IX / is A-iterable. Therefore h.MX ; IX /; fX i 2 Qmax : By Theorem 6.76, fg witnesses ˘++ .!1
t u

The analysis of L.R/Qmax given by the assumption of the existence of a huge cardinal can now be carried out just assuming ADL.R/ . For example we obtain the following theorem as an immediate consequence of Theorem 6.80, Theorem 6.30, Theorem 6.31, and Theorem 6.53. Theorem 6.81. Assume AD holds in L.R/. Suppose G  Qmax is L.R/-generic. Then L.R/ŒG  ZFC and in L.R/ŒG: (1) P .!1 / D P .!1 /G and IG D INS ; (2) every set of reals of cardinality !1 is of measure 0; (3) the reals cannot be decomposed as an !1 union of meager sets; (4) the nonstationary ideal on !1 is !1 -dense; (5) the function fG witnesses ˘++ .!1
t u

6.2 Variations for obtaining !1 -dense ideals

363

Corollary 6.82. Assume ZF C AD is consistent. Then so is ZFC C “The nonstationary ideal on !1 is !1 -dense”:

t u

We continue our analysis of Qmax by proving another absoluteness theorem which suggests that the Qmax model is simply a conditional form of the Pmax model. In Chapter 7 we shall consider other conditional variations of Pmax . The proof of this absoluteness theorem uses the generic elementary embedding associated to the stationary tower forcing. The argument can be adapted to prove the absoluteness theorems for Pmax without using Theorem 2.61. See the remarks preceding Theorem 4.63. Remark 6.83. The most general absoluteness theorem for Qmax requires a restriction on the …2 formulas. With this restriction we shall obtain an absoluteness theorem where only ˘+ .!1
t u

Definition 6.84. Suppose A is an alphabet for a first order language and that A contains 2 and a unary predicate U . A formula  of L.A/ is a U -restricted …2 formula if there is a †0 -formula .x; y; z/ in L.A n ¹U º/ such that  D 8x8y9zŒ.x/ !

.x; y; z/

where .x/ is the atomic formula U.x/.

t u

Theorem 6.85. Suppose there are ! many Woodin cardinals with a measurable above them all. Suppose F W !1 ! H.!1 / is a function which witnesses ˘+ .!1
364

6 Pmax variations

and that hH.!2 /; ŒF INS ; X; 2 W X  R; X 2 L.R/i  : Then

Qmax

hH.!2 /; ŒfG IG ; X; 2 W X  R; X 2 L.R/iL.R/

 :

Proof. From the large cardinal assumptions, AD holds in L.R/. Therefore by Theorem 6.80, Qmax is nontrivial in the sense required for the analysis of Qmax . Fix A  R with A 2 L.R/.  is a ŒF INS -restricted …2 sentence and so  D 8x8y9zŒ.x/ !

.x; y; z/

where  is the atomic formula expressing x 2 ŒF INS and is a †0 formula in the language for hH.!2 /; X; 2 W X  R; X 2 L.R/i: We assume that the only unary predicate occurring in corresponds to A. Let ı0 be the least Woodin cardinal and let 0 be the least strongly inaccessible cardinal above ı0 . Thus by Theorem 2.13, the set A is
 :

then hH.!2 /; ŒF INS ; X; 2 W X  R; X 2 L.R/i  : Then there exists G  Qmax such that: (1) G is L.R/-generic; (2) P .!1 /  L.R/ŒG; (3) fG D F . Proof. By Theorem 6.80, Qmax is nontrivial in the sense required for the analysis of Qmax . Suppose g 2 ŒF NS : We associate to the function g a filter Fg  Qmax defined to be the set of h.M; I /; f i 2 Qmax such that there is an iteration j W .M; I / ! .M ; I  / such that the set

¹˛ < !1 j F .˛/ D j  .f /.˛/º

is closed and unbounded in !1 . Suppose g is generic for Qmax ; i. e. that there is an L.R/-generic filter G0  Qmax such that g D fG0 . Then as in the proof of Lemma 6.35, Fg D G0 . Fix D  Qmax such that D is dense and D 2 L.R/. Suppose G  Qmax is L.R/generic. Then by Lemma 6.34 and by Lemma 6.30(1) the following sentences holds in L.R/ŒG:

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367

(1.1) For all g 2 ŒfG INS , Fg \ D ¤ ;. (1.2) For all g 2 ŒfG INS , Fg is a filter in Qmax . (1.3) For all g 2 ŒfG INS , for all a  !1 , a 2 L.g; x/ for some x 2 R. The first sentence is expressible by a ŒfG INS -restricted …2 sentence in the structure hH.!2 /; 2; ŒfG INS ; DiL.R/ŒG ; the second sentence is expressible by a ŒfG INS -restricted …2 sentence in the structure hH.!2 /; 2; ŒfG INS ; Qmax iL.R/ŒG ; and the third sentence is expressible by a ŒfG INS -restricted …2 sentence in the structure hH.!2 /; 2; ŒfG INS iL.R/ŒG : Thus by the hypothesis of the theorem the three sentences hold in V . Therefore for all g 2 ŒF INS , the filter Fg is L.R/-generic and further ŒgINS  L.R/ŒFg : For any A  !1 there exists g 2 ŒF INS such that A 2 L.F; g/. Let G D FF . Thus it follows that L.P .!1 // D L.R/ŒG and this proves the theorem.

t u

The next theorem shows that the Qmax -extension satisfies a restricted version of the homogeneity condition satisfied by the Pmax -extension (cf. Theorem 5.67). We fix some notation. Suppose t 2 R and that g  Coll.!;
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6 Pmax variations

Theorem 6.87. Assume AD holds in L.R/. Suppose L.P .!1 // D L.R/ŒG where G  Qmax is L.R/-generic. Suppose X  P .!1 / is a nonempty set such that X 2 L.P .!1 //; and such that X is ordinal definable in L.P .!1 // with parameters from R [ ¹ŒfG INS º: Then there exist t 2 R and a term   !1  Coll.!;
A D j.a/ j W .M; I / ! .M ; I  /

is the iteration such that j.f / D fG . By genericity we can suppose that in L.R/, h.M; I /; f i Qmax Œ ; w; ; ŒfG INS  where is the term for j.a/. Let t 2 R code h.M; I /; f i. Suppose g  Coll.!;
6.2 Variations for obtaining !1 -dense ideals

369

be the iteration of h.M; I /; f i such that for all ˇ < !1 jˇ;ˇ C1 .j0;ˇ .f //./ D Fg ./ where

M

 D !1 ˇ : This uniquely specifies the iteration. We note that for each ˇ < !1 , hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ < i 2 LŒt Œg \ Coll.!; L.R/ŒG  ŒA ; w; ; ŒfG INS : t u

370

6.2.3

6 Pmax variations 2

Qmax

We define and briefly analyze a variant of Qmax which is analogous to 2 Pmax . We denote this partial order by 2 Qmax . We give this example to illustrate how extensions with various ideal structures can be easily obtained by modifying Pmax . Assuming AD holds in L.R/ we shall prove that if G  2 Qmax is L.R/-generic then in L.R/ŒG, INS is not saturated, sat.INS / is !1 -dense and further for each S 2 sat.INS / n INS INS jS is !1 -dense. Before defining 2 Qmax we prove that AC holds in the Qmax -extension of L.R/. Lemma 6.88 is the analog of Lemma 5.16, used to prove that AC holds in the Pmax -extension. Here the situation is even simpler. Lemma 6.88. Suppose that h.N ; J /; gi < h.M; I /; f i in Qmax and let x0 2 N \ R code M Let C be the set of of the Silver indiscernibles of LŒx below !1N and let C 0 be the limit points of C . Suppose that ¹s; t º  .P .!1 //M n I is such that both !1M0 n s … I and !1M0 n t … I . Then there exists an iteration j W .M; I / ! .M ; I  / such that j 2 N ,

¹˛ j j.f /.˛/ ¤ g.˛/º 2 J; 0

and such that for all  2 C , if and only if

 2 j.s/  C 2 j.t /

where  C is the least element of C above  . Proof. The proof is essentially a trivial modification of the proof of Lemma 6.23. Construct the iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1M1 i; in N , by induction such that for all  2 C 0 if g. / is M -generic for Coll.!;  / then j;C1 .j0; .f //. / D g. / 0

and such that for all  2 C ,

j0; .s/ 2 G

if and only if j0;ˇ .t / 2 Gˇ C

where ˇ D  .

6.2 Variations for obtaining !1 -dense ideals

371

By the boundedness lemma, Lemma 4.6, for all  2 C , if k W .M; I / ! .M ; I  / is given by any iteration of length  then k.!1M / D : Therefore the requirements do not interfere with each other. As in the proof of Lemma 6.23, ¹˛ j j.f /.˛/ ¤ g.˛/º 2 J; and so j0;omega1 W .M; I / ! .M!1 ; I!1 / t u

is as desired.

As a corollary to Lemma 6.88 and the basic analysis of L.R/Qmax we obtain the following lemma. The proof is essentially identical to that of Lemma 5.17, using Lemma 6.88 in place of Lemma 5.16. Lemma 6.89. Assume AD L.R/ . Suppose G  Qmax is L.R/-generic. Then L.R/ŒG 

AC :

t u

Lemma 6.89 combined with Theorem 6.78 yields the following variation of Theorem 6.80. Corollary 6.90. Assume AD holds in L.R/. Then for each set A  R with A 2 L.R/, there is a condition h.M; I /; f i 2 Qmax such that (1) M  ZFC C

AC ,

(2) A \ M 2 M, (3) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i, (4) .M; I / is A-iterable.

t u

Definition 6.91. 2 Qmax is the set of finite sequences hM; I; J; f; Y i which satisfy the following. (1) M  ZFC C

AC .

(2) In M, I and J are normal !1 -dense ideals on !1 with I  J . (3) .M; I / is iterable. (4) h.M; J /; f i 2 Qmax . (5) Y  J n I and jY j  !1 in M.

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6 Pmax variations

(6) For each a; b 2 Y , a M b 2 I or a \ b 2 I . (7) For each a 2 Y , h.M; Ia /; f i 2 Qmax where Ia is the ideal I ja as computed in M. The ordering on 2 Qmax is analogous to Qmax . hM1 ; I1 ; J1 ; f1 ; Y1 i < hM0 ; I0 ; J0 ; f0 ; Y0 i if M0 2 M1 ; M0 is countable in M1 and there exists an iteration j W .M0 ; ¹I0 ; J0 º/ ! .M0 ; ¹I0 ; J0 º/ such that: (1) j.f0 / D f1 ; (2) M0 2 M1 and j 2 M1 ; (3) I0 D I1 \ M0 and J0 D J1 \ M0 ; (4) j.Y0 /  Y1 . Remark 6.92.

t u

(1) The requirement (2) of Definition 6.91 implies that J D I ja

for some a 2 P .!1 /M n I . Necessarily, by (5), a \ b 2 I for all b 2 Y . Further iterations of .M; ¹I; J º/ correspond to iterations of .M; I / and so (3) implies that .M; ¹I; J º/ is iterable. (2) Given (3) of Definition 6.91, (4) and (7) become first order conditions on M. For example (4) simply asserts that J is an !1 -dense ideal and g is a function t u related to J in the usual fashion; i. e. g 2 YColl .J /. Lemma 6.93. Suppose that hM; I; J; f; Y i 2 2 Qmax : Suppose that j1 W .M; ¹I; J º/ ! .M1 ; ¹I1 ; J1 º/ and j2 W .M; ¹I; J º/ ! .M2 ; ¹I2 ; J1 º/ are iterations of .M; ¹I; J º/ such that j1 .f / D j2 .f /. Then M1 D M2 and j1 D j2 .

6.2 Variations for obtaining !1 -dense ideals

Proof. Let

373

a D ¹˛ < !1M j .0; 0/ 2 f .˛/º:

Since h.M; J /; f i 2 Qmax , it follows that a…J and that

!1M n a … J:

Therefore M  “a is a stationary, co-stationary, subset of !1 .” Since j1 .f / D j2 .f / it follows that j1 .a/ D j2 .a/. The lemma follows by Lemma 5.15.

t u

Using Corollary 6.90 we trivially obtain the nontriviality of 2 Qmax as required for the analysis of the 2 Qmax -extension. Lemma 6.94. Assume AD holds in L.R/. Then for each set A  R with A 2 L.R/; there is a condition hM; I; J; f; Y i 2 2 Qmax such that (1) jY =I j D !1 in M, (2) A \ M 2 M, (3) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i, (4) .M; I / is A-iterable. Proof. By Corollary 6.90, there is a condition h.M0 ; I0 /; f0 i 2 Qmax such that .M0 ; I0 / satisfies (2)–(4). We may suppose, by modifying f0 is necessary, that for all ˛ < !1M0 , f0 .˛/ is a maximal filter in Coll.!; ˛/. For each ˇ < !1M0 let Sˇ D ¹˛ < !1M0 j ˇ < 1 C ˛ and .0; ˇ/ 2 f0 .˛/º: Define

f W !1M0 ! H.!1 /M0

as follows. Suppose ˛ < !1M0 . Let ˇ be such that ˛ 2 Sˇ . Then f .˛/ D ¹p 2 Coll.!; ˛/ j .0; ˇ/a p 2 f0 .˛/º: Let (1.1) M D M0 , (1.2) Y D ¹Sˇ j ˇ > 0º,

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6 Pmax variations

(1.3) I D I0 , (1.4) J D I0 jS0 . It follows that hM; I; J; f; Y i 2 2 Qmax and is as required.

t u

The iteration lemmas for 2 Qmax are proved by minor modifications in the arguments used to prove the iteration lemmas for Qmax . The only difference is that the iteration lemmas for 2 Qmax are more awkward to state. Lemma 6.95. Suppose that hM0 ; I0 ; J0 ; f0 ; Y0 i 2 2 Qmax ; hM1 ; I1 ; J1 ; f1 ; Y1 i 2 2 Qmax ; and that hM0 ; I0 ; J0 ; f0 ; Y0 i 2 .H.!1 //M1 : Suppose that Y  Y1 , Y 2 M1 and that M1  jY =I1 j D !1 : Then there exists an iteration j W .M0 ; ¹I0 ; J0 º/ ! .M0 ; ¹I0 ; J0 º/ such that j 2 M1 and such that the following hold. (1) ¹˛ < !1M1 j j.f0 /.˛/ ¤ f1 .˛/º 2 I1 . (2) I0 D M0 \ I1 and J0 D M0 \ J1 . (3) j.Y0 /=I1  Y =I1 . Proof. The key point is the following. Suppose jQ W .M0 ; ¹I0 ; J0 º/ ! .MQ 0 ; ¹IQ0 ; JQ0 º/ is a countable iteration and that Q

g  Coll.!; !1M0 / is MQ 0 -generic. Then (1.1) there exists an iteration kQ W .MQ 0 ; JQ0 / ! .MQ 1 ; JQ1 / such that

Q kQ ı jQ.f0 /.!1M0 / D g;

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375

(1.2) for each S 2 jQ.Y0 / there exists an iteration kQS W .MQ 0 ; IQ0 / ! .MQ 1 ; IQ1 / such that

Q kQS ı jQ.f0 /.!1M0 / D g

and such that

Q !1M0 2 kQS .S /:

With this simple observation, the desired iteration, j , is easily constructed in M1 by the usual book-keeping arguments used in the proofs of the earlier iteration lemmas, cf. the proof of Lemma 4.36. The point is that one must associate elements of j.Y0 /, as they are generated in the course of the iteration, to elements of Y . t u We require two other lemmas. The proofs are easy variations of the proofs of Lemma 6.94 and Lemma 6.95. We again leave the details to the reader. Lemma 6.96. Assume AD holds in L.R/. Suppose that hM0 ; I0 ; J0 ; f0 ; Y0 i 2 2 Qmax and that a0 2 J0 . Then there exists hM1 ; I1 ; J1 ; f1 ; Y1 i 2 2 Qmax such that hM1 ; I1 ; J1 ; f1 ; Y1 i < hM0 ; I0 ; J0 ; f0 ; Y0 i and such that

j.a0 / n 5Y1 2 I1

where j W .M0 ; ¹I0 ; J0 º/ ! .M1 ; ¹I1 ; J1 º/ is the .unique/ iteration such that j.f0 / D f1 .

t u

Lemma 6.97. Assume AD holds in L.R/. Suppose that hM0 ; I0 ; J0 ; f0 ; Y0 i 2 2 Qmax and that a0 2 P .!1 /M0 n J0 . Then there exists hM1 ; I1 ; J1 ; f1 ; Y1 i 2 2 Qmax and b  j.a0 / such that hM1 ; I1 ; J1 ; f1 ; Y1 i < hM0 ; I0 ; J0 ; f0 ; Y0 i; b 2 J1 n I1 , and such that

b \ a 2 I1

for all a 2 j.Y0 /, where j W .M0 ; ¹I0 ; J0 º/ ! .M1 ; ¹I1 ; J1 º/ is the .unique/ interation such that j.f0 / D f1 .

t u

376

6 Pmax variations

Using the proof of Lemma 6.95 and of its generalization to sequences of conditions, the analysis of the 2 Qmax extension can be carried out in a manner quite similar to that for the Qmax -extension. The results are summarized in the next theorem where we use the following notation. Suppose G  2 Qmax is L.R/-generic. Let fG D [¹f j hM; I; J; f; Y i 2 Gº: For each condition hM; I; J; Y; f i 2 G there is a unique iteration j W .M; ¹I; J º/ ! .M ; ¹I  ; J  º/ such that j.f / D fG . We let Y  denote j.Y /. Let (1) P .!1 /G D [¹P .!1 / \ M j hM; I; J; Y; f i 2 Gº, (2) IG D [¹I  j hM; I; J; f; Y i 2 Gº, (3) JG D [¹J  j hM; I; J; f; Y i 2 Gº, (4) YG D [¹Y  j hM; I; J; f; Y i 2 Gº. Theorem 6.98. Assume ADL.R/ . Suppose G  2 Qmax is L.R/-generic. Then L.R/ŒG  ZFC and in L.R/ŒG: (1) P .!1 / D P .!1 /G ; (2) IG D INS ; (3) for each set A 2 JG there exists Y  YG such that jY j D !1 and such that A n 5Y 2 IG I (4) YG is predense in .P .!1 / n IG ; /; (5) For each S 2 P .!1 / n JG , ¹˛ j p 2 fG .˛/º n S 2 JG for some p 2 Coll.!; !1 /; (6) For each S 2 YG and for each T  S such that T … IG , ¹˛ j p 2 fG .˛/º n T 2 IG for some p 2 Coll.!; !1 /; (7) JG D sat.IG /.

6.2 Variations for obtaining !1 -dense ideals

377

Proof. The proofs that P .!1 / D P .!1 /G , IG D INS and that L.R/ŒG  AC are routine adaptations of earlier arguments. (3) follows from (1), Lemma 6.96, and the genericity of G. (4) follows from (3) given that JG n IG is predense in .P .!1 / n IG ; / which in turn follows from (1), Lemma 6.97, and the genericity of G. (5) and (6) are immediate consequence of (1) and the definition of 2 Qmax . It remains to prove (7). By (1), Lemma 6.97, and the genericity of G, for all A 2 P .!1 / n JG and for all Y  YG such that jY j D !1 , there exists B  A such that B 2 YG and such that B \ S 2 IG for all S 2 Y . Thus for all A 2 P .!1 / n JG , IG jA is not saturated. Therefore sat.IG / is defined and sat.IG /  JG : We finish by calculating sat.IG /. By (6), YG  sat.IG /. However by (3), any normal ideal containing YG [ IG must contain JG . Therefore JG  sat.IG / and so JG D sat.IG /:

t u

As an immediate corollary to Theorem 6.98 be obtain the following theorem. Theorem 6.99. Assume ADL.R/ . Suppose that G  2 Qmax is L.R/-generic. Then in L.R/ŒG: (1) INS is not !2 -saturated, (2) sat.INS / is !1 -dense, (3) for each S 2 sat.INS /, the ideal INS jS is !1 -dense.

t u

6.2.4 Weak Kurepa trees and Qmax The absoluteness theorems suggest that in the model L.R/Qmax one should have all the consequences for hH.!2 /; 2i which follow from the largest fragment of Martin’s Maximum which is consistent with the existence of an !1 -dense ideal on !1 .

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6 Pmax variations

It is therefore perhaps curious that there is a weak Kurepa tree on !1 in L.R/Qmax . This is the principal result of this section. This result together with the results of the next section show that the existence of a weak Kurepa tree is independent of the proposition that the nonstationary ideal is !1 -dense. See Remark 3.57. The following holds in the extension obtained by any Pmax -variation unless one explicitly prevents it:  For each A  !1 there exists x 2 R such that x # … L.A; x/: For the Pmax -extension this is a corollary of Theorem 5.73(5). 1 Lemma 6.100 ( 2 -Determinacy). Suppose that for each A  !1 there exists x 2 R such that x # … L.A; x/:

Suppose that A  !1 and let A D sup¹.!2 /LŒZ j Z  !1 ; A 2 LŒZ; and RLŒA D RLŒZ º: Then A < !2 . Proof. We first prove the following. (1.1) Suppose that  !2 is a countable set. Then !1 is inaccessible in L. /. Choose A0  !1 such that (2.1) 2 LŒA0 , (2.2) !1 D .!1 /LŒA0  , 1 (2.3) LŒA0    2 -Determinacy.

By Jensen’s Covering Lemma, if A  !1 and x 2 R are such that x # exists and x … L.A; x/, then A# exists. Therefore, by the hypothesis of the lemma, A#0 exists and so !2 is an indiscernible of LŒA0 . We work in LŒA0 . Let  2 LŒA0  be a countable set of uniform indiscernibles of LŒA0  such that for some x0 2 R \ LŒA0 , #

2 L.; x0 /: Let ˛ be the ordertype of  . We can suppose that ! ˛ D ˛ by increasing  if necessary. Let M 2 LŒA0  be a countable transitive set such that (3.1) x0 2 M , (3.2) ˛ < !1M , (3.3) M  ZFC C “There exist ˛ measurable cardinals ”, (3.4) M is iterable (by linear iterations using the normal measures in M ).

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379

Since 1 LŒA0    2 -Determinacy;

the transitive set M exists. It follows that there exists an iteration by the normal measures in M , j W M ! M ; such that  is M -generic for product Prikry forcing. Thus !1 is inaccessible in M  . However x0 2 M and so 2 M  Œ : This proves (1.1). Fix A and fix x 2 R such that x # … L.A; x/: Assume toward a contradiction that A D !2 : Then for every set B  !1 ,

.A;B/ D !2

where .A;B/ D sup¹.!2 /LŒZ j Z  !1 ; .A; B/ 2 LŒZ; and RLŒAŒB D RLŒZ º: Thus we can assume that !1 D .!1 /LŒA and that .A;x/ D !2 . Let 0 be an infinite set of indiscernibles of LŒx with 0  !2 n !1 . Let Z  !1 witness that .A;x/ > sup. 0 /. Thus there exists a countable set 2 LŒZ such that (4.1)  !2LŒZ , (4.2) 0  , (4.3) x 2 LŒ , (4.4) is countable in LŒ . By (1.1), !1 is inaccessible in LŒ  and so by Jensen’s Covering Lemma, x # 2 LŒ   LŒZ: This contradicts that RLŒZ D RLŒAŒx .

t u

This (essentially) rules out one method for attempting to have weak Kurepa trees in L.R/P where P is any Pmax variation we have considered so far. Remark 6.101. (1) There are Pmax -variations which yield models in which any previously specified set of reals is !1 -borel in the simplest possible manner, given

380

6 Pmax variations

X  R with (say) X 2 L.R/, one obtains in L.R/P that  [ \ XD B˛;ˇ ˛

ˇ >˛

for some sequence hB˛;ˇ W ˛ < ˇ < !1 i of borel sets. 1 If X is the complete † 3 set then in such an extension there exists A  !1 such # that for all t 2 R, t 2 LŒAŒt . Simply choose A such that hx˛;ˇ W ˛ < ˇ < !1 i 2 LŒA where for each ˛ < ˇ < !1 , x˛;ˇ 2 H.!1 / is a borel code of B˛;ˇ .

(2) There is an interesting open question. Suppose that INS is !2 -saturated and that P .!1 /# exists. For each A  !1 let A be as defined in Lemma 6.100. Must t u A < !2 ? To prove that there are weak Kurepa trees in L.R/Qmax , it is necessary to to find a condition h.M; I /; f i 2 Qmax and a tree T 2 M of rank !1 in M such that if h.N ; J /; gi is a condition in Qmax and M 2 N with M countable in N then there is an iteration j W .M; I / ! .M ; I  / in N with the following properties. (1) J \ M D I  . (2) j.f / D g modulo J . (3) There is a cofinal branch b of j.T / such that b … M . The next lemma identifies the requirements which we shall use. Lemma 6.102 (ZFC ). Suppose that f is a function which witnesses ˘+ .!1
6.2 Variations for obtaining !1 -dense ideals

381

(6) For each limit ˛ < !1 , ¹ˇ < !1 j xˇ˛ 2 T º contains a club in !1 where xˇ˛ D [¹h˛ .s/ j s  f .ˇ/º: Proof. This is a routine construction.

t u

The role of g in the conditions specified in Lemma 6.102 is simply to control the sets T˛ , for example it follows that T˛  Tˇ whenever ˛ < ˇ. Let T be the set of hM; I; f; .T; g; h/i such that (1) h.M; I /; f i 2 Qmax , (2) f witnesses ˘++ .!1
382

6 Pmax variations

The following is the key to the construction. Suppose j W .M0 ; I0 / ! .M; I / is a countable iteration of .M0 ; I0 / and that b is a M-generic branch of j.T0 /. Suppose G is MŒb-generic for Coll.!; !1M / and let j  W .M; I / ! .M ; I  / be the corresponding generic elementary embedding. Then there exists G  such that  G  is M -generic for Coll.!; !1M / and such that b 2 j  .j  .j.T0 /// where j  W .M ; I  / ! .M ; I  /; where M is the generic ultrapower of M given by G  , and where j  is the corresponding elementary embedding. In this we are identifying generic ultrapowers with their transitive collapses (as usual). Given this the construction is straightforward. The existence of G  follows from the mutual genericity of b and G relative to M and the properties of j.T0 ; g0 ; h0 / D .T; g; h/ in M. There are two relevant points. (2.1) .T; g; h/ satisfies in M the conditions (1)–(6) of Lemma 6.102 and further .T  ; g  ; h / satisfies these conditions in M where .T  ; g  ; h / is the image of .T; g; h/ under the iteration associated to the generic ultrapower given by G. (2.2) T 2 M and

g D g  j!1M :

Thus by the mutual genericity of b and G, it follows that b is a M -generic branch of T . Therefore by clause (5) of the conditions set forth in Lemma 6.102, there exists an M -generic G  such that b D ¹h˛ .s/ j s 2 G  º where ˛ D !1M .

t u

From the previous lemmas and the basic analysis of Qmax it follows that there is a weak Kurepa tree in L.R/Qmax . Recall that if G  Qmax is L.R/-generic then the associated function fG witnesses ˘++ .!1
6.2 Variations for obtaining !1 -dense ideals

383

Proof. (2) is an immediate consequence of (1). We prove (1) which really is an immediate consequence of Lemma 6.103 and the basic analysis of L.R/ŒG given in Theorem 6.30. By Theorem 6.30 (and Theorem 6.80) there exist ¹.T0 ; g0 ; h0 /; D0 º 2 M0 and h.M0 ; I0 /; f0 i 2 G; such that j0 ..T0 ; g0 ; h0 // D .T; g; h/ and j0 .D0 / D D where j0 W .M0 ; I0 / ! .M0 ; I0 / is the iteration such that j0 .f0 / D fG . By Theorem 6.80 there exists a condition h.M1 ; I1 /; f1 i < h.M0 ; I0 /; f0 i such that f1 witnesses ˘++ .!1
6.2.5

KT

Qmax

As our next example of a variant of Qmax we define a partial order KT Qmax . The partial order KT Qmax is obtained from Qmax by simply changing the definition of the order. Our goal is to produce a model in which the nonstationary ideal on !1 is !1 -dense and in which there are no weak Kurepa trees on !1 . By Lemma 6.51, if there is an !1 -dense ideal on !1 then there is a Suslin tree. Thus one cannot obtain a model in which there is an !1 -dense ideal on !1 and in which there are no weak Kurepa trees, by sealing trees. We shall also state as Theorem 6.121, the absoluteness theorem for the KT Qmax extension which is analogous to the absoluteness theorem (Theorem 6.84) which we proved for the Qmax -extension.

384

6 Pmax variations

Definition 6.105. Let KT Qmax be the partial order obtained from Qmax as follows: KT

Qmax D Qmax ;

but the order on KT Qmax is the following strengthening of the order on Qmax . A condition h.M1 ; I1 /; f1 i < h.M0 ; I0 /; f0 i if h.M1 ; I1 /; f1 i < h.M0 ; I0 /; f0 i relative to the order on Qmax and if addition the following holds. Let j W .M0 ; I0 / ! .M0 ; I0 / is the (unique) iteration such that j.f0 / D f1 . Suppose b  !1M1 , b 2 M1 and b \ ˛ 2 M0 for all ˛ < !1M1 . Then b 2 M0 .

t u

The iteration lemmas necessary for the analysis of KT Qmax are an immediate corollary of the following lemmas. Lemma 6.106 (ZFC + ˘+ .!1
of length !1 such that: (1) F D j.f / on a club in !1 ; (2) if b  !1 is a set such that b \ ˛ 2 M for all ˛ < !1 , then b 2 M : Proof. For each a 2 H.!1 / let M.a/ D L˛ .b [ ¹aº/ where b is the transitive closure of a and ˛ is the least ordinal such that L˛ .b/ is admissible. Since F witnesses ˘+ .!1
(1.1) Gˇ is M.Mˇ ; h ˛ W ˛ < ˇi/-generic for Coll.!; !1 ˇ /,

6.2 Variations for obtaining !1 -dense ideals

385

(1.2) if j0;ˇ .!1M0 / D ˇ and if M

F .ˇ/  Coll.!; !1 ˇ / is a filter which is M.Mˇ ; h ˛ W ˛ < ˇi/-generic then Gˇ D F .ˇ/: This iteration is easily constructed. Since F witnesses ˘+ .!1
t u

386

6 Pmax variations

Lemma 6.107 (ZFC + ˘+ .!1
(iii) !1

MkC1

D !1

,

(iv) IkC1 \ Mk D Ik , (v) fkC1 D fk , (vi) if C 2 Mk is closed and unbounded in !1M0 then there exists D 2 MkC1 such that D  C , D is closed and unbounded in C and such that D 2 LŒx for some x 2 R \ MkC1 . Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i of length !1 such that: (1) F D j.f0 / on a club in !1 ; (2) if b  !1 is a set such that b \ ˛ 2 [Mk for all ˛ < !1 , then b 2 [Mk :

t u

Suppose G  KT Qmax is L.R/-generic. Let fG D [¹f j h.M; I /; f i 2 G for some M; I º: For each condition h.M; I /; f i 2 G there is a unique iteration j W .M; I / ! .M ; I  / such that j.f / D fG . Let IG D [¹I  j h.M; I /; f i 2 G for some M; f º and let  P .!1 /G D [¹P .!1 /M j h.M; I /; f i 2 Gº: Using Lemma 6.106 and Lemma 6.107 the analysis of Qmax generalizes to yield the analogous results for KT Qmax . However for this we assume the existence of a huge cardinal so that Lemma 6.47 holds. This gives a suitably rich collection of conditions h.M; I /; f i 2 Qmax such that ˘ holds in M. Within these conditions Lemma 6.106 and Lemma 6.107 can be applied. We note that the conclusion of Lemma 6.106 is false in L.R/Qmax . This shows that some additional assumption is required, in particular Lemma 6.106 cannot be proved from just ˘+ .!1
6.2 Variations for obtaining !1 -dense ideals

Theorem 6.108. Assume there is a huge cardinal. Then mogeneous. Suppose G  KT Qmax is L.R/-generic. Then

KT

387

Qmax is !-closed and ho-

L.R/ŒG  ZFC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal !1 -dense ideal on !1 ; (3) IG is the nonstationary ideal; (4) there are no weak Kurepa trees; (5) for every A  !1 there exists B  !1 such that A 2 LŒB and such that for all S  !1 if S \  2 LŒB for all  < !1 then S 2 LŒB. Proof. By Lemma 6.47, for every set A  R with A 2 L.R/ there is a condition h.M; I /; f i 2 Qmax such that (1.1) A \ M 2 M, (1.2) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i, (1.3) .M; I / is A-iterable, (1.4) ˘ holds in M, (1.5) f witnesses ˘++ .!1
6 Pmax variations

388

Suppose h.M0 ; I0 /; f0 i 2 G and let j0 W .M0 ; I0 / ! .M0 ; I0 / be the iteration such that j0 .f0 / D fG . Suppose b  !1 and that b \ ˛ 2 M0 for all ˛ < !1 . Then b 2 M0 . Choose h.M1 ; I1 /; f1 i 2 G, and b1 2 M1 such that h.M1 ; I1 /; f1 i < h.M0 ; I0 /; f0 i and such that j1 .b1 / D b where j1 W .M1 ; I1 / ! .M1 ; I1 / is the iteration such that j1 .f1 / D fG . Let k W .M0 ; I0 / ! .M; I / be the iteration such that k.f0 / D f1 . Thus k 2 M1 and j1 .M/ D M0 . Further b1 \  2 M !1M .

for all  < Therefore by the definition of the order in KT Qmax , b1 2 M. This implies that b 2 M0 . We now prove (5). Fix X0  !1 with X0 2 L.R/ŒG. By (1) there is a condition h.M0 ; I0 /; f0 i 2 G and a set b0 

!1M0

such that b0 2 M0 and j.b0 / D X0 where j W .M0 ; I0 / ! .M0 ; I0 /

is the unique iteration such that j.f0 / D fG . We work in L.R/ and we assume that the condition h.M0 ; I0 /; a0 i forces that X0 D j.b0 / is a counterexample to (5). Let z 2 R be any real such that M0 2 LŒz and M0 is countable in LŒz. For each i  !, Let i be the i th Silver indiscernible of LŒz. Let k W L! Œz ! L! Œz be the canonical embedding such that cp(k) = 0 and let L! Œz D ¹k.f /.0 / j f 2 L! Œzº: Let U be the L! Œz-ultrafilter on 0 given by k, U D ¹A  0 j A 2 L! Œz; 0 2 k.A/º: Thus L! Œz Š Ult.L! Œz; U / and k is the associated embedding. For each X  P .0 / \ L! Œz if X 2 L! Œz and jX j  0 in L! Œz then U \ X 2 L! Œz. Therefore .L! Œz; U / is naturally iterable.

6.2 Variations for obtaining !1 -dense ideals

389

Let g  Coll.!; <0 / be L! Œz-generic. Let N D L! ŒzŒg. Therefore 0 D !1N and the ultrafilter U defines an ideal I on !1N with I  N . Further for each X 2 N if jX j  !1N in N then I \ X 2 N . As in the proof of Theorem 6.64, if S   then Coll .!; S / the restriction of Coll.!; < / to S . Thus if ˛ < ˇ then Coll.!;
For each ˇ < !1N let Tˇ D ¹˛ j .0; ˇ/ 2 Fg .˛/º: !1N i

2 N and hTˇ W ˇ < !1N i is a sequence of pairwise disjoint Thus hTˇ W ˇ < sets which are positive relative to I . Fix a set S  0 such that S 2 L! Œz, S is stationary in L! Œz and S … U . Thus S  !1N , S 2 N , S is stationary in N , and S … I . For each or each ˇ < !1N , let Sˇ D Tˇ n S . Thus hSˇ W ˇ < !1N i is a sequence in N of pairwise disjoint I -positive sets each disjoint from S . By the proof of Lemma 7.7, there is an iteration j0 W .M0 ; I0 / ! .M1 ; I1 / such that j0 2 N and such that:

390

6 Pmax variations

(2.1) for each s  !1M1 , if s … I1 then S˛ n s 2 INS for some ˛ < !1 ; (2.2) if b  !1M1 is a set in N such that for all ˛ < !1M1 , b \ ˛ 2 M1 ; then b 2 M1 . Thus I \ M1 D I1 . Let f1 D j0 .f0 / and let b1 D j0 .b0 /. We come to the key points. First, N D L! Œy1  where y1  !1N and second, the proof Lemma 6.106 can be applied to .N; I /. Let h.M2 ; I2 /; f i be any condition in KT Qmax such that N 2 M2 , N is countable in M2 and such that ˘ holds in M2 . Then there is an iteration k  W .N; I / ! .N  ; I  / in M2 such that (3.1) k  .!1N / D .!1 /M2 , (3.2) I2 \ N  D I  , (3.3) if b  !1M2 is a set in M2 such that b \ ˛ 2 N  for all ˛ < !1 , then b 2 N : Let f2 D k  .f1 /, b2 D k  .b1 / and let y2 D k  .y1 /. Thus (4.1) h.M2 ; I2 /; f2 i < h.M0 ; I0 /; f0 i, (4.2) b2 D j.b0 / where j is the embedding given by the iteration of .M0 ; I0 / which sends f0 to f2 , (4.3) b2  .!1 /M2 , (4.4) y2  .!1 /M2 , (4.5) b2 2 .LŒy2 /M2 , (4.6) if b  !1M2 is a set in M2 such that for all ˛ < !1M2 , b \ ˛ 2 LŒy2 ; then b 2 LŒy2 . Now suppose G  KT Qmax is L.R/-generic and that h.M2 ; I2 /; f2 i 2 G. Let X0 D j0 .b0 / where j0 is the elementary embedding given by the iteration of .M0 ; I0 / which sends f0 to fG . Similarly let j2 W .M2 ; f2 / ! .M2 ; I2 / be the iteration such that j2 .f2 / D fG . Let Y0 D j.y2 /. Now by the claim proved above, in L.R/ŒG if B  !1 is a set such that for all ˛ < !1 , B \ ˛ 2 M2 ;

6.2 Variations for obtaining !1 -dense ideals

391

then B 2 M2 . By elementarity, since (4.6) holds in M2 , if B  !1 , B 2 M2 and if B \ ˛ 2 LŒY0  for all ˛ < !1 , then B 2 LŒY0 . Therefore if B  !1 , B 2 L.R/ŒG and if B \ ˛ 2 LŒY0  for all ˛ < !1 , then B 2 LŒY0 . This is a contradiction since X0 2 LŒY0  and Y0  !1 . t u Remark 6.109. Theorem 6.108(5) is a useful approximation to ˘ and this principle serves successfully in place of ˘ in the proofs of Lemma 6.106 and Lemma 6.107 (cf. Lemma 6.118). Theorem 6.108(5) is in some sense a feature of the KT Qmax -extension which is analt u ogous to that of the Pmax extension given in Theorem 5.73(5). Theorem 6.108 can be proved just assuming ADL.R/ . We briefly sketch the argument which in essence involves exploiting Theorem 6.108(5). First one refines the partial order, Qmax , defining a partial order KT Qmax which is the appropriate analog of KT Qmax . Definition 6.110. Let KT Qmax  Qmax be the partial order obtained from Qmax as follows. KT Qmax is the set of .hMk W k < !i; f / 2 Qmax such that for all a 2 [¹Mk j k < !º if a 

!1M0

and if a \ ˛ 2 M0

!1M0 ,

for all ˛ < then a 2 M0 . The order on KT Qmax is the following strengthening of  the order from Qmax . A condition .hNk W k < !i; g/ < .hMk W k < !i; f / if .hNk W k < !i; g/ < .hMk W k < !i; f / relative to the order on Qmax and if the following holds. Let j W hMk W k < !i ! hMk W k < !i M

be the (unique) iteration such that j.f / D g. Suppose b  !1 0 , b 2 [¹Nk j k < !º and

b \ ˛ 2 M0 M

for all ˛ < !1 0 . Then b 2 M0 . Lemma 6.106 easily generalizes to the following lemma.

t u

392

6 Pmax variations

Lemma 6.111 (ZFC + ˘+ .!1
of length !1 such that: (1) F D j.f / on a club in !1 ; (2) if b  !1 is a set such that b \ ˛ 2 M0 for all ˛ < !1 , then b 2 M0 :

t u

The analysis of Qmax generalizes to KT Qmax using the proof of Lemma 6.111 and using Theorem 6.113 to obtain the necessary conditions. One obtains Theorem 6.113 by modifying the proof of Theorem 6.64. (6) is the key requirement, the other requirements are automatically satisfied by the condition produced in the proof of Theorem 6.64. The modification of the proof of Theorem 6.64 involves proving the following strengthening of Lemma 6.63. Lemma 6.112 (For all x 2 R, x # exists). Suppose N is a transitive model of ZFC of height !1 and A  !1 is a cofinal set such that A\˛ 2N for all ˛ < !1 . Suppose B  !1 , B  A and that .N; A; B/ is constructible from a real. Let z 2 R be such that .N; A; B/ 2 LŒz and such that .N; A; B/ is definable from .!1V ; z/ in LŒz. Then there exist x 2 R and a function f W H.!1 / ! H.!1 / such that f is …11 .x/ and such that the following hold. (1) For all ˛ 2 B, if .g; h/ 2 H.!1 / and if a) g is N -generic for Coll .!; A \ ˛/, b) h is N Œg-generic for Coll .!; ¹˛º/, then f .g; h/ is an N ŒhŒg-generic filter for Coll .!; S / where S D A \ ¹ j ˛ <  < ˇº and ˇ is the least element of B above ˛.

6.2 Variations for obtaining !1 -dense ideals

393

(2) Suppose ı is an indiscernible of L.x/ and ı < !1 . Suppose H  Coll .!; B \ ı/ is L.x/-generic and g  Coll .!; A \ ı/ is N -generic. Suppose that for all ˛ 2 B \ ı, gjColl .!; S / D f .g; h/ where h D H jColl .!; ¹˛º/, S D A \ ¹ j ˛ <  < ˇº and ˇ is the least element of B above ˛. Finally, suppose b  ı, b 2 L.x/ŒH  and b \  2 N Œg for all  < ı. Then b 2 N Œg. Proof. Let x 2 R with z recursive in x and let f W H.!1 / ! H.!1 / …11 .x/

be any definable function which satisfies (1). These exist by Lemma 6.63. It follows that f must satisfy (2).

t u

Using Lemma 6.112 in the proof of Theorem 6.64 yields the requisite strengthening of Theorem 6.65. Theorem 6.113. Assume AD holds in L.R/. Then for every set A  R with A 2 L.R/ there is a condition .hMk W k < !i; f / 2 Qmax such that the following hold. (1) A \ M0 2 M0 . (2) hH.!1 /M0 ; A \ M0 ; 2i  hH.!1 /; A; 2i. (3) hMk W k < !i is A-iterable. (4) ˘ holds in M0 . (5) f witnesses ˘+ .!1
t u

6 Pmax variations

394

We illustrate the use of Theorem 6.113 in the analysis of KT Qmax . Suppose that .hMk W k < !i; f / 2 Qmax , .hNk W k < !i; g/ 2 Qmax , .hMk W k < !i; f / 2 .H.!1 //N0 and that .hNk W k < !i; g/ satisfies the conditions (1)–(6) of Theorem 6.113 with A D ;. By Lemma 6.111, there exists in N0 an iteration j W hMk W k < !i ! hMk W k < !i such that in N0 on a club in

!1N0

j.f / D g and such that if b  !1N0 is a set in N0 satisfying b \  2 M0

for all  < !1N0 then b 2 M0 . Suppose b  !1N0 , and that

b 2 [¹Nk j k < !º; b \  2 M0

for all  < !1N0 . Then by condition (6), b 2 N0 and so b 2 M0 . Thus .hMk W k < !i; f / < .hNk W k < !i; j.f // in KT Qmax . The basic analysis of KT Qmax is easily carried out. This yields the following theorem. Suppose G  KT Qmax is L.R/-generic. Let fG D [¹f j .hMk W k < !i; f / 2 G for some hMk W k < !iº: For each condition .hMk W k < !i; f / 2 G there is a unique iteration j W hMk W k < !i ! hMk W k < !i: such that j.f / D fG . This is the unique iteration such that j.f / D fG . Let  IG D [¹.INS /M1 j .hMk W k < !i; f / 2 Gº and let



P .!1 /G D [¹P .!1 /M j h.M; I /; f i 2 Gº:

Theorem 6.114. Assume ADL.R/ . Then KT Qmax is !-closed and homogeneous. Suppose G  KT Qmax is L.R/-generic. Then L.R/ŒG  ZFC and in L.R/ŒG:

6.2 Variations for obtaining !1 -dense ideals

395

(1) P .!1 /G D P .!1 /; (2) IG is a normal !1 -dense ideal on !1 ; (3) IG is the nonstationary ideal; (4) fG witnesses ˘++ .!1
t u

This suffices for the consistency result. With just a little more work one can easily prove the following lemmas which are the relevant versions of Lemma 6.77 and Lemma 6.79. This in turn leads to absoluteness theorems for the KT Qmax -extension of L.R/. Lemma 6.115. Assume ADL.R/ and suppose G  KT Qmax is L.R/-generic. Then in L.R/ŒG, for every set A 2 P .R/ \ L.R/ the set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº t u

contains a club, where MX is the transitive collapse of X .

The proof of Lemma 6.116 follows that of Theorem 6.78 using Lemma 6.115 in place of Lemma 6.77. Lemma 6.116. Assume AD holds in L.R/. Suppose G  Then in L.R/ŒG the following holds. Suppose  > !2 , L .R/ŒG  ZFC ;

KT

Qmax is L.R/-generic.

and that L .R/ †1 L.R/: Suppose X  L .R/ŒG is a countable elementary substructure with G 2 X . Let MX be the transitive collapse of Y and let IX D .INS /MX : Then for each A  R such that A 2 X \ L.R/, .MX ; IX / is A-iterable.

t u

Putting everything together we obtain Theorem 6.117 which is a strengthening of Theorem 6.80. The additional property (7) comes from Theorem 6.114(5).

396

6 Pmax variations

Theorem 6.117. Assume AD holds in L.R/. Then for each set A  R with A 2 L.R/, there is a condition h.M; I /; f i 2 Qmax such that the following hold. (1) M  ZFC . (2) I D .INS /M . (3) A \ M 2 M. (4) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i. (5) .M; I / is A-iterable. (6) f witnesses ˘++ .!1
ha˛ W ˛ < !1 i

of countable transitive sets with the following property. For all A  !1 there exists a set C  !1 , closed and unbounded in !1 , such that for all ˛ 2 C if ˛ is a limit point of C then .A \ ˛; C \ ˛/ 2 a˛ : Lemma 6.118 (ZFC + ˘+ .!1
of length !1 such that:

6.2 Variations for obtaining !1 -dense ideals

397

(1) F D j.f / on a club in !1 ; (2) if b  !1 is a set such that b \ ˛ 2 M for all ˛ < !1 , then b 2 M : Proof. As in the proof of Lemma 6.106 we use the following notation. For each a 2 H.!1 / let M.a/ D L˛ .b [ ¹aº/ where b is the transitive closure of a and ˛ is the least ordinal such that L˛ .b/ is admissible. Let B  !1 be such that (1.1) .F; M/ 2 LŒB, (1.2) for all S  !1 if

S \  2 LŒB

for all  < !1 then S 2 LŒB. Since F 2 LŒB, !1LŒB D !1 and so ˘+ holds in LŒB. Let ha˛ W ˛ < !1 i 2 LŒB be a sequence which witnesses ˘+ in LŒB. Let h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i 2 LŒB be an iteration of .M; I / of length !1 such that for all ˇ < !1 , M

(2.1) Gˇ is M.Mˇ ; ha˛ W ˛ < ˇi/-generic for Coll.!; !1 ˇ /, M

(2.2) if F .ˇ/ is M.Mˇ ; ha˛ W ˛ < ˇi/-generic for Coll.!; !1 ˇ / and if then Gˇ D F .ˇ/.

j0;ˇ .!1M0 / D ˇ

This iteration is easily constructed in LŒB. A key property of the iteration is the following one. Suppose ˇ < !1 and that t 2 aˇ . Then t 2 Mˇ C1 or t … M!1 : This follows the genericity requirement of (2.1). Since F witnesses ˘+ .!1
398

6 Pmax variations

(3.1) A \  2 a . (3.2) F ./ is M.A \ /-generic for Coll.!; /. (3.3) j0; .!1M0 / D  and .b \ ; h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  i/ 2 M.A \ /: By (2.1) and (2.2), for each such , G D F ./ and so b\ 2 M or b\ … M C1 . But if b \  … M C1 then b \  … M!1 which is a contradiction. Hence b \  2 M . Thus for a stationary set of  < !1 , b \  2 M and so b 2 M!1 . In summary we have proved that if b  !1 , b 2 LŒB and if b \  2 M!1 for all  < !1 , then b 2 M!1 . Now suppose b  !1 , b 2 V , and that b \  2 M!1 for all  < !1 . Then

b \  2 LŒB

for all  < !1 since M!1 2 LŒB. Therefore b 2 LŒB by the key property of B, and so b 2 M!1 . Therefore the iteration has the desired properties in V . t u There are absoluteness theorems corresponding to the details to the reader. Theorem 6.121 corresponds to Theorem 6.85.

KT

Qmax . We state one, leaving

Definition 6.119. ˆ˘ : For all X  !1 there is a sequence ha˛ W ˛ < !1 i of elements of H.!1 / such that for all Y  !1 if Y \ ˇ 2 L.X; ha˛ W ˛ < !1 i/ for all ˇ < !1 then contains a club in !1 .

¹˛ < !1 j Y \ ˛ 2 a˛ º t u

The sentence ˆ˘ is a weakening of the principle used in place of ˘ in Lemma 6.118. It is also sufficient to prove the requisite iteration lemmas for KT Qmax . ˆ˘ is implied by ˘+ . The absoluteness theorem for KT Qmax requires the following iteration lemma which is easily proved using Lemma 6.66 and the proof of Lemma 6.118.

6.2 Variations for obtaining !1 -dense ideals

399

Lemma 6.120 (ˆ˘ ). Suppose F W !1 ! H.!1 / is a function which witnesses ˘ .!1
(i) M  ZFC, (ii) I 2 M and I is the tower of ideals I t u

Recall that a tree T  ¹0;1º
KT Q

hH.!2 /; ŒfG IG ; B; X; 2 W X  R; X 2 L.R/iL.R/

max

 :

t u

400

6 Pmax variations

We end this section with a sketch of the proof of Theorem 5.75. For this it is convenient to make the following definition. A tree T  ¹0;1º
t u

It follows, by absoluteness and reflection, that the set of weakly special trees of cardinality !1 is †1 definable in the structure hH.!2 /; 2i using !1 as a parameter. This leads to a strengthening of the sentence ˆ˘ . Definition 6.123. ˆC ˘ : For all A  !1 there exists B  !1 such that (1) A 2 LŒB, (2) the tree TB is weakly special where TB D ¹0;1º
t u

is provably equivalent to the assertion that a certain …2 By the remarks above, sentence holds in the structure, hH.!2 /; 2i:

6.2 Variations for obtaining !1 -dense ideals

401

Therefore by the absoluteness theorem, Theorem 4.64, ˆC ˘ (if appropriately consistent) is a consequence of the axiom ./. Note that while ˆ˘ is consistent with CH, (˘C implies ˆ˘ ); if for all A  !1 , A# exists, then ˆC ˘ implies :CH. Theorem 5.75 is an immediate corollary of the following theorem. Theorem 6.124. Assume the axiom ./. Then ˆC ˘ holds. Proof. By Theorem 4.60, it suffices to prove the following. Suppose h.M0 ; I0 /; a0 i 2 Pmax : Then there exist h.M1 ; I1 /; a1 i 2 Pmax and b1 2 .P .!1 //M1 such that (1.1) h.M1 ; I1 /; a1 i < h.M0 ; I0 /; a0 i, (1.2) a1 2 LŒb1 , (1.3) Tb1 is weakly special in M1 where Tb1 D .¹0;1º
!1M

then d 2 LŒb. j0 W .M0 ; I0 / ! .M0 ; I0 /

be an iteration such that j0 2 M and such that I0 D I \ M0 : Let x 2 R code M. By Theorem 5.34 there exist a transitive inner model N containing the ordinals and ı < !1 such that

6 Pmax variations

402

(3.1) N  ZFC, (3.2) x 2 N , (3.3) ı is a Woodin cardinal in N . Let g0 be N -generic for the partial order .Coll.!1 ; R//N and let g1 be N Œg0 -generic for Coll.!;
j.j0 / W .M0 ; I0 / ! .M0 ; I0 /

is an iteration in N such that I0 D .INS /N \ M0 : Further there exists b1 2 N such that (5.1) b1  !1N , (5.2) j.j0 /.a0 / 2 LŒb1 , (5.3) if b 2 P .!1 /N is a set such that b \ ˛ 2 LŒb1  for all ˛ < !1N , then b 2 LŒb1 : Let

N

T D ¹0;1º
6.2 Variations for obtaining !1 -dense ideals

403

N

(6.3) if s 2 ¹0;1º!1 \ N Œg0 Œg1 Œg is a branch of T then s 2 N Œg0 Œg1 . Let g2  P be N Œg0 Œg1 -generic and let g3  .Coll.!1 ; 1=2 and such that .X \ O/ > 0 for all open sets O  Œ0;1 with X \ O ¤ ;. The latter condition serves to make A separative. The order on A is by set inclusion. Suppose G  A is V -generic and in V ŒG let X D \¹P

V ŒG

j P 2 Gº

V ŒG

where P denotes the closure of P computed in V ŒG. This is P as computed in V ŒG. Then X has measure 1=2 and every member of X is random over V . Suppose I is a uniform, countably complete, ideal on !1 and F W !1 ! P .Œ0;1/: Let YA .F; I / be the set of all pairs .S; P / such that the following hold. (1) S  !1 and S … I . (2) P  Œ0;1 and P 2 A. (3) Suppose hPk W k < !i is a maximal antichain in A below P . Then ¹˛ 2 S j F .˛/ 6 Pk for all k < !º 2 I: (4) If Q  P is a perfect set of measure > 1=2 then ¹˛ 2 S j F .˛/  Qº … I: (5) For all ˛ 2 S , .F .˛// D 1=2. Suppose I is a uniform normal ideal on !1 and that F is a function such that YA .F; I / is nonempty. Suppose .S1 ; P1 / 2 YA .F; I /, .S2 ; P2 / 2 YA .F; I / and that S1  S2 . Then P1  P2 . Therefore if G  P .!1 / n I is a filter in .P .!1 / n I; / then HG D ¹P 2 A j .S; P / 2 YA .F; I / for some S 2 Gº generates a filter in A. Lemma 6.125 (ZFC ). Suppose I is a uniform normal ideal on !1 and F W !1 ! P .Œ0;1/ is a function such that YA .F; I / is nonempty. Suppose .S1 ; P1 / 2 YA .F; I /. (1) Suppose P2 is a perfect subset of P1 and P2 2 A. Let S2 D ¹˛ 2 S1 j F .˛/  P2 º: Then .S2 ; P2 / 2 YA .F; I /.

6.2 Variations for obtaining !1 -dense ideals

405

(2) Suppose S2  S1 and S2 … I . Then there exists .S3 ; P3 / 2 YA .F; I / such that S3  S 2 . Proof. We first prove (1). To show that .S2 ; P2 / 2 YA .F; I / we have only to prove that condition (3) in the definition of YA .F; I / holds for .S2 ; P2 /. The other clauses are an immediate consequence of the fact that .S1 ; P1 / 2 YA .F; I /. We may assume that .P2 / < .P1 / for otherwise there is nothing to prove. Let hXi W i < !i be a maximal antichain in A below P2 . Let hZi W i < !i be a maximal antichain in A of conditions below P1 which are incompatible with P2 . The key point is that we may assume that for each i < !, .Zi \ P2 / < 1=2; if .Z \ P2 / D 1=2 then there exists a condition W 2 A such that (1.1) W < Z, (1.2) .W \ P2 / < 1=2. Clearly ¹Xi j i < !º [ ¹Zi j i < !º is a maximal antichain below P1 . Since .S1 ; P1 / 2 YA .F; I /, for I -almost all ˛ 2 S1 , there exists i < ! such that either F .˛/  Xi or F .˛/  Zi . For every ˛ 2 S2 and for all i < !, .F .˛// D 1=2, F .˛/  P2 and .P2 \ Zi / < 1=2. Therefore for I -almost all ˛ 2 S2 , F .˛/  Xi for some i < !. Therefore condition (3) holds for .S2 ; P2 / and so .S2 ; P2 / 2 YA .F; I /: This proves (1). We prove (2). Suppose G  P .!1 / n I is V -generic for .P .S1 / n I; /. Let j W V ! .M; E/ be the associated generic elementary embedding. Since the ideal I is normal it follows that !1 belongs to the wellfounded part of .M; E/. Since .S1 ; P1 / 2 YA .F; I / it follows that HG is V -generic for A where HG D ¹Q 2 A j j.F /.!1 /  Qº: By part (1) of the lemma this induces a complete boolean embedding  W RO.AjP1 / ! RO..P .S1 / n I; // where AjP1 denotes the suborder of A obtained by restricting to the conditions below P1 . Let b D ^¹c 2 RO.AjP1 / j S2  .c/º and let hXi W i < !i be a maximal antichain below b of conditions in A. For each i < ! let Ti D ¹˛ 2 S2 j F .˛/  Xi º: For each i < !, if Ti … I then .Ti ; Xi / 2 YA .F; I /. Therefore it suffices to show that for some i < !, Ti … I . Note that if Q 2 AjP1 and T  S are such that T  .Q/ then ¹˛ 2 T j F .˛/ 6 Qº 2 I: t u This follows from the definition of HG . Hence Ti … I for all i < !.

6 Pmax variations

406

Lemma 6.126 (ZFC ). The following are equivalent. (1) There is a sequence hP˛ W ˛ < !1 i of perfect subsets of Œ0;1 each of positive measure such that if B  Œ0;1 is a set of measure 1 then P˛  B for some ˛ < !1 . (2) There is a sequence hP˛ W ˛ < !1 i of perfect subsets of Œ0;1 each of positive measure such that if P  Œ0;1 is a perfect set of positive measure then P˛  P for some ˛ < !1 . (3) There is a sequence hP˛ W ˛ < !1 i of perfect subsets of Œ0;1 each of positive measure such that if P  Œ0;1 is a perfect set of positive measure then for each  > 0 there exists ˛ < !1 such that P˛  P and .P n P˛ / < . (4) There is a sequence hB˛ W ˛ < !1 i of borel subsets of Œ0;1 such that each B˛ is of measure 1 and such that if B  Œ0;1 is of measure 1 then B˛  B for some ˛ < !1 . Proof. These are elementary equivalences. We fix some notation. For each closed interval J  Œ0;1 with distinct endpoints let J W J ! Œ0;1 be the affine, order preserving, map which sends J onto Œ0;1. Suppose X  .0; 1/. Let XJ D J1 ŒX . Thus XJ is the subset of J given by scaling X to J . Let X  D \¹J ŒX \ J  j J  Œ0;1 is a closed interval with rational endpointsº and let X  D [¹XJ j J  Œ0;1 is a closed interval with rational endpointsº: It follows that if .X / D 1 then .X  / D 1 and if .X / > 0 then

.X  / D 1: The fact that X  is of measure one is a consequence of the Lebesgue density theorem applied to Œ0;1 n X  . We note that if P and B are borel subsets of Œ0;1 such that P  B  then P   B. Let hP˛ W ˛ < !1 i witness (1). For each ˛ < !1 let B˛ D P˛ . Therefore for each ˛ < !1 , .B˛ / D 1. Suppose B  Œ0;1 and .B/ D 1. Therefore there exists ˛ < !1 such that P˛  B  and so B˛  B since B˛ D P˛ . This proves that (1) implies (4). Trivially, (4) implies (1). We next show that (1) implies (2). Fix hP˛ W ˛ < !1 i. We may assume that for each ˛ < !1 and for each open set O  .0; 1/, if O \ P˛ ¤ ; then

.P˛ \ O/ > 0:

6.2 Variations for obtaining !1 -dense ideals

407

Let Q  Œ0;1 be a perfect (nowhere dense) set of positive measure. Since Q has positive measure, Q is of measure 1. Fix ˛ < !1 such that P˛  Q . Q is an F and so there exist closed (proper) intervals I  J  Œ0;1 with rational endpoints such that P˛ \ I ¤ ; and such that P˛ \ I D QJ \ P˛ \ I D Q \ P˛ \ I: This implies that J .P˛ \ I / D Q \ J .P˛ \ I / and so J .P˛ \ J / \ J .I /  Q and .J .P˛ \ J / \ J .I // > 0. There are only !1 many sets of the form J .P˛ \ J / \ J .I / where I  J  Œ0;1 are closed subintervals with rational endpoints, ˛ < !1 and I \ P˛ ¤ ;. . Therefore these sets collectively witness (2). Finally we show that (2) implies (3). Let hP˛ W ˛ < !1 i be a sequence of perfect subsets of Œ0;1 each of positive measure such that the sequence witnesses (2). Suppose Q  Œ0;1 is a perfect set of positive measure. For each ˇ < !1 let Xˇ D [¹P˛ j ˛ < ˇ and P˛  Qº: We claim that for all sufficiently large ˇ, .Q n Xˇ / D 0. This is immediate. Suppose ˇ < !1 and .Q n Xˇ / > 0. Then there exists ˛ < !1 such that P˛  Q n Xˇ and so .Xˇ / < .X / for some  < !1 . The claim follows. Let hQ˛ W ˛ < !1 i enumerate the perfect subsets of Œ0;1 which can be expressed as a finite union of the P˛ ’s. Thus hQ˛ W ˛ < !1 i witnesses (3). t u Lemma 6.127 (ZFC ). Assume ˘+ .!1
6 Pmax variations

408

Proof. We first prove that (1) implies (3). Fix F and I . It follows immediately from the definition of YA .F; I / that if B  Œ0;1 is a set of measure 1 then F .˛/  B for some ˛ < !1 . The point is that the set ¹Q 2 A j Q  Bº is dense in A. Therefore by Lemma 6.126, (3) holds. We finish by proving that (3) implies (2). Let I be a normal ideal on !1 . Let f W !1 ! H.!1 / be a function which witnesses ˘+ .!1 p for some p 2 f .˛/º: Thus on a club in !1 , F .˛/ is a perfect set of measure 1=2. It is straightforward to u t verify that .!1 ; Œ0;1/ 2 YA .F; I /. Definition 6.128. M Qmax consists of finite sequences h.M; I /; f; F; Y i such that: (1) h.M; I /; f i 2 Qmax ; (2) M  ZFC ; (3) f witnesses ˘++ .!1
F W !1M ! P .Œ0;1/I

(5) Y 2 M is the set YA .F; I / as computed in M, and .!1M ; Œ0;1M / 2 Y . The order on M Qmax is given as follows. Suppose that ¹h.M1 ; I1 /; f1 ; F1 ; Y1 i; h.M2 ; I2 /; f2 ; F2 ; Y2 iº  M Qmax : Then h.M2 ; I2 /; f2 ; F2 ; Y2 i < h.M1 ; I1 /; f1 ; F1 ; Y1 i if h.M2 ; I2 /; f2 i < h.M1 ; I1 /; f1 i in Qmax and if j W .M1 ; I1 / ! .M1 ; I1 /

6.2 Variations for obtaining !1 -dense ideals

409

is the corresponding iteration, (1) j.F1 / D F2 , (2) j.Y1 / D Y2 \ M1 .

t u

We prove the basic iteration lemmas for M Qmax . There are two iteration lemmas, one for models and one for sequences of models. The latter is necessary to show that M Qmax is !-closed. As usual its proof is an intrinsic part of the analysis of M Qmax . We need a preliminary lemma. Lemma 6.129 (ZFC ). Suppose h.M; I /; f; F; Y i 2 M Qmax and Q  Œ0;1 is a perfect set with measure greater than 1=2. Suppose .S; P / 2 Y and

.Q \ P / > 1=2: Suppose that A 2 M,

A  .P .!1 / n I /M ;

and A is open, dense in .P .!1 / n I; /M below S . Then there exists .S  ; P  / 2 Y such that

.Q \ P  / > 1=2 and such that S  2 A. Proof. Fix .S; P / 2 Y and fix Q  Œ0;1 such that

.Q \ P / > 1=2: The key point is that by Lemma 6.125, the set D D ¹P  j .S  ; P  / 2 Y for some S  2 Aº is open dense in AM below P . Let hPi W i < !i 2 M be maximal antichain of conditions below P such that Pi 2 D for all i < !. M is wellfounded and so by absoluteness hPi W i < !i is a maximal antichain in A below P . Therefore for some i < !,

.Q \ Pi / > 1=2 t u

and the lemma follows.

With this lemma the main iterations lemmas are easily proved. As usual it is really the proofs of these iteration lemmas which are the key to the analysis of M Qmax . Lemma 6.130 (ZFC C ˘+ .!1
410

6 Pmax variations

Suppose g W !1 ! H.!1 / is a function which witnesses ˘+ .!1
M˛C1

j˛C1;˛C2 .F˛C1 /.!1

/  P˛

where for each ˇ < !1 , Fˇ D j0;ˇ .F /. The iteration is easily constructed by induction on ˛. Lemma 6.129 guarantees that (1.2) can be satisfied at every stage. The use of Lemma 6.129 is as follows. Fix  < !1 and suppose h.M˛ ; I˛ /; G˛ ; j˛;ˇ W ˛ < ˇ   C 1i is given. Let fC1 ; FC1 , and YC1 be the images of f; F , and Y under j0;C1 . Thus h.MC1 ; IC1 /; fC1 ; FC1 ; YC1 i 2 M Qmax : Suppose .S; P / 2 YC1 and .P \ P / > 1=2. Suppose A 2 MC1 and A is open dense in the partial order .P .!1 / n IC1 ; /M C1 : By Lemma 6.129, there exists .S  ; P  / 2 YC1 such that S   S , S  2 A and

.P \ P  / > 1=2: The model MC1 is countable and so there exists GC1  P .!1 / n IC1 such that GC1 is MC1 -generic for .P .!1 / n IC1 ; /M C1 and such that for all .S; P / 2 YC1

6.2 Variations for obtaining !1 -dense ideals

411

if S 2 GC1 then .P \ P / > 1=2. The filter GC1 is MC1 -generic and so G D ¹P 2 AM C1 j .S; P / 2 YC1 for some S 2 GC1 º is a filter in AM C1 which is MC1 -generic. (Clearly G generates a generic filter which is all we require. By Lemma 6.125, G literally is the filter it generates since .!1 ; Œ0;1/M C1 2 YC1 .) However for each P 2 G,

.P \ P / > 1=2: It follows that \¹P j P 2 Gº  P : This is an elementary property of the generic for Amoeba forcing. Let XG D \¹P j P 2 Gº: Then .XG / D 1=2 and .X \P / D 1=2. But if O  Œ0;1 is open and O \XG ¤ ; then .XG \ O/ ¤ 0. Therefore XG D XG \ P  : Finally

M C1

/  XG

M C1

/  P :

jC1;C2 .FC1 /.!1 and so

jC1;C2 .FC1 /.!1

This verifies that condition (1.2) can be met at every relevant stage. We consider the effect of condition (1.1). Since g witnesses ˘+ .!1
412

6 Pmax variations

(2.1) Suppose hQk W k < !i is a maximal antichain in A below P . Then ¹˛ 2 S j j0;!1 .F /.˛/ 6 Qk for all k < !º 2 J: (2.2) If Q  P is a perfect set of measure > 1=2 then ¹˛ 2 S j j0;!1 .F /.˛/  Qº … J: The other requirements .S; P / must satisfy follow by absoluteness. We first prove (2.1). The key point is that there exists a club C0  !1 such that for all ˛ 2 C0 , D˛ D ¹P 2 AM˛ j P  Qk for some k < !º is dense in AM˛ . The existence of C0 follows from clause (1.2) in the construction of the iteration. Let X  H.!2 / be a countable elementary substructure such that h.M˛ ; I˛ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i and such that D 2 X where D D ¹P 2 AM!1 j P  Qk for some k < !º: Let ˛ D X \ !1 and let MX be the transitive collapse of X . g witnesses ˘+ .!1
6.2 Variations for obtaining !1 -dense ideals

413

Lemma 6.131 (ZFC C ˘+ .!1
(iii) !1

MkC1

D !1

,

(iv) Fk D F0 and fk D f0 , (v) IkC1 \ Mk D Ik , (vi) Yk D YkC1 \ Mk , (vii) if C 2 Mk is closed and unbounded in !1M0 then there exists D 2 MkC1 such that D  C , D is closed and unbounded in C and such that D 2 LŒx for some x 2 R \ MkC1 . Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i such that (1) ¹˛ j j.f0 /.˛/ D g.˛/º contains a club in !1 , (2) YA .j.F0 /; J / \ Mk D j.Yk /. Proof. By Corollary 4.20 the sequence h.Mk ; Ik / W k < !i is iterable. Given this the proof of the lemma is essentially identical to the proof of Lemma 6.130. Let hP˛ W ˛ < !1 i be a sequence of conditions in A which is dense. Let hh.Mk˛ ; Ik˛ / W k < !i; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i be an iteration of h.Mk ; Ik / W k < !i such that the following hold.

414

6 Pmax variations M˛

(1.1) For all ˛ < !1 if ˛ D !1 0 and if g.˛/ is [¹Mk˛ j k < !º-generic for Coll.!; ˛/ then G˛ is the corresponding generic filter. (1.2) For all ˛ < !1 , M0˛C1

j˛C1;˛C2 .F0˛C1 /.!1

/  P˛ :

where for each ˛ < !1 , F0˛ D j0;˛ .F0 /. This is the iteration analogous to that specified in the proof of Lemma 6.131. Given this iteration the remainder of the proof is the same. In constructing this iteration the only point to check here is that Lemma 6.129 can still be applied. It suffices to show the following. Suppose j0;ˇ W h.Mk0 ; Ik0 / W k < !i ! h.Mkˇ ; Ikˇ / W k < !i is a countable iteration of h.Mk ; Ik / W k < !i D h.Mk0 ; Ik0 / W k < !i and suppose Q 2 A. Then there exists an iteration jˇ;ˇ C1 W h.Mkˇ ; Ikˇ / W k < !i ! h.Mkˇ C1 ; Ikˇ C1 / W k < !i of length 1 such that ˇ

M0

jˇ;ˇ C1 .Fˇ /.!1

/Q

where Fˇ D j0;ˇ .F0 /. We verify this in the special case that ˇ D 0; i. e. given Q 2 A we construct an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i of length 1 such that j.F0 /.!1M0 /  Q. The general case is identical. Fix Q and construct by induction on k a sequence h.Sk ; Pk / W k < !i such that for all k 2 !, (2.1) .Q \ Pk / > 1=2, (2.2) .Sk ; Pk / 2 Yk , (2.3) SkC1  Sk , (2.4) The set ¹S 2 .P .!1 //Mk j Si  S for some i 2 !º is Mk -generic for .P .!1 / n Ik ; /Mk .

6.2 Variations for obtaining !1 -dense ideals

415

Lemma 6.129 is used in the construction as follows. Suppose .Sk ; Pk / 2 Yk and A 2 MkC1 is a dense open set in .P .!1 / n IkC1 ; /MkC1 : Suppose .Q \ Pk / > 1=2. YkC1 \ Mk D Yk and so .Sk ; Pk / 2 YkC1 . Therefore by Lemma 6.129 applied to MkC1 , there exists .SkC1 ; PkC1 / 2 YkC1 such that

.Q \ PkC1 / > 1=2, SkC1  Sk and such that SkC1 2 A. For each k 2 !, h.Mk ; Ik /; fk i 2 Qmax and fk D fkC1 . This is a key point for it implies that if A 2 Mk is a dense open set in .P .!1 / n Ik ; /Mk ; then A is predense in .P .!1 / n IkC1 ; /MkC1 : Therefore the genericity conditions (2.4) are easily met and so the sequence h.Sk ; Pk / W k < !i exists. For each k 2 ! let Gk D ¹S 2 .P .!1 //Mk j Si  S for some i 2 !º and let Hk D ¹P 2 AMk j .S; P / 2 Yk for some S 2 Gk º: Thus for each k 2 !,

Hk D ¹P 2 AMk j P \ MkC1 2 HkC1 º and for all P 2 Hk , .Q \ P / > 1=2. For each k < !, Gk is Mk -generic and so for each k < !, Hk is Mk -generic for AMk . Let j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i

be the iteration given by [¹Gk j k < !º and let X D \¹P j P 2 H0 º D \¹P j P 2 [¹Hk jk 2 !ºº: Therefore j.F0 /  X  Q and so the iteration is as desired. We make the usual associations. Suppose G  M Qmax is L.R/-generic. Then (1) fG D [¹f j h.M; I /; f; F; Y i 2 Gº, (2) FG D [¹F j h.M; I /; f; F; Y i 2 Gº, (3) IG D [¹j  .I / j h.M; I /; f; F; Y i 2 Gº, (4) YG D [¹j  .Y / j h.M; I /; f; F; Y i 2 Gº, (5) P .!1 /G D [¹M \ P .!1 / j h.M; I /; f; F; Y i 2 Gº, where for each h.M; I /; f; F; Y i 2 G, j  W .M; I / ! .M ; I  / is the (unique) iteration such that j.f / D fG .

t u

416

6 Pmax variations

The basic analysis of M Qmax follows from these lemmas in a by now familiar fashion. The results of this we give in the following theorem. The analysis requires that M Qmax is suitably nontrivial. More precisely one needs that for each set A  R with A 2 L.R/ there exists h.M; I /; f; F; Y i 2 M Qmax such that hH.!1 /M ; A \ H.!1 /M ; 2i  hH.!1 /; A; 2i and such that .M; I / is A-iterable. By Lemma 6.47 and Lemma 6.127, this follows from the existence of a huge cardinal. Theorem 6.132. Assume that for each set A  R with A 2 L.R/ there exists h.M; I /; f; F; Y i 2 M Qmax such that (i) hH.!1 /M ; A \ H.!1 /M ; 2i  hH.!1 /; A; 2i, (ii) .M; I / is A-iterable. Then M Qmax is !-closed and homogeneous. Suppose G  M Qmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal !1 -dense ideal on !1 ; (3) IG is the nonstationary ideal; (4) YG D YA .FG ; INS /; (5) fG witnesses ˘++ .!1
6.2 Variations for obtaining !1 -dense ideals

417

Proof. The proof that M Qmax is !-closed follows closely the proof that Qmax is !closed. Suppose hpk W k < !i is a strictly decreasing sequence of conditions in M Qmax and that for each k < !, pk D h.Mk ; Ik /; fk ; Fk ; Yk i: Let f D [¹fk j k < !1 º and let F D [¹Fk j k < !1 º: For each k < ! let and let

jk W .Mk ; Ik / ! .Mk ; Ik / pk D h.Mk ; Ik /; f; jk .Fk /; jk .Yk /i

where jk is the iteration such that j.fk / D f . By boundedness it follows that hpk W k < !i is a sequence of conditions in M Qmax which satisfies the conditions (i)–(vi) of Lemma 6.131. By the nontriviality of M Qmax there exists a condition h.N ; J /; g; G; Y i 2 M Qmax such that

hpk W k < !i 2 .H.!1 //M :

By Lemma 6.131 there exists an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i such that j 2 N and such that in N , (1.1) ¹˛ j j.f /.˛/ D g.˛/º contains a club in !1 , (1.2) for all k < !,

YA .j.F /; J / \ Mk D j.Yk /:

Thus h.N ; J /; j.f /; j.F /; Zi 2 M Qmax and for all k < !, h.N ; J /; j.f /; j.F /; Zi < pk where

Z D .YA .j.F /; J //N :

In a similar fashion the other claims are proved by just adapting the proofs of the corresponding claims for Qmax . Because of the requirement (2) in the definition of M Qmax , (5) is immediate from (1). (4) is an immediate consequence of (1) and the definition of the order on M Qmax . (6) follows from (4) by the definition of YA .FG ; INS /. t u

418

6 Pmax variations

There is a version of M Qmax analogous to Qmax for which the analysis can be carried out just assuming ADL.R/ . This version is a little tedious to define and we leave the details to the reader. The net effect of this is the following theorem that M Qmax is suitably nontrivial just assuming ADL.R/ . This is analogous to Theorem 6.80. Theorem 6.133. Assume ADL.R/ . Then for each set A  R with A 2 L.R/ there exists h.M; I /; f; F; Y i 2 M Qmax such that (1) hH.!1 /M ; A \ H.!1 /M ; 2i  hH.!1 /; A; 2i, t u

(2) .M; I / is A-iterable. Combining the two previous theorems we obtain the following theorem. Theorem 6.134. Assume ADL.R/ . Suppose G  M Qmax is L.R/-generic. Then L.R/ŒG  ZFC and in L.R/ŒG: (1) the nonstationary ideal on !1 is !1 -dense; (2) ˘++ .!1
(3) there is a sequence hB˛ W ˛ < !1 i of borel subsets of Œ0;1 such that each B˛ is of measure 1 and such that if B  Œ0;1 is set of measure 1 then B˛  B for t u some ˛ < !1 . There are absoluteness theorems for M Qmax analogous to the absoluteness theorems for Qmax . These require the following preliminary lemmas. With these lemmas in hand the proof of the absoluteness theorem, Theorem 6.139, is an easy variation of the proof of the corresponding theorem for Qmax , Theorem 6.85. We leave the details as an exercise. We generalize the definition of YA .F; INS / to the setting of the stationary tower. Suppose that ı is strongly inaccessible and that F W !1 ! P .Œ0;1/: Let YA .F; ı/ be the set of all pairs .S; P / such that the following hold. (1) S 2 Q 1=2 then ¹a 2 S j F .a \ !1 /  Qº is stationary in S . (5) For all a 2 S , .Q/ D 1=2 where Q D F .a \ !1 /. The relationship between YA .F; INS / and YA .F; ı/ is summarized in the following lemma which is an immediate consequence of the definitions. Lemma 6.135. Suppose that ı is strongly inaccessible and that F W !1 ! P .Œ0;1/: Then YA .F; INS / D ¹.S; P / j .S; P / 2 YA .F; ı/ and S  !1 º:

t u

The next two lemmas, Lemma 6.136 and Lemma 6.137, are used to prove the iteration lemma, Lemma 6.138, just as Lemma 6.125 and Lemma 6.129 are used to prove the basic iteration lemmas for M Qmax , Lemma 6.130. Lemma 6.136. Suppose ı is strongly inaccessible F W !1 ! P .Œ0;1/ is a function such that YA .F; ı/ is nonempty. Suppose .S1 ; P1 / 2 YA .F; ı/. (1) Suppose P2 is a perfect subset of P1 and P2 2 A. Let S2 D ¹a 2 S1 j F .a \ !1 /  P2 º: Then .S2 ; P2 / 2 YA .F; ı/. (2) Suppose that S2  S1 in Q (i) M  ZFC, (ii) I 2 M and I is the tower of ideals I t u

Theorem 6.139 is an absoluteness theorem for M Qmax . Again the proof is an easy adaptation of earlier arguments and stronger absoluteness theorems can be proved. For this theorem one uses the iteration lemma, Lemma 6.138, modifying the proof of the corresponding absoluteness theorem for Qmax , Theorem 6.85. The situation here is simpler since there are no restricted …2 sentences to deal with. Theorem 6.139 (˘+ .!1
MQ

hH.!2 /; X; 2 W X  R; X 2 L.R/iL.R/

6.3

max

 :

t u

Nonregular ultrafilters on !1

We consider ultrafilters on !1 . Definition 6.140. Suppose that U is a uniform ultrafilter on !1 . (1) The ultrafilter U is nonregular if for each set W  U of cardinality !1 there exists an infinite set Z  W such that \Z ¤ ;:

422

6 Pmax variations

(2) The ultrafilter U is weakly normal if for any function f W !1 ! !1 ; either ¹˛ j ˛  f .˛/º 2 U or there exists ˇ < !1 such that ¹˛ j f .˛/ < ˇº 2 U:

t u

We begin with the basic relationship between the existence nonregular ultrafilters on !1 and the existence of weakly normal ultrafilters on !1 . This relationship is summarized in the following theorem of Taylor .1979/. This theorem is the analog for !1 of the theorem that if is measurable then there is a normal measure on . Theorem 6.141 (Taylor). Suppose that U is a uniform ultrafilter on !1 . (1) Suppose that U is weakly normal. Then U is nonregular. (2) Suppose that U is nonregular. Then there exists a function f W !1 ! !1 such that U  is weakly normal where U  D ¹A  !1 j f 1 ŒA 2 U º:

t u

The relative consistency of the existence of nonregular ultrafilters on !1 first established by Laver. Laver proved that if there exists an !1 -dense uniform ideal on !1 and ˘ holds, then there exists a nonregular ultrafilter on !1 . Huberich improved Laver’s theorem proving the theorem without assuming ˘. Thus in L.R/Qmax there is a nonregular ultrafilter on !1 . The basic method for producing nonregular ultrafilters on !1 is to produce them from suitably saturated normal ideals on !1 . The approach is due to Laver and involves the construction of indecomposable ultrafilters on the quotient algebra, P .!1 /=I: Definition 6.142. Suppose that B is a countably complete boolean algebra. An ultrafilter U B is indecomposable if for all X  B, _X 2 U if and only if _Y 2 U for some countable set Y  X .

t u

The fundamental connection between normal ideals on !1 and nonregular ultrafilters on !1 is given in the following lemma due to Laver.

6.3 Nonregular ultrafilters on !1

423

Lemma 6.143 (Laver). Suppose I  P .!1 / is a normal uniform ideal. Let B D P .!1 /=I: (1) Suppose that U  B is an ultrafilter which is indecomposable. Let W D ¹A  !1 j ŒAI 2 U º: Then W is a weakly normal ultrafilter on !1 . (2) Suppose that W is a weakly normal ultrafilter on !1 such that W \ I D ;. Let U D ¹ŒAI j A 2 W º: Then U is an indecomposable ultrafilter on B.

t u

The following theorem was first proved by Laver assuming ˘ and then by Huberich, .Huberich 1996/, without any additional assumptions. Theorem 6.144 (Huberich). Let B D RO.Coll.!; !1 //: Then there is an ultrafilter U on B which is indecomposable.

t u

We prove the following stronger version. Suppose ı is an ordinal. Then Add.!; ı/ is the Cohen partial order for adding ı many Cohen reals. Theorem 6.145. Let ı be an ordinal and let B D RO .Coll.!; !1 /  Add .!; ı// : Then there is an ultrafilter U on B which is indecomposable. Proof. Let P D Coll.!; !1 /  Add .!; ı/. More formally P is the set of pairs .f; g/ such that f is a finite partial function from ! to !1 and g is a finite partial function from !  ı to ¹0;1º. For each q D .f; g/ in P let ˛q be the largest ordinal in the range of f . Fix a cardinal such that !1 ; ı < . For each countable elementary substructure X  V such that P 2 X let PX D P \ X . Thus PX is a countable partial order. For each such X  V let FX D ¹_D j D  PX and D is dense º where the join, _D, is computed in B. Let F D [¹FX j X  V ; X 2 P!1 .V / and P 2 X º: We prove that if S  F is finite then ^S ¤ 0 in B. Suppose hb0 ; : : : ; bn i is a finite sequence of elements of F . For each i  n let Xi  V be a countable elementary substructure containing P and let Di  PXi be a dense subset such that bi D _Di . By

424

6 Pmax variations

reordering if necessary we may assume that Xi \ !1  Xj \ !1 : for all i  j . The key point is the following. Suppose X  V , X is countable and P 2 X . Suppose q 2 P and ˛q < X \ !1 . Here ˛q is the ordinal defined above. Then there is a condition q0 2 PX such that if q1 < q0 and q1 2 PX then there is a condition p 2 P such that p < q, p < q1 and ˛p < X \ !1 . Using this it is straightforward to construct a sequence h.p0 ; q0 /; : : : ; .pn ; qn /i of pairs of conditions in P such that for all i  j : (1.1) pi < qi ; (1.2) qi 2 Di ; (1.3) ˛pi < Xi \ !1 ; (1.4) pj  pi . Thus pn  ^¹bi j i  nº. Let F be the filter in B generated by the finite meets of elements of F . Let U be an ultrafilter on B extending F . We prove that U is indecomposable. Suppose X  B and _X 2 U . Let Y be the set of conditions q 2 P such that q  b for some b 2 X. Let W be the set of conditions q 2 P such that q ^ b D 0 in B for all b 2 X. Let D D Y [ W . Thus D is dense in P . Let Z  V be a countable elementary substructure such that ¹P ; Dº  Z. Let D D D \ Z. Thus D is dense in PZ and so _D 2 F  U . Let b D _.Y \ Z/ and let c D _.W \ Z/ Thus _D D b _ c. Further c  _W and ._W /^._X/ D 0. Thus c … U and so b 2 U . But b D _.Y \Z/ and Z is countable. Therefore b  _X for some countable set X  X. Thus U is indecomposable. t u An immediate corollary of Lemma 6.143 is the following theorem of .Huberich 1996/.

Theorem 6.146 (Huberich). Assume there is an !1 -dense ideal on !1 . Then there is a t u nonregular ultrafilter on !1 . Corollary 6.147. Assume there is an !1 -dense ideal on !1 . Suppose ı is a cardinal and that G  Add .!; ı/ is V -generic. Then in V ŒG there is a nonregular ultrafilter t u on !1 .

6.3 Nonregular ultrafilters on !1

425

The following theorem is now immediate. Theorem 6.148. Assume ZF C AD is consistent. Then so are (1) ZFC C “The nonstationary ideal on !1 is !1 -dense”. (2) ZFC C “There is a nonregular ultrafilter on !1 ”. (3) ZFC C “There is a nonregular ultrafilter on !1 ” C “ 2@0 is large”.

t u

The following theorem, in conjunction with Theorem 6.148, completes the analysis of the consistency strength of the assertion that there exists an !1 -dense ideal on !1 . The proof of this theorem involves the core model induction which is also the method used to prove Theorem 5.111 and as noted is beyond the scope of this book. Theorem 6.149. Suppose that I is a normal, uniform, ideal on !1 such that I is !1 dense. Then L.R/  AD: t u Corollary 6.150. The following are equiconsistent: (1) ZF C AD. (2) ZFC C “INS is !1 -dense”. (3) ZFC C “There is a normal, uniform, !1 -dense ideal on !1 ”.

t u

The consistency strength of the existence of a nonregular ultrafilter on !1 is not known.

Chapter 7

Conditional variations In this chapter we define two conditional variations of Pmax . The models obtained are in essence simply conditional versions of the Pmax -extension, i. e. the models maximize the collection of …2 sentences which can hold in the structure hH.!2 /; INS ; 2i given that some specified sentence holds. The Qmax -extension is an example of such a variation. It conditions the extension on the assertion that the nonstationary ideal is !1 -dense. There is an analogy for these conditional variations with variations of Sacks forcing. Suppose  is a …13 sentence which is true in V and that there is a function f W ! ! ! which eventually dominates all those functions which are constructible. Then  is true in LP where P is Laver forcing. This can be proved by a modification of Mansfield’s argument. Thus the Laver extension of L realizes all possible …13 sentences conditioned on the existence of fast functions. These variations of Pmax also yield models in which conditional forms of Martin’s Maximum hold. For example we shall define a variation Bmax such that in L.R/Bmax the Borel Conjecture holds together with a large fragment of Martin’s Axiom.

7.1

Suslin trees

Throughout this section, a tree, T , is a Suslin tree if T is an !1 -Suslin tree; i. e. if T is an .!1 ; !1 /-tree which satisfies the countable chain condition. We define a variation of Pmax which we shall denote Smax . Our goal is to have that Suslin trees exist in the resulting generic extension of L.R/. We give the sentence relative to which we shall condition the final model. Definition 7.1. ˆS : For all X  !1 there is a transitive model M such that (1) M  ZFC , (2) ˘ holds in M , (3) X 2 M , (4) for every tree T 2 M , if T is a Suslin tree in M then T is a Suslin tree in V . u t The sentence ˆS is implied by ˘ and it will hold after any (sufficiently long) forcing iteration where cofinally often ˘ holds and Suslin trees are preserved. In the model which we obtain, a strong form of ˆS actually holds.

7.1 Suslin trees

427

Definition 7.2. ˆC S : For every set X  !1 there exists Y  !1 such that X 2 LŒY  and such that every tree T 2 LŒY  which is a Suslin tree in LŒY  is a Suslin tree in V. t u This strong version of ˆS seems quite subtle in the context of large cardinals. For example assuming for all A  !1 , A# exists; it is not obvious that it can even hold. We prove that if for all A  !1 , A# exists then ˆC S implies :CH. Lemma 7.3. Suppose that A  !1 is a set such that R  LŒA and such that A# exists. Then there is a tree T 2 LŒA such that T is a Suslin tree in LŒA and such that T has a cofinal branch. Proof. We naturally view any .!1 ; !1 / tree as an order on !1  ! such that for each ˛ 2 !1 , ¹˛º  ! is the set of nodes in T on the ˛ th level. We restrict our considerations to trees with only infinite levels and which are splitting. We may suppose that for all ˛ < !1 , ˛  !1LŒA\˛ : Let T D .!1  !; Let Z  be the set of .h.f0 ; s  .h.m0 / //i; T / such that (4.1) .h.f0 ; h/i; S / 2 YBC .I /, (4.2) s 2 !
The elements of Z  provide the witnesses necessary to show that 0/ .h.f0 ; h.m /i; T / 2 YBC .I /: 0

t u

Lemma 7.36. Suppose I is a uniform normal ideal on !1 . Suppose .h.fi ; hi / W i < ni; S / 2 YBC .I /; .h.fn ; hn /i; S / 2 YBC .I /; and that T  S is a set such that T … I . Then there exists m < ! such that .h.fi ; h.m/ i / W i < n C 1i; T / 2 YBC .I /: Proof. For each .i; m/ 2 .n C 1/  ! define Hi;m 2 ! ! by Hi;m .j / D hi .k C m/ where k is such that 2  j C 1 < 2kC1 . By Lemma 7.35, we can suppose that for each i < n C 1, and for each m < !, k

.h.fi ; h.m/ i /i; T / 2 YBC .I /: Assume the lemma fails. Thus by Lemma 7.30, for each m 2 !,  .h.fi ; h.m/ i / W i < n C 1i; T / … YBC .I /:  .I / as a refinement of ZBC .I /, Therefore as a consequence of the definition of YBC for each m 2 ! there exists a sequence

hOim W i < n C 1i of open sets such that (1.1) T n ¹˛ 2 T j fi .˛/ 2 Oim for some i < n C 1º 2 I , (1.2) for all i < n C 1, Oim is Hi;m -small. The ideal I is countably complete and so there is a set T1  T such that T nT1 2 I and such that for all m 2 ! and for all ˛ 2 T1 , fi .˛/ 2 Oim for some i < n C 1.

7.2 The Borel Conjecture

461

For each i < n C 1 let gi 2 ! ! be such that (2.1) .h.fi ; gi /i; S / 2 ZBC .I /, (2.2) for sufficiently large k 2 !, gi .m/  hi .k/ if m < 5k . By Lemma 7.33, for each i < n C 1 and for sufficiently large k 2 !, [¹Oim j m > kº is gi.k/ -small. For each i < n C 1 let Bi D \¹[¹Oim j m > kº j k 2 !º and so for each i < n C 1, Bi is Œgi E -small. Therefore, since for each i < n C 1, .h.fi ; gi /i; S / 2 ZBC .I /; there is a set T2  T1 such that T1 n T2 2 I and such that for all i < n C 1 and for all ˛ 2 T2 , fi .˛/ … Bi . Fix ˛ 2 T2 . Then ˛ 2 T1 and so for all m 2 !, fi .˛/ 2 Oim for some i < n C 1. Thus for some i < n C 1, the set ¹m 2 ! j fi .˛/ 2 Oim º is infinite and so fi .˛/ 2 Bi which is a contradiction. Therefore for some m 2 !, .h.fi ; h.m/ i / W i < n C 1i; T / 2 YBC .I /.

t u

Lemma 7.36 can be recast as follows. This reformulation is in essence what is required to prove the iteration lemmas. Lemma 7.37. Suppose that h.M; I /; a; Y i 2 Bmax , .h.fi ; hi / W i < ni; S / 2 Y , .h.fn ; hn /i; S / 2 Y , and that hOi W i < ni is a finite sequence of open sets such that each Oi is hi -small. For each i < n let hIki W k < !i be a sequence of open intervals in .0;1/ with rational endpoints such that the sequence witnesses that Oi is hi -small. Suppose A 2 M, A  .P .!1 / n I /M ; and A is dense below S in .P .!1 / n I; /M . Suppose m0 2 !. There exists m > m0 and there exists T 2 A such that (1) T  S , (2) .h.fi ; h.m/ i / W i < n C 1i; T / 2 Y , (3) fi .˛/ … Iki for all k < m, for all i < n, and for all ˛ 2 T . Proof. This follows from Lemma 7.36 by absoluteness. Fix m0 2 !. Let T be the tree of attempts to build the sequences hIki W k < !i to refute the lemma. So T is the set of hti W i < ni such that for some m > m0 :

462

7 Conditional variations

(1.1) For all i < n, ti D h.rki ; ski / W k < mi where for all k < m, a) 0  rki < ski  1, b) rki 2 Q, c) ski 2 Q, d) .ski rki / < 1=.hi .k/ C 1/. (1.2) For all T  S with T 2 A, either .h.fi ; h.m/ i / W i < n C 1i; T / … Y; or for some i < n and for some ˛ 2 T , fi .˛/ 2 [¹.rki ; ski / j k < mº: The ordering on T is by (pointwise) extension, hsi W i < ni  hti W i < ni if ti  si for all i < n. Clearly T 2 M. Suppose T has an infinite branch. Then by absoluteness, T has an infinite branch in M. We work in M and assume toward a contradiction that T has an infinite branch. Any such branch yields for each i < n a sequence h.rki ; ski / W k < !i of open intervals in .0;1/ with rational endpoints such that for all i < n and for all k < !, jski rki j < 1=.hi .k/ C 1/: These sequences have the additional property that for all T 2 A such that T  S and for all m < ! either .h.fi ; h.m/ i / W i < n C 1i; T / … Y; or for some i < n and for some ˛ 2 T , fi .˛/ 2 [¹.rki ; ski / j k < mº: For each i < n let OQ i D [¹.rki ; ski / j k < !º: Thus for each i < n, OQ i is hi -small. Let T0 D ¹˛ 2 S j fi .˛/ … OQ i for all i < nº: Since .h.fi ; hi / W i < ni; S / 2 Y ,

T0 … I:

A is dense below S and so there exists T 2 A such that T  T0 . By Lemma 7.36, there exists m < ! such that .h.fi ; h.m/ i / W i < n C 1i; T / 2 Y: This is a contradiction and so T is wellfounded in M. Hence T is wellfounded in V . t u

7.2 The Borel Conjecture

463

The iteration lemmas are proved using the following lemmas which in turn follow rather easily from the previous lemmas. Lemma 7.38. Suppose h.M; I /; a; Y i 2 Bmax . Suppose .h.fi ; hi / W i < ni; S / is an element of Y and suppose h.fi ; hi ; Si / W i < !i is a sequence extending h.fi ; hi ; S / W i < ni such that for each i < ! if i n then .h.fi ; hi /i; Si / 2 Y . Suppose hBi W i < !i is a sequence of borel sets such that each i < !, if i < n then Bi is hi -small and if i n then Bi is Œhi E -small. Then there is an iteration j W .M; I / ! .M ; I  / of length 1 such that (1) for all i < !, if !1M 2 j.Si / then j.fi /.!1M / … Bi , (2) !1M 2 j.S /. Proof. Let hAi W n  i < !i enumerate the sets in M which are dense in .P .!1 / n I /M . Using Lemma 7.37 it is straightforward to build sequences hTi W i < !i;

hIji W i; j < !i;

and

hNi W i < !i

such that for all i < ! the following hold. Let Z D ¹i < ! j Si \ Ti ¤ ;º: (1.1) Ni D 0 and Ti D S for i < n. (1.2) If i n then Ti 2 Ai and either Ti  Si or Ti \ Si D ;. (1.3) If i n then TiC1  Ti  S , Ni 2 ! and Ni < NiC1 . (1.4) Iji is an open interval with rational endpoints and

.Iji / < 1=.hi .Ni C j / C 1/: (1.5) Bi  [¹Iji j j < !º. .Ni /

(1.6) .h.fj ; hj

/ W j 2 Z \ i i; Ti / 2 Y .

(1.7) If i < n then for all ˛ 2 Tn , for all j < i , fj .˛/ … [¹Ikj j k < Nn º: (1.8) If i n then for all ˛ 2 TiC1 , for all j < i , if j 2 Z then fj .˛/ … [¹Ikj j k < NiC1 º:

464

7 Conditional variations

We first construct hTi W i  ni;

hIji W i < n; j < !i;

and

hNi W i  ni:

For this we need only specify hIji W i < n; j < !i; Tn and Nn . For each i < n let hIji W j < !i be a sequence of open intervals with rational endpoints such that Bi  [¹Iji j j < !º and such that for all j < !,

.Iji / < 1=.hi C 1/: By Lemma 7.37, there exist L0 2 ! and T 0 2 An such that (2.1) T 0  Sn or T 0 \ Sn D ;, (2.2) T 0  S , 0

(2.3) .h.fi ; hi.L / / W i < ni; T 0 / 2 Y , (2.4) for all k < L0 , for all i < n, fi .˛/ … Iki for all ˛ 2 T 0 . Let Tn D T 0 and let Nn D L0 . We next suppose m n and that hTi W i  mi;

hIji W i < m; j < !i;

and

hNi W i  mi

i are given. For each i < m and k < ! let Jki D IkCN . Therefore m m/ / W i 2 Z \ mi; Tm / 2 Y .h.fi ; h.N i

and for each i < m, the sequence hJki W k < !i witnesses that Oi is .hi.Nm / /-small where Oi D [¹Jki j k < !º: By Lemma 7.37, there exist L0 2 ! and T 0 2 AmC1 such that (3.1) T 0  SmC1 or T 0 \ SmC1 D ;, (3.2) T 0  Tm , 0

m / .L / / / W i 2 Z \ mi; T 0 / 2 Y , (3.3) .h.fi ; .h.N i

(3.4) for all k < L0 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 0 .

7.2 The Borel Conjecture

465

By Lemma 7.37 again, there exist L00 2 ! and T 00 2 AmC1 such that (4.1) T 00  T 0 , 00

m / .L / / / W i 2 Z \ m C 1i; T 00 / 2 Y , (4.2) .h.fi ; .h.N i

(4.3) for all k < L00 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 00 . Of course if T 0 \ SmC1 D ; then one can simply let T 00 D T 0 and L00 D L0 . Set TmC1 D T 00 and NmC1 D Nm C L00 . Choose a sequence hJk W k < !i such .N / that hJk W k < !i witnesses that Bm is hm mC1 -small. The sequence exists since Bm is Œhm E -small. For each k < ! set Ikm D Jk . Therefore by induction the sequences exist. Let G be the filter generated by ¹Ti j i < !º. Thus G is M-generic. Let j W .M; I / ! .M ; I  / be the associated iteration of length 1. It follows from (1.5), (1.7), and (1.8) that for all t u i < !, if !1M 2 j.Si / then j.fi /.!1M / … Bi . There is an analogous version of the previous lemma for sequences of models. We shall apply this lemma only to sequences which are iterable. However the lemma holds for sequences which are not necessarily iterable and it is this more general version which we shall prove, (for no particular reason). Lemma 7.39. Suppose that h.Mk ; Ik / W k < !i is a sequence such that for each k < !, Mk is a countable transitive model of ZFC, Ik 2 Mk and such that in Mk , Ik M is a uniform normal ideal on !1 k . For each k < ! let Yk D .YBC .Ik //Mk : Suppose that for all k < !, (i) Mk 2 MkC1 , (ii) jMk jMkC1 D .!1 /MkC1 , Mk

(iii) !1

MkC1

D !1

,

(iv) IkC1 \ Mk D Ik , (v) Yk D YkC1 \ Mk , M

(vi) for each A 2 Mk such that A  P .!1 k / \ Mk n Ik , if A is predense in .P .!1 / n Ik /Mk then A is predense in .P .!1 / n IkC1 /MkC1 .

466

7 Conditional variations

Suppose .h.fi ; hi / W i < ni; S / is an element of Y0 and suppose h.fi ; hi ; Si / W i < !i is a sequence extending h.fi ; hi ; S / W i < ni such that for each i < ! if i n then .h.fi ; hi /i; Si / 2 Yi . Suppose hBi W i < !i is a sequence of borel sets such that each i < !, if i < n then Bi is hi -small and if i n then Bi is Œhi E -small. Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i of length 1 such that (1) for all i < !, if !1M0 2 j.Si / then j.fi /.!1M0 / … Bi , (2) !1M0 2 j.S /. Proof. Let hAi W n  i < !i enumerate the sets A 2 [¹Mk j k 2 !º such that if A 2 Mk then

A  .P .!1 / n Ik /Mk

and A is predense in

.P .!1 / n Ik ; /Mk :

By (vi) in the hypothesis of the lemma we can suppose that for each i < !, Ai 2 Mi : Following the proof of Lemma 7.38 it is straightforward, using Lemma 7.37 and (v), to build sequences hTi W i < !i, hIji W i; j < !i and hNi W i < !i such that for all i < ! the following hold. Let Z D ¹i < ! j Si \ Ti ¤ ;º: (1.1) Ni D 0 and Ti D S for i < n. (1.2) If i n then Ti 2 Ai and either Ti  Si or Ti \ Si D ;. (1.3) If i n then TiC1  Ti  S , Ni 2 ! and Ni < NiC1 . (1.4) Iji is an open interval with rational endpoints and

.Iji / < 1=.hi .Ni C j / C 1/: (1.5) Bi  [¹Iji j j < !º. .Ni /

(1.6) .h.fj ; hj

/ W j 2 Z \ i i; Ti / 2 Yi .

7.2 The Borel Conjecture

467

(1.7) If i < n then for all ˛ 2 Tn , for all j < i , fj .˛/ … [¹Ikj j k < Nn º: (1.8) If i n then for all ˛ 2 TiC1 , for all j < i , if j 2 Z then fj .˛/ … [¹Ikj j k < NiC1 º: We first construct hTi W i  ni;

hIji W i < n; j < !i;

and hNi W i  ni:

For this we need only specify hIji W i < n; j < !i; Tn and Nn . For each i < n let hIji W j < !i be a sequence of open intervals with rational endpoints such that Bi  [¹Iji j j < !º and such that for all j < !,

.Iji / < 1=.hi C 1/: By Lemma 7.37, there exist L0 2 ! and T 0 2 An such that (2.1) T 0  Sn or T 0 \ Sn D ;, (2.2) T 0  S , 0

(2.3) .h.fi ; hi.L / / W i < ni; T 0 / 2 Yn , (2.4) for all k < L0 , for all i < n, fi .˛/ … Iki for all ˛ 2 T 0 . Let Tn D T 0 and let Nn D L0 . We next suppose m n and that hTi W i  mi;

hIji W i < m; j < !i;

and

hNi W i  mi

i are given. For each i < m and k < ! let Jki D IkCN . m Therefore m/ .h.fi ; h.N / W i 2 Z \ mi; Tm / 2 Ym i

and for each i < m, the sequence hJki W k < !i witnesses that Oi is .hi.Nm / /-small where Oi D [¹Jki j k < !º: By (v), Ym D YmC1 \ Mm and so

m/ .h.fi ; h.N / W i 2 Z \ mi; Sm / 2 YmC1 : i

468

7 Conditional variations

By Lemma 7.37, there exist L0 2 ! and T 0 2 MmC1 such that (3.1) T 0  !1M0 and T 0  a for some a 2 AmC1 , (3.2) T 0  SmC1 or T 0 \ SmC1 D ;, (3.3) T 0  Tm , 0

m / .L / (3.4) .h.fi ; .h.N / / W i 2 Z \ mi; T 0 / 2 Ym , i

(3.5) for all k < L0 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 0 . By Lemma 7.37 once more, there exist L00 2 ! and T 00 2 MmC1 such that (4.1) T 00  T 0 , 00

m / .L / / / W i 2 Z \ m C 1i; T 00 / 2 YmC1 , (4.2) .h.fi ; .h.N i

(4.3) for all k < L00 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 00 . Of course, as in the proof of Lemma 7.38, if T 0 \ SmC1 D ; then one can simply let T 00 D T 0 and L00 D L0 . Set TmC1 D T 00 and NmC1 D Nm C L00 . Choose a sequence hJk W k < !i such .N / that hJk W k < !i witnesses that Bm is hm mC1 -small. The sequence exists since Bm is Œhm E -small. For each k < ! set Ikm D Jk . Therefore by induction the sequences exist. Let G be the filter generated by ¹Ti j i < !º. Thus G is [¹Mi j i < !º-generic. Let j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i be the associated iteration of length 1. It follows from (1.5), (1.7) and (1.8) that for all t u i < !, if !1M0 2 j.Si / then j.fi /.!1M0 / … Bi . With these lemmas the main iterations lemmas are easily proved. As usual it is really the proofs of these iteration lemmas which are the key to the analysis of Bmax . Lemma 7.40 (CH). Suppose h.M; I /; a; Y i 2 Bmax and that J is a normal uniform ideal on !1 . Then there is an iteration j W .M; I / ! .M ; I  / such that (1) J \ M D I  , (2) j.Y / D YBC .J / \ M .

7.2 The Borel Conjecture

469

Proof. Let hSk;˛ W k < !; ˛ < !1 i be a sequence of pairwise disjoint J -positive sets such that !1 D [¹Sk;˛ j k < !; ˛ < !1 º: Let hs˛ W ˛ < !1 i be an enumeration (with repetition) of all finite sequences of open subsets of .0;1/ such that for each finite sequence s of open subsets of .0;1/, and for each .k; ˛/ 2 !  !1 , ¹ı 2 Sk;˛ j s D sı º is a set which is J -positive. Let hB˛ W ˛ < !1 i be an enumeration of all the borel subsets of .0;1/. Let x be a real which codes M and let C  !1 be a closed unbounded set of ordinals which are admissible relative to x. Fix a function F W !  !1M ! M such that (1.1) F is onto, (1.2) for all k < !, F jk  !1M 2 M, (1.3) for all A 2 M if A has cardinality !1M in M then A  ran.F jk  !1M / for some k < !. The function F is simply used to anticipate elements in the final model. Our situation is similar to that in the proof of Lemma 7.7. Suppose j W .M; I / ! .M ; I  / is an iteration. Then we define j.F / D [¹j.F jk  !1M / j k < !º and it is easily verified that M is the range of j.F /. This follows from (1.3). Implicit in what follows is that for ˇ 2 C if j W .M; I / ! .M ; I  / is an iteration of length ˇ then j.!1M / D ˇ. This is a consequence of Lemma 4.6(1). We construct an iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i of M of length !1 using the function F to provide a book-keeping device for dealing with elements of j0;!1 ..P .!1 / n I /M / and for dealing with elements of j0;!1 .Y /.

470

7 Conditional variations

More precisely construct by induction, an iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i as follows. Suppose ı < !1 and that h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  ıi: Fix .k; / 2 !  !1 such that ı 2 Sk; . If ı … C or if  ı then choose Gı to be any Mı -generic filter. If ı 2 C and if  < ı there are three cases. We first suppose that j0;ı .F /.k; / D .h.fi ; hi / W i < ni; S / and that .h.fi ; hi / W i < ni; S / 2 j0;ı .Y /. Suppose sı D hOi W i < ni is a sequence of length n such that for each i < n, Oi is hi -small. Let h.fi ; hi ; Si / W i < !i be a sequence extending the sequence h.fi ; hi ; S / W i < ni such that for all i < !, .h.fi ; hi /i; Si / 2 j0;ı .Y / and such that for all

.h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y /;

.h.f 0 ; h0 /i; S 0 / D .h.fi ; hi /i; Si / for infinitely many i < !. Let hBi0 W i < !i be a sequence of borel sets extending hOi W i < ni such that for all i n, Bi0 is Œhi E -small and such that for all ˛ < ı if .h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y / and if B˛ is h0 -small then for some j > n, B˛ D Bj0 and .h.f 0 ; h0 /i; S 0 / D .h.fj ; hj /i; Sj /: By Lemma 7.38, there exists an iteration j W .Mı ; Iı / ! .MıC1 ; IıC1 / of length 1 such that Mı

(2.1) for all i < !, if !1 Mı

(2.2) !1

M

2 j.Si / then j.fi /.!1 ı / … Bi0 ,

2 j.S /.

Let jı;ıC1 D j and let Gı be the associated Mı -generic filter. The remaining cases are similar. Choose j W .Mı ; Iı / ! .MıC1 ; IıC1 /

7.2 The Borel Conjecture

471

of length 1 such that for all .h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y / if

Mı

!1 then

2 j.S 0 / M

j.f 0 /.!1 ı / … B˛

for all ˛ < ı such that B˛ is h0 -small. Let jı;ıC1 D j and let Gı be the associated Mı -generic filter. If j0;ı .F /.k; / 2 .P .!1 / n Iı /Mı ; then choose j such that in addition to the requirement above, Mı

!1

2 j.S /

where S D j0;ı .F /.k; /. In each of these last two cases j exists by Lemma 7.38. This completes the inductive construction of the iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i: It is straightforward to verify that this iteration is as required. The first case of the construction at the inductive step guarantees that j0;!1 .Y /  ZBC .J / and this implies that j0;!1 .Y /  YBC .J /: The second case guarantees J \ M!1 D I!1 and so j0;!1 .Y / D YBC .J / \ M!1 :

t u

The analysis of the Bmax -extension requires the generalization of Lemma 7.40 to sequences of models. We state this lemma only for the sequences that arise, specifically those sequences of structures coming from descending sequences of conditions in Bmax . Suppose that hpk W k < !i is a sequence of conditions in Bmax such that for all k < !, pkC1 < pk : We let hpk W k < !i be the associated sequence of conditions which is defined as follows. For each k < ! let h.Mk ; Ik /; ak ; Yk i D pk and let

jk W .Mk ; Ik / ! .Mk ; Ik /

472

7 Conditional variations

be the iteration obtained by combining the iterations given by the conditions pi for i > k. Thus jk is uniquely specified by the requirement that jk .ak / D [¹ai j i < !º: For each k < !, pk D h.M ; Ik /; jk .ak /; jk .Yk /i: We note that by Corollary 4.20, the sequence h.Mk ; Ik / W k < !i is iterable (in the sense of Definition 4.8). Lemma 7.41 (CH). Suppose hpk W k < !i is a sequence of conditions in Bmax such that for each k < ! pkC1 < pk : Let hpk W k < !i be the associated sequence of Bmax conditions and for each k < ! let h.Mk ; Ik /; ak ; Yk i D pk : Suppose that J is a normal uniform ideal on !1 . Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i such that for all k < !; (1) J \ Mk D Ik , (2) YBC .J / \ Mk D j.Yk /. Proof. By Corollary 4.20, the sequence h.Mk ; Ik / W k < !i is iterable. The lemma follows by a routine modification of the proof of Lemma 7.40 using Lemma 7.39 in place of Lemma 7.38. t u Theorem 7.43 establishes the nontriviality of Bmax in the sense required for the analysis of L.R/Bmax . The proof requires Theorem 7.18, Theorem 7.42 and the transfer principle supplied by Theorem 5.36.

7.2 The Borel Conjecture

473

Theorem 7.42. Suppose that ı is a Woodin cardinal. Suppose A  R and that every set of reals which is projective in A is ı C -weakly homogeneously Suslin. Then there is an iterable structure .M; I / such that M  ZFC C CH and such that (1) M 

AC .I /,

(2) A \ M 2 M, (3) hH.!1 /M ; A \ Mi  hH.!1 /; Ai, (4) .M; I / is A-iterable. Proof. Suppose that G  Coll.!1 ; 7.2 The Borel Conjecture

477

is a function where Œ!
t u

Suppose G  PU is V -generic. Suppose B  .0;1/ is a borel set in V . Let BG denote the borel set defined by interpreting B in V ŒG. Lemma 7.48. Suppose U is a selective ultrafilter on ! and that h 2 ! ! . Suppose p 2 ! ! is such that (1) p.0/ D 0, (2) p.k/ < p.k C 1/ for all k 2 !, (3) limk!1 .p.k C 1/ p.k// D 1. Let h 2 ! ! be the function such that for all j 2 !, h .j / D h.k/ where k 2 ! and p.k/  j < p.k C 1/. Suppose that G  PU is V -generic and that O 2 V ŒG is an open set such that O  .0;1/ and O is h-small. Then there exists an open set W 2 V such that in V , W is h -small and such that in V ŒG, O  WG :

478

7 Conditional variations

Proof. Let 0 be a term for O. We work in V . We may suppose that .;; !/  0  .0;1/ and that .;; !/  “0 is h-small”: Let 1 be a term for an h-small cover of  . Again we may suppose that .;; !/  “1 is an h-small cover of 0 ”: We prove that there exists an open set W  .0;1/ such that W is h -small and such that .;; /  0  WG for some 2 U . By homogeneity this suffices. Let I be the set of open subintervals of .0;1/. By Lemma 7.47, for each s 2 Œ!
.Hs .k//  2=.h.k/ C 1/: For each k 2 ! n 0 let Nk 2 ! be such that for all j Nk , p.j C 1/ p.j / > 3 2kC2 : Let N0 D 0. By increasing the Nk , k > 0, if necessary, we may suppose that for all k < !, Nk < NkC1 : For each m 2 ! let m 2 U be such that .t; m /  1 .k/  H t .k/ for all t  m C 1 and for all k  NmC1 . The ultrafilter U is selective and so there exists  2 U such that for all k 2 !, j  \ ŒNk ; NkC1 /j  1; and such that for all s 2 Œ   mº  m where m D [s. For each k 2 ! let ak be the least element of  n Nk . Let J be the set of intervals of the form Hs .j / such that for some k 2 !, Nk  j < NkC1 and such that

s  .NkC1 \  / [ ¹akC1 º:

7.2 The Borel Conjecture

Let W D [J. We claim that W is h -small and that .;;  /  0  WG : We first prove that Suppose

.;;  /  0  WG : .s; /  .;;  /

and that j < [s. Let k 2 ! be such that Nk  j < NkC1 : We may suppose that

 ¹k 2  j k > [sº:

There are two cases. First, suppose that akC1 2 s: Then .s; /  1 .j /  H t .j / where t D s \ .akC1 C 1/. However H t .j / 2 J and so .s; /  1 .j /  WG : Second, suppose that akC1 … s: Then .s; /  1 .j /  H t .j / where t D s \ akC1 . Again H t .j / 2 J and so again .s; /  1 .j /  WG : Thus

.;;  /  0  WG :

We finish by proving that W is h -small. We note that since for all k 2 !, j  \ ŒNk ; NkC1 /j  1; it follows that for all k 2 !, jP .  \ NkC1 /j  2kC1 : Suppose m 2 ! and let k 2 ! be such that Nk  m < NkC1 : Let Jm be the set of intervals of the form Hs .m/ such that s  .NkC1 \  / [ ¹akC1 º: Thus J D [¹Jm j m 2 !º:

479

480

7 Conditional variations

Further for each m 2 !,

jJm j  2kC2

and each interval in Jm has length at most 2=.h.m/ C 1/ where k 2 ! is such that

Nk  m < NkC1 :

 be the collection of intervals For each m 2 !, let Jm  interval .a; b/ 2 Jm , Jm contains the intervals,

.a; .a C b/=2/; 

Let J D

 [¹Jm

obtained as follows. For each

.a C .b a/=4; b .b a/=4/;

and

..a C b/=2; b/:

j m < !º. Thus W D [J  :

Suppose m 2 ! and let k 2 ! be such that Nk  m < NkC1 : Each interval in

 Jm

has length at most 1=.h.m/ C 1/ and  j  3 2kC2 : jJm

For each j 2 ! such that

p.m/  j < p.m C 1/;

we have that h .j / D h.m/. Further since m Nk , p.m C 1/ p.m/ 3 2kC2 : It follows that W is h -small.

t u

Suppose G  PU is V -generic. Let aG D [¹s j .s; / 2 Gº and let hG W ! ! ! be the enumeration function of aG . We note the following. Suppose that I is a normal, uniform, ideal on !1 and that P is ccc. Suppose that G  P is V -generic. Then in V ŒG the ideal I defines three ideals, (1) I0 which is the ideal generated by I , I0 D ¹A  !1 j A  B for some B 2 I º; (2) I1 which is the -ideal generated by I0 , (3) I2 which is the normal ideal generated by I0 . Under certain circumstances, these three ideals can coincide.

7.2 The Borel Conjecture

481

Lemma 7.49. Suppose U is a selective ultrafilter on ! and that for all X  U , if jX j D !1 then there exists 2 U such that n  is finite for all  2 X . Suppose I is a normal uniform ideal on !1 . Suppose G  PU is V -generic. Let I.G/ be the ideal generated by I in V ŒG. Then in V ŒG: (1) I.G/ is a normal uniform ideal on !1 ; (2) suppose f W !1 ! .0;1/ is an injective function such that f 2 V , then V ŒG .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

for some n 2 !. Proof. Suppose F 2 V ŒG is a function, F W !1 ! !1 such that F .˛/ < ˛ for all ˛ > 0. Suppose A  !1 , A 2 V ŒG and A … I.G/ . We may suppose 0 … A. We must show that there exists B such that B  A, B … I.G/ and such that F jB is constant. Let F 2 V PU be a term for the function F and let A 2 V P be a term for the set A. Fix a condition .s0 ; 0 / 2 G. We may suppose that .s0 ; 0 /  A … I.G/ : 

We work in V . Let A be the set of ˛ < !1 such that there exists a condition .s; / < .s0 ; 0 / with the property that .s; /  ˛ 2 A : Since .s0 ; 0 /  A … I.G/ ; 

it follows that A … I . For each ˛ 2 !1 choose a condition .s˛ ; ˛ / < .s0 ; 0 / and an ordinal ˛ < ˛ such that .s˛ ; ˛ /   .˛/ D ˛ ; and such that if ˛ 2 A then .s˛ ; ˛ /  ˛ 2 A ; 

and if ˛ … A then

.s˛ ; ˛ /  ˛ … A :

Let 2 U be such that for all ˛ 2 A, \ .! n ˛ / is finite.

482

7 Conditional variations

For each ˛ 2 A let n˛ 2 ! be such that n n˛  ˛ . The ideal I is normal. Therefore there exists a set B  A such that B … I and there exists .s; n; / 2 Œ!
.s; n n/   .˛/ D :

By the genericity of G we may suppose that .s; n n/ 2 G: This proves (1). We prove (2). Fix a function f W !1 ! .0;1/ such that f 2 V and such that f is injective. Assume that for each n 2 !, V ŒG : .h.f; h.n/ G /i; !1 / 62 .ZBC .I.G/ //

Then for each n 2 ! there exist an open set On and a set An 2 I.G/ such that On is h.n/ G -small and such that ¹˛ < !1 j f .˛/ … On º  An : I.G/ is the ideal generated by I and PU is ccc. Therefore there must exist A 2 I such that for all n < ! and for all ˛ 2 !1 n A, f .˛/ 2 On . Let X D ¹f .˛/ j ˛ 2 Aº. Thus X 2 V , jX j D !1 and X is ŒhG E -small in V ŒG. This is a contradiction. Therefore for some n 2 !, V ŒG : .h.f; h.n/ G /i; !1 / 2 .ZBC .I.G/ //

A similar argument shows the following. Suppose that p 2 ! ! \ V and that (1.1) p.0/ D 0, (1.2) p.k/ < p.k C 1/ for all k 2 !, (1.3) limk!1 .p.k C 1/ p.k// D 1. Let hG;p 2 ! ! be the function such that for all j 2 !, hG;p .j / D h.k/ where k 2 ! and p.k/  j < p.k C 1/. Then for some n < !, V ŒG .h.f; h.n/ : G;p /i; !1 / 2 .ZBC .I.G/ //

It follows that for some n 2 !, V ŒG : .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

t u

7.2 The Borel Conjecture

483

Lemma 7.50. Suppose U is a selective ultrafilter on ! and that I is a normal uniform ideal on !1 . Suppose G  PU is V -generic. Let I.G/ be the normal ideal generated by I in V ŒG. Then in V ŒG: (1) I.G/ \ V D I ; (2) .YBC .I //V D V \ .YBC .I.G/ //V ŒG ; (3) suppose f W !1 ! .0;1/ is an injective function such that f 2 V , then V ŒG .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

for some n 2 !. Proof. (1) is an immediate consequence of the fact that PU is ccc. We prove (2). Suppose B  .0;1/ is a borel set in V . Let BG be the interpretation of B in V ŒG. It suffices to prove that for all .h.fi ; hi / W i < ni; S / 2 .YBC .I //V ; .h.fi ; hi / W i < ni; S / 2 .ZBC .I.G/ //V ŒG : Granting this (2) follows from the definition of .YBC .I.G/ //V ŒG as a subset of .ZBC .I.G/ //V ŒG . The claim that .YBC .I //V  .ZBC .I.G/ //V ŒG follows from Lemma 7.48. To illustrate how we suppose .h.f; h/i; S / 2 .YBC .I //V and prove that .h.f; h/i; S / 2 .ZBC .I.G/ //V ŒG : Fix p0 2 ! ! such that .h.f; h /i; S / 2 .YBC .I //V where: (1.1) p0 .0/ D 0 and p0 .k/ < p0 .k C 1/ for all k < !; (1.2) for some m 2 !, m > 1 and p0 .k/ D k m for all sufficiently large k < !;

484

7 Conditional variations

(1.3) for all i < n and for all j < !, hi .j / D hi .k/ where k < ! is such that p0 .k/  j < p0 .k C 1/. Suppose O 2 V ŒG is an open set such that O is h-small. By Lemma 7.48, there exists an open set W 2 V such that W is h -small and such that O  WG : In V , ¹˛ 2 S j f .˛/ … W º is I -positive. Hence in V ŒG, ¹˛ 2 S j f .˛/ … Oº is I.G/ positive since by (1) I.G/ \ V D I: Therefore .h.f; h/i; S / 2 .ZBC .I.G/ //V ŒG : The general case is similar. Finally we prove (3). Fix f W !1 ! .0;1/ such that f is injective and such that f 2 V . Suppose V ŒG0  is a ccc extension of V such that V ŒG0   MA C “.2@0 /V < 2@0 ”: Let U0 2 V ŒG0  be a selective ultrafilter such that U  U0 and such that in V ŒG0 , for all X  U0 , if jX j D !1 then there exists 2 U0 such that \ .! n  / is finite for all  2 X. Suppose G1  PU0 is V ŒG0 -generic. By Lemma 7.47(2), G1 \ PU is V -generic. Therefore without loss of generality we may suppose that G1 \ PU D G. Let I.G0 / be the normal ideal generated by I in V ŒG0  and let I.G0 ;G1 / be the ideal generated by I.G0 / in V ŒG0 ŒG1 . By Lemma 7.49, there exists n 2 ! such that V ŒG0 ;G1  : .h.f; h.n/ G /i; !1 / 2 .YBC .I.G0 ;G1 / //

Therefore V ŒG .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

since I.G/  I.G0 ;G1 / . Combining Theorem 7.44 and Lemma 7.50 we obtain the following corollary.

t u

7.2 The Borel Conjecture

485

Theorem 7.51. Assume ADL.R/ . Then L.R/Bmax  ZFC C Borel Conjecture: Proof. By Theorem 7.44 it suffices to prove the following. Suppose h.M0 ; I0 /; a0 ; Y0 i 2 Bmax and that

f0 W !1M0 ! .0;1/

is an injective function such that f0 2 M0 . Then there exists a condition h.M1 ; I1 /; a1 ; Y1 i 2 Bmax such that h.M1 ; I1 /; a1 ; Y1 i < h.M0 ; I0 /; a0 ; Y0 i and such that for some h 2 M1 , .h.j.f0 /; h/i; !1M1 / 2 Y1 ; where j is the iteration of .M0 ; I0 / such that j.a0 / D a1 . Fix f0 and h.M0 ; I0 /; a0 ; Y0 i. Let z 2 R code M0 and let N be a transitive inner model of ZFC C CH such that Ord  N , z 2 N and such that for some ı < !1 , N  “ı is a Woodin cardinal”: We also require that !1 is strongly inaccessible in N . Since AD holds in L.R/, N exists by Theorem 5.34. By Lemma 7.40, there exists an iteration j W .M0 ; I0 / ! .M0 ; I0 / such that j 2 N and such that (1.1) .INS /N \ M0 D I0 , (1.2) j.Y / D .YBC .INS //N \ M0 . Let U 2 N be a selective ultrafilter on ! and let G0  .PU /N be N -generic. Let G1  .Coll.!1 ; t u

8.1 Condensation Principles

497

Corollary 8.8. Suppose that M is a transitive model of ZFC and that j WM !N is an elementary embedding of M into a transitive set N . Suppose that M  Axiom of Condensation: Then M  N .

t u

Corollary 8.9 (Axiom of Condensation). Suppose that j W V ! M  V ŒG is a generic elementary embedding. Then V DM t u

and j is the identity. Theorem 8.10. Suppose that A  ı and that condensation holds for A. (1) Suppose that B   and that B 2 LŒA. Then condensation holds for B. (2) Suppose that P ./  LŒA: j j

Then 2

C

D jj .

t u

Corollary 8.11. Assume the Axiom of Condensation holds in V . Then GCH holds. u t D. Law has improved Corollary 8.11, proving that ˘ follows from the Axiom of Condensation, Law .1994/. The proof yields a different proof of Corollary 8.11; the original proof used Namba forcing. Theorem 8.12 (Law). Assume the Axiom of Condensation holds in V . Then ˘ holds. Proof. Suppose that j WM !N is an elementary embedding such that (1.1) M and N are transitive, (1.2) M  ZC C †1 -Replacement, (1.3) N D ¹j.f /.!1M / j f 2 M º, (1.4) .H.!2 //M 2 N .

498

8 | principles for !1

Then M  ˘: To see this fix f0 W !1M ! M such that f0 2 M and such that h.H.!2 //M ;
499

Let .AX ; X ; ıX / be the image of .A; ; ı/ under the transitive collapse of X . The key point is that j.X / witnesses condensation in N for j.AX / and so by absoluteness, AX 2 N; since ¹j.˛/ j ˛ < ıX º is closed under j.X /. But this implies that .H.!2 //M 2 N since .H.!2 //M 2 LıX ŒAX . Thus .N; M; j / satisfies (1.1)–(1.4) and so M ˘ which implies that ˘ holds in V .

t u

Remark 8.13. The proof of Theorem 8.12 easily adapts to prove directly that the Axiom of Condensation implies that for any (uncountable) regular cardinal ı, ˘ holds at ı on any stationary subset of ı. It is open whether the Axiom of Condensation implies ˘C or whether it implies t u principles such as !1 . Natural models in which the Axiom of Condensation holds are provided by AD. Theorem 8.14. Assume AD holds in L.R/ and let M D H.!1 / \ .HODŒx/L.R/ where x 2 R. Then M  ZFC C Axiom of Condensation:

t u

We shall need a strong form of condensation. This we now define. Definition 8.15. Suppose that M is a transitive set closed under the G¨odel operations and that F W Ord \ M ! M is a bijection. The function F witnesses strong condensation for M if for any X  hM; F; 2i; FX D F j.Ord \ MX / where FX and MX are the images of F and M under the transitive collapse of X .

t u

We say that strong condensation holds for M if there exists a function F W Ord \ M ! M which witnesses strong condensation for M . The Axiom of Strong Condensation is the axiom which asserts that strong condensation holds for H. / for all uncountable .

500

8 | principles for !1

Remark 8.16. (1) The definition of strong condensation imposes some unnecessary requirements on M . A slightly more general definition could be given by specifying as a witness, a wellordering of M . (2) We shall essentially only be concerned with strong condensation for transitive sets of the form H.ı/ where ı is an uncountable cardinal (actually !3 in most cases). t u We note that in the definition of a witness for strong condensation it is necessary only to consider elementary substructures which lie in V as opposed to the case of witnesses for condensation where it is necessary to consider elementary substructures which are generic over V . This is verified in the following theorem. Theorem 8.17. Suppose that M is a transitive set closed under the G¨odel operations and that F W Ord \ M ! M is a bijection. Suppose that N is a transitive inner model such that (1) N  ZC C †1 -Replacement, (2) ¹M; F º  N , (3) F witnesses strong condensation for M in N . Then F witnesses strong condensation for M .

t u

As an immediate corollary of Theorem 8.6 and Theorem 8.17 one obtains; Corollary 8.18. Suppose that M is a transitive set closed under the G¨odel operations and that F W Ord \ M ! M is a bijection which witnesses strong condensation for M . Suppose that A  Ord and that A 2 M . Then condensation holds for A. t u Suppose that strong condensation holds for H. / for some cardinal > !1 . Then ¹X  j ordertype.X / D !1 º is not stationary in P . /. Therefore there are no Ramsey cardinals below . This is in contrast to condensation which can hold below the least measurable cardinal. Theorem 8.19. Assume AD holds in L.R/ and that x 2 R. Let N D HODL.R/ Œx: Suppose that  is an uncountable cardinal of N which is below the least weakly comt u pact cardinal of N . Then strong condensation holds for .H. //N in N .

|

NS 8.2 Pmax

501

Remark 8.20. (1) Theorem 8.19 generalizes to other inner models of AD, satisfying “V D L.P .R//”, provided that a particular form of AD is assumed, see Theorem 9.9. (2) Suppose that the Axiom of Condensation holds. Does strong condensation hold for H.!2 /? (3) Suppose that A  Ord and that for each uncountable cardinal of LŒA, strong condensation holds in LŒA for H. /LŒA . Suppose that A# exists. Then there exists ˛ < !1 and a set A  ˛ such that LŒA D LŒA : (4) Does condensation or strong condensation capture the combinatorial essence of inner models like L? One test question is the following.  Suppose that N is a transitive inner model of ZFC containing the ordinals such that for each uncountable cardinal of N , strong condensation holds in N for H. /N . Suppose that covering fails for N in V . Must there exist a real x such that N  LŒx‹ Note that if N D LŒA for some A  Ord, then by (3) and Jensen’s Covering Lemma, the answer is yes. t u

|

NS 8.2 Pmax |

NS  We define Pmax as a variation of Pmax . For this definition and the subsequent analysis we shall use a generalization of the partial orders PU to the case where U is an ultrafilter on !1 , cf. the discussion preceding Lemma 7.47.

Definition 8.21. Suppose that U is an ultrafilter on !1 . PU is the set of pairs .s; f / such that s  !1 is finite and such that f W Œ!1 
t u

502

8 | principles for !1

Thus PU is a generalization of Prikry forcing to the case of ultrafilters on !1 . The standard properties of Prikry forcing, suitably rephrased, hold for PU . This is summarized in the following lemmas which generalize Lemma 7.47. Lemma 8.22 (Prikry property). Suppose that U is an ultrafilter on !1 . Suppose that .s; f / 2 PU and b 2 RO.PU /. Then there exists .s; f  / 2 PU such that .s; f  /  b t u or such that .s; f  /  b 0 . Lemma 8.23 (Geometric Condition). Suppose that M is a transitive model of ZC, U 2 M and that M  “U is a uniform ultrafilter on !1 ”: M Suppose  !1 is an infinite cofinal set of ordertype !. Suppose that for all f W Œ!1M 
|

NS 8.2 Pmax

503

We note that by Theorem 6.28 it is possible for the following to hold.  For each uniform ultrafilter U on !1 there exists a normal saturated ideal I on !1 such that RO.PU / Š P .!1 /=I: The most elegant method for achieving |0NS would be to obtain the following.  For some ultrafilter U on !1 , U extends the club filter and RO.PU / Š P .!1 /=INS : Unfortunately this is not possible. Lemma 8.24. Suppose that I is a normal ideal on !1 , U is a uniform ultrafilter on !1 and that RO.PU / Š P .!1 /=I: Then I \ U ¤ ;.

t u

A weaker requirement would be that for some ultrafilter U on !1 , U extends the club filter and RO.PU / Š B where B is an !2 -complete boolean subalgebra of P .!1 /=INS . Even this is not possible. Lemma 8.25. Suppose that I is a normal ideal on !1 , U is a uniform ultrafilter on !1 and that RO.PU / Š B where B is an !2 -complete boolean subalgebra of P .!1 /=I . Then I \ U ¤ ;. Proof. Fix a function F W !1 ! Œ!1 ! such that F induces the given isomorphism RO.PU / Š B  P .!1 /=I: We may suppose that for each limit ordinal ˛ < !1 , F .˛/ is cofinal in ˛. For each ordinal ˛, let F˛  P .˛/ be the tail filter given by F .˛/. Let M D L ŒF; U  where  is least such that !1 <  and such that L ŒF; U   ZC:

504

8 | principles for !1

Similarly for each ˛ < !1 let M˛ D L˛ ŒF j˛; F˛  where ˛ is least such that ˛ < ˛ and such that L˛ ŒF j˛; F˛   ZC: Clearly, for all ˛ < !1 , ˛ < !1 . Suppose that G  .P .!1 / n I; / is V -generic. Then the generic ultraproduct Y hM˛ ; F˛ i=G Š hM; U i: Thus there exists a set A  !1 such that !1 n A 2 I and such that for all ˛ 2 A, (1.1) ˛ < !1 , (1.2) ˛ D .!1 /M˛ . Therefore for any formula .x0 ; x1 /, M  ŒF; U \ LŒF; U  if and only if ¹˛ j M˛  ŒF j˛; U˛ º … I; where for each limit ordinal ˛ < !1 , U˛ D M˛ \ F˛ : Let F be the filter dual to I . Assume toward a contradiction that F  U: Then for any formula .x0 ; x1 /, M  ŒF; U \ LŒF; U  if and only if ¹˛ j M˛  ŒF j˛; U˛ º 2 U: This contradicts Tarski’s theorem on the undefinability of truth.

t u

These lemmas however do not rule out the following. There is a set Y of triples .U; I; B/ such that (1) U is a uniform ultrafilter on !1 which extends the club filter, (2) I is a normal uniform saturated ideal on !1 , (3) B is an !2 -complete boolean subalgebra of P .!1 /=I , (4) RO.PU / Š B,

|

NS 8.2 Pmax

505

and such that Y satisfies the condition, INS D \¹I j .U; I; B/ 2 Y º: If the isomorphisms witnessing (4) are induced by a single function F W !1 ! Œ!1 ! then this function yields a function witnessing |NS . |

NS This is how we shall obtain |NS in the Pmax -extension except the ultrafilters U will be generic over the model, see Theorem 8.84. In fact there will exist an .!1 ; 1/distributive partial order PF (defined from F ) for adding U such that

RO.PF  PU / Š B  P .!1 /=INS ; see Lemma 8.76 and Corollary 8.88. We continue to fix some notation. Definition 8.26. Suppose that F W !1 ! Œ!1 ! and that U is a uniform ultrafilter on !1 . (1) For each function h W Œ!1 
t u

506

8 | principles for !1

Suppose that F and U are as in Definition 8.26. In general IU;F is not a proper ideal. Suppose that IU;F is a proper ideal and that F  P .!1 / n IU;F is a V -normal ultrafilter (occurring in a set generic extension of V ). Let .M; E/ D Ult.V; F / and let j W V ! .M; E/ be the corresponding elementary embedding. Since IU;F is a normal ideal, !2V  OrdM ; i. e. !2V is contained in the wellfounded part of M . Thus j.F /.!1V / 2 Œ!1V ! : The key point is that by Lemma 8.23, it follows that j.F /.!1V / is V -generic for PU . Let GF denote the V -generic filter GF  PU determined by

j.F /.!1V /.

Thus GF D ¹p 2 PU j Zp;F 2 F º:

This motivates the next definition. Definition 8.27. Suppose that F W !1 ! Œ!1 ! ; U is a uniform ultrafilter on !1 , and that IU;F is a proper ideal. Let RU;F be the set of pairs .S; p/ such that (1) S  !1 and S … IU;F , (2) p 2 PU , (3) if G  PU is V -generic and p 2 G then there exists a V -normal ultrafilter F  P .!1 / n IU;F such that S 2 F , such that F is set generic over V ŒG and such that G D GF :

t u

The following lemma is an immediate consequence of the definitions. Lemma 8.28. Suppose that F W !1 ! Œ!1 ! and U is a uniform ultrafilter on !1 such that IU;F is a proper ideal. Suppose that F  P .!1 / n IU;F is a V -normal ultrafilter which is set generic over V . Then for each S 2 F there exists p 2 GF such that V : .S; p/ 2 RU;F

|

NS 8.2 Pmax

507

Proof. Fix S 2 F . Since GF  PU is V -generic, either there exists p 2 GF as desired or the following must hold, (1.1) if

FO  P .!1 / n IU;F

is a V -normal ultrafilter, set generic over V , such that S 2 FO , then GFO ¤ GF : The relevant point is that (1.1) is a first order property of the pair .S; GF /. But F is a counterexample to this.

t u

The next lemma gives a simple characterization of RU;F . Lemma 8.29. Suppose that F W !1 ! Œ!1 ! and U is a uniform ultrafilter on !1 such that IU;F is a proper ideal. Suppose that S 2 P .!1 / n IU;F and that p 2 PU . Then .S; p/ 2 RU;F if and only if for all q  p,

Zq;F \ S … IU;F :

Proof. The lemma easily follows from the definitions and Lemma 8.23 which gives the geometric condition which characterizes when a cofinal ! sequence in !1V is V -generic for PU . If .S; p/ 2 RU;F then it is immediate that for all q  p, Zq;F \ S … IU;F : Now suppose that .S; p/ … RU;F . Then by the definability of forcing, there must exist q0  p such that if G  PU is a V -generic filter, with q0 2 G, then G ¤ GF for any V -normal ultrafilter, F , such that (1.1) F  .P .!1 / n IU;F /V , (1.2) S 2 F , (1.3) F is set generic over V . It follows that in V , Zq0 ;F \ S 2 IU;F . |

t u

NS  We define Pmax . The definition is closely related to that of Pmax which is given as Definition 5.41.

508

8 | principles for !1 |

NS Definition 8.30. Pmax is the set of pairs

.h.Mk ; Yk / W k < !i; F / such that hMk W k < !i is iterable and such that the following hold for all k < !. (1) Mk is a countable transitive model of ZFC. Mk

MkC1

(2) Mk 2 MkC1 ; !1

D !1

(3) [¹Mk j k 2 !º 

 AC .

.

(4) Strong condensation holds in Mk for Mk \ V where  is the least inaccessible cardinal of Mk . (5) F 2 M0 and

F W !1M0 ! Œ!1M0 ! :

(6) For each (nonzero) limit ordinal ˛ < !1 , sup.F .˛// D ˛: (7) Yk 2 Mk and Mk

a) for each U 2 Yk , U is a uniform ultrafilter on !1 b) for each U 2 Yk , .IU;F /

Mk

in Mk ,

is a proper ideal and

M .!1 k ; p/

2 .RU;F /Mk

where p D .1PU /Mk . (8) Yk D ¹U \ Mk j U 2 YkC1 º. (9) For each U 2 YkC1 , a) .IU;F /MkC1 \ Mk D .IW;F /Mk , b) .RU;F /MkC1 \ Mk D .RW;F /Mk , where W D U \ Mk . (10) \¹.IU;F /Mk j U 2 Yk º D Mk \ .INS /MkC1 . Mk

(11) Let Ik 2 Mk be the ideal on !1

which is dual to the filter,

Fk D \¹U j U 2 Yk º: Then

\¹.IU;F /Mk j U 2 Yk º  Ik :

(12) hMk W k < !i is iterable.

|

NS 8.2 Pmax

509

|

NS is defined as follows. A condition The ordering on Pmax O O .h.Mk ; Yk / W k < !i; FO / < .h.Mk ; Yk / W k < !i; F / if hMk W k < !i 2 MO 0 , hMk W k < !i is hereditarily countable in MO 0 and there exists an iteration j W hMk W k < !i ! hMk W k < !i such that:

(1) j.F / D FO ; (2) hMk W k < !i 2 M0 and j 2 M0 ; (3) For all k < !,

j.Yk / D ¹U \ Mk j U 2 YO0 º

and

M

O

INS kC1 \ Mk D .INS /M1 \ Mk I (4) For each U 2 YO0 , O



a) .IU;FO /M0 \ Mk D .IW;FO /Mk , O



b) .RU;FO /M0 \ Mk D .RW;FO /Mk , where W D U \ Mk . Remark 8.31.

t u

(1) Suppose that |

NS .h.Mk ; Yk / W k < !i; F / 2 Pmax :

Then for all k < !, .INS /MkC1 \ Mk D .[¹.INS /Mi j i < !º/ \ Mk : (2) An immediate consequence of condition (6) is that !1LŒF  D !1M0 : Therefore if

|

NS .h.Mk ; Yk / W k < !i; F / 2 Pmax

and if

j W hMk W k < !i ! hMk W k < !i

is a countable iteration, then j is uniquely determined by j.F /. This is by condition (3) and by Lemma 5.43. u t |

NS is suitably nontrivial assuming ADL.R/ by proving an We shall prove that Pmax iteration lemma for structures of the form .M; I/ where

I D .Q Lemma 8.37. Suppose that F W !1 ! Œ!1 ! is a function such that for every uniform ultrafilter, U , on !1 , the normal ideal IU;F is proper. Then there is a normal uniform ideal I on !1 such that I D \¹IU;F j U 2 Y º; where Y is the set of uniform ultrafilters on !1 which are disjoint from I .and so extend the filter dual to I /. Proof. Let ˇ!1 denote the set of all uniform ultrafilters on !1 . We define by induction on ˛ a normal ideal I˛ as follows: I0 D \¹IU;F j U 2 ˇ!1 º and for all ˛ > 0, I˛ D \¹IU;F j U 2 ˇ!1 and for all  < ˛, I \ U D ;º: It follows easily by induction that if ˛1 < ˛2 then I˛1  I˛2 : Thus for each ˛, I˛ is unambiguously defined as the intersection of a nonempty set of uniform normal ideals on !1 . The sequence of ideals is necessarily eventually constant. Let ˛ be least such that I˛ D I˛C1 and let I D I˛ . Thus I is a uniform normal ideal on !1 such that I D \¹IU;F j U 2 Y º; where Y is the set of uniform ultrafilters on !1 which extend the filter dual to I .

t u

We note the following lemma which is an immediate corollary of Definition 8.35. Lemma 8.38. Suppose that h.M0 ; I0 ; a0 /; Y0 ; F0 i 2 M|NS and that W 2 Y0 . Let ı0 2 M0 be the Woodin cardinal associated to I0 and let Q0 D .Q In fact the next theorem shows that must less determinacy is required to obtain the |NS nontriviality of M|NS from which the nontriviality of Pmax follows. The first theorem is in essence a “lightface” version of Theorem 8.19.

8 | principles for !1

528

Theorem 8.43. Suppose that x 2 R, y 2 R, x 2 LŒy, and that LŒy  12 .x/-Determinacy: Then: (1) !2LŒy is a Woodin cardinal in HODLŒy . x (2) Let  be the least inaccessible cardinal of HODLŒy . Then strong condensation x LŒy LŒy holds for .HODx / in HODx . t u Remark 8.44. The hypothesis  For each x 2 R there exists h.M; I; a/; Y; F i 2 M|NS with x 2 M, t u

1 is equivalent to  2 -Determinacy.

From Theorem 8.43 one obtains a little more than just that for every x 2 R there exists h.M; I; a/; Y; F i 2 M|NS with x 2 M. One can require for example that modest large cardinals exist in M, above the Woodin cardinal of M associated to I. 1 Theorem 8.45 ( 2 -Determinacy). For each x 2 R there exists

.M; I; ı/ 2 H.!1 / such that (1) x 2 M, (2) M is transitive and M  ZFC C “ı is a Woodin cardinal”, (3) I D .I p0 j p 2 M.p0 ;0/ º ¤ ;:

Proof. Fix A and let B be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; A; 2i: Let B  be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; B; ¹q0 º; 2i: Thus B  2 L.R/. By Theorem 8.42 applied to B  , there exist a countable transitive model M and an ordinal ı 2 M such that the following hold. (1.1) M  ZFC. (1.2) ı is a Woodin cardinal in M . (1.3) B  \ M 2 M and hV!C1 \ M; B  \ M; 2i  hV!C1 ; B  ; 2i. (1.4) B  \ M is ı C -weakly homogeneously Suslin in M . (1.5) Suppose  is the least inaccessible cardinal of M . Then strong condensation holds for M in M . |

NS M / . (1.6) q0 2 .Pmax

Let hD˛ W ˛ < !1M i |

NS M enumerate all the dense subsets of .Pmax / which are first order definable in the structure hH.!1 /M ; A \ M; 2i:

By Lemma 8.55 there exists a filter |

NS M g  .Pmax / such that the following hold in M where for each p 2 g,  jp;g W hM.p;k/ W k < !i ! hM.p;k/ W k < !i is the iteration given by p.

|

NS 8.2 Pmax

553

(2.1) q0 2 g. (2.2) For each ˛ < !1M ,

g \ D˛ ¤ ;:

(2.3) Suppose that p 2 g. For each W 2 jp;g .Y.p;k/ /, there exists U 2 Yg such that  U \ M.p;k/ D W:

(2.4) For each U 2 Yg , the normal ideal IU;Fg is proper. (2.5) Suppose that .p; k/ 2 g  !. For each U 2 Yg , 

 a) IU;Fg \ M.p;k/ D .IW;Fg /M.p;k/ , 

 D .RW;Fg /M.p;k/ , b) RU;Fg \ M.p;k/  . where W D U \ M.p;k/

By Lemma 8.56, there exists Y 2 M such that Y  Yg and such that in M : (3.1) For each U 2 Yg there exists U  2 Y such that for all .p; k/ 2 g  !,   U \ M.p;k/ D U  \ M.p;k/ I

(3.2) Let I be the ideal dual to the filter F D \¹U j U 2 Y º; then \¹IU;Fg j U 2 Y º  I: Let a D .aI /M where I D ¹.IU;Fg /M j U 2 Y º: Let be the least strongly inaccessible cardinal of M above ı. By .1:3/, B  \ M 1 is not † 1 in M and so by (1.4), exists. and let X0  M

be an elementary substructure such that X0 2 M , X0 is countable in M and such that ¹B \ M; Y; gº 2 X0 : Let M0 be the transitive collapse of X0 . Let .a0 ; Y0 ; F0 ; g0 / be the image of .a; Y; Fg ; g/ under the collapsing map and let I0 be the image of .I p0 j p 2 M.p0 ;0/ º ¤ ;: By genericity we may suppose that p0 2 G. Let B  !1 be the interpretation of  by G, The key point is that |

NS \ M.p0 ;0/ j p0 < pº 2 M.p0 ;1/ ¹p 2 Pmax

and so by (1.2) and (1.3),

M.p0 ;0/

B \ !1 Let

2 M.p0 ;1/ :

 jp0 ;G W hM.p0 ;k/ W k < !i ! hM.p W k < !i 0 ;k/

be the iteration given by G. By (1.1)–(1.3), again using the fact that |

NS ¹p 2 Pmax \ M.p0 ;0/ j p0 < pº 2 M.p0 ;1/

it follows that B D jp0 ;G .b/ M.p

;0/

where b D B \ !1 0 . This proves (1). A similar argument shows that L.R/ŒG  !1 -DC: The remaining claims, (2) and (3), are immediate consequences of (1) and the defini|NS tion of the order on Pmax . t u

556

8 | principles for !1 |

NS We now begin the analysis of the nonstationary ideal on !1 in the Pmax -extension. Our goal is to show that the ideal is saturated. We begin with the following lemma |NS -extension. which is the analog of Lemma 6.77 for the Pmax

Lemma 8.59. Assume ADL.R/ and suppose |

NS G  Pmax

is L.R/-generic. Then in L.R/ŒG, for every set A 2 P .R/ \ L.R/ the set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Proof. The proof is identical to that of Lemma 6.77 using the basic analysis provided by Theorem 8.58 (i. e. using Theorem 8.58 in place of Theorem 6.74) and using the |NS is !-closed (Corollary 8.52 in place of Theorem 6.73). t u fact that Pmax Remark 8.60. An immediate corollary of Lemma 8.59 is the following. Assume AD |NS is L.R/-generic. Then in L.R/ŒG, INS is semiholds in L.R/ and that G  Pmax saturated. The verification is a routine application of Lemma 4.24. t u Assume AD holds in L.R/ and that |

NS G  Pmax

is L.R/-generic. Then it is not difficult to show that in L.R/ŒG, the set YG is empty. However one can force over L.R/ŒG to make YG nonempty. The resulting model is |NS . We shall define and briefly itself a generic extension of L.R/ for a variant of Pmax |NS analyze this variant which we denote Umax . |NS The basic property of Umax is the following. Suppose that AD holds in L.R/ and that |NS G  Umax is L.R/-generic. Then L.R/ŒG D L.R/ŒgŒY  where in L.R/ŒG; |

NS is L.R/-generic, (1) g  Pmax

(2) L.R/Œg is closed under !1 sequences in L.R/ŒG, (3) Y D Yg , (4) INS D \¹IU;Fg j U 2 Y º, (5) for all U 2 Y , INS \ U D ;. |

NS . We now define Umax

|

NS 8.2 Pmax

557

|

NS is the set of pairs .p; f / such that Definition 8.61. Umax

|

NS , (1) p 2 Pmax

(2) f 2 M.p;0/ and f W .!3 /M.p;0/ ! Y.p;0/ is a surjection. |

NS The ordering on Umax is defined as follows:

.p1 ; f1 / < .p0 ; f0 / |

NS if p1 < p0 in Pmax and for all ˛ 2 dom.f0 /,

 j.f0 .˛// D f1 .j.˛// \ M.p 0 ;0/

where j W hM.p0 ;k/ W k < !i ! hM.p0 ;k/ W k < !i is the unique iteration such that j.F.p0 / / D F.p1 / .

t u

|

|

NS NS Suppose that G  Umax is a filter. Then G projects to define a filter FG  Pmax . |NS The filter G is semi-generic if the projection FG is a semi-generic filter in Pmax . We |NS is a semi-generic filter. Let fix some more notation. Suppose that G  Umax

fG D ¹jp;FG .f / j .p; f / 2 Gº: Thus fG is a function with domain, dom.fG / D sup¹!2LŒA j A 2 P .!1 /FG º: For each ˛ 2 dom.fG /,

fG .˛/  P .!1 /FG

and fG .˛/ is an ultrafilter in P .!1 /FG . |

NS -extension of L.R/ is a routine generalization of the analThe analysis of the Umax |NS ysis of the Pmax -extension of L.R/. We summarize the basic results in the next theorem the proof of which we leave as an exercise for the dedicated reader.

|

NS Theorem 8.62. Assume AD L.R/ . Suppose G  Umax is L.R/-generic. F D FFG and let Y D .YFG /L.R/ŒG :

Then in L.R/ŒG: |

NS ; (1) FG is L.R/-generic for Pmax

(2) P .!1 /FG D P .!1 /; (3) dom.fG / D !2 ; (4) for all ˇ < !2 , f .ˇ/ 2 Y ;

Let

558

8 | principles for !1

(5) for each U 2 Y , IU;F is a proper ideal and .!1 ; p/ 2 RU;F where p D 1PU ; (6) INS D \¹IU;F j U 2 Y º; (7) for each U 2 Y , INS \ U D ;.

t u |

NS Suppose that for each x 2 R there exists p 2 Pmax such that

x 2 M.p;0/ : We fix some more notation. Suppose that |

NS G  Pmax

is a semi-generic filter. Then 

 IG D [¹M.p;0/ \ .INS /M.p;1/ j p 2 Gº

where for each p 2 G,  jp;G W hM.p;k/ W k < !i ! hM.p;k/ W k < !i

is the iteration given by G. A set |

NS   Pmax  H.!1 /

defines a term for a dense subset of .P .!1 /G n IG ; / if the following conditions are satisfied. (1)  is a set of pairs .p; b/ such that M.p;0/

b  !1 and such that

b 2 M.p;0/ n .INS /M.p;1/ : |

NS (2) For each .p0 ; b0 / 2 Pmax  H.!1 / such that

M.p0 ;0/

b0  !1 and such that

b0 2 M.p0 ;0/ n .INS /M.p0 ;1/ ;

there exists .p1 ; b1 / 2  such that p1 < p0 and such that b1  j.b0 / where  j W hM.p0 ;k/ W k < !i ! hM.p W k < !i 0 ;k/

is the (unique) iteration such that j.F.p0 / / D F.p1 / . |

NS is a semi-generic filter. Then Suppose G  Pmax

G D ¹jp;G .b/ j p 2 G and .p; b/ 2  º: If the filter G is sufficiently generic then G is dense in the partial order, .P .!1 /G n IG ; /:

|

NS 8.2 Pmax

559

|

NS 1 Lemma 8.63 ( 2 -Determinacy). Suppose that   Pmax  H.!1 / defines a term

|

NS for a dense subset of .P .!1 /G n IG ; / and that q0 2 Pmax . Suppose that strong condensation holds for H.!3 /. Then there is a semi-generic filter

|

NS G  Pmax

and a set Y0  YG such that the following hold where for each p 2 G,  W k < !i jp;G W hM.p;k/ W k < !i ! hM.p;k/

is the iteration given by G. (1) q0 2 G. (2) For each U 2 Y0 ,

G n IU;FG

is dense in .P .!1 /G n IU;FG ; /. (3) Suppose that .p; k/ 2 G  !. For each W 2 jp;G .Y.p;k/ /, there exists U 2 Y0 such that  D W: U \ M.p;k/ (4) For each U 2 Y0 , the normal ideal IU;FG is proper. (5) Suppose that .p; k/ 2 G  !. For each U 2 Y0 , 

 D .IW;FG /M.p;k/ , a) IU;FG \ M.p;k/ 

 b) RU;FG \ M.p;k/ D .RW;FG /M.p;k/ ,  where W D U \ M.p;k/ .

(6) Suppose that U0 2 YG , U1 2 Y0 and that U0 \ P .!1 /G D U1 \ P .!1 /G : Then U0 2 Y0 . Proof. Fix a function f W !3 ! H.!3 / which witnesses strong condensation for H.!3 /. Define a function h W !3 ! H.!3 / as follows. Let hD˛ W ˛ < !i

8 | principles for !1

560

|

NS enumerate all the dense subsets of Pmax which are first order definable in the structure

hH.!1 /; ; 2i: We require that for each limit ordinal ˛, ¹Dˇ j ˇ < ˛º contains all the dense sets which are definable with parameters from ¹f .ˇ/ j ˇ < ˛º: Let X be the set of t  ! such that t codes a pair .ˇ; p/ where ˇ < !1 and |NS . p 2 Pmax For each ˛ < !3 let ˛ D ! ˛. Thus h˛ W ˛ < !3 i is the increasing enumeration of the limit ordinals (with 0) less than !3 . Suppose ˛ < !3 then for each k < ! h.˛ C k C 2/ D f .˛/: Suppose ˛ < !3 and f .˛/ … X . Then h.˛ / D f .˛/: Suppose ˛ < !3 and f .˛/ 2 X . Let .ˇ; p/ be the pair coded by f .˛/. Then h.˛ / D f .˛  / where ˛  is least such that f .˛  / 2 Dˇ and such that f .˛  /  p. Finally for each ˛ < !3 , ² 1 if f .˛/ 2  ; h.˛ C 1/ D 0 otherwise. Just as in the proof of Lemma 8.55, h witnesses strong condensation for H.!3 /. The additional feature we have obtained here is that (using the notation from the proof of Lemma 8.55) for each  2 S,  \ M 2 M : Let hp˛ W ˛ < !1 i be as constructed in the proof of Lemma 8.55 using the function h and the sequence hD˛ W ˛ < !1 i: Let G 

|NS Pmax

be the filter, |

NS j for some ˛ < !1 ; p˛ < pº: G D ¹p 2 Pmax

Thus G is a semi-generic filter, YG ¤ ;, and the following hold. (1.1) Suppose that .p; k/ 2 G  !. For each W 2 jp;G .Y.p;k/ /, there exists U 2 YG such that  D W: U \ M.p;k/

|

NS 8.2 Pmax

(1.2) For each U 2 YG , the normal ideal IU;FG is proper. (1.3) Suppose that .p; k/ 2 G  !. For each U 2 YG , 

 D .IW;FG /M.p;k/ , a) IU;FG \ M.p;k/ 

 D .RW;FG /M.p;k/ , b) RU;FG \ M.p;k/  . where W D U \ M.p;k/

Let Y0 be the set of U 2 YG such that G n IU;FG is dense in .P .!1 /G n IU;FG ; /. Y0 satisfies the requirements of the lemma provided for each p 2 G,  jp;G .Y.p;0/ / D ¹U \ M.p;0/ j U 2 Y0 º:

For each  < !3 let

M D ¹h.ˇ/ j ˇ < º

and let h D hj: Let S be the set of  < !3 such that (2.1) M is transitive, (2.2) hˇ 2 M for all ˇ < , (2.3) hM ; h ; 2i  ZFC n Powerset, M

(2.4) !2

M

exists and !2

2 M ,

(2.5) q0 2 H.!1 /M , M

(2.6) H.!1 /M D ¹h.ˇ/ j ˇ < !1  º. Let Q D [¹jp;G .Y.p;k/ / j .p; k/ 2 G  !º: Define a partial order on Q by W1 < W2 if W2  W1 . Suppose that U 2 YG . Then ¹W 2 Q j W  U º is a maximal filter in Q. Suppose that  2 S and !1 <  < !2 . Then Q 2 M :

561

562

8 | principles for !1

Suppose that F Q is a filter which is M -generic and let U D UF . We shall prove that G n IU;FG is dense in .P .!1 /G n IU;FG ; /. We first prove that the relevant filters exists. More precisely suppose that D  P .Q/ is set of dense subsets of Q such that jDj  !1 . We prove that there exists a filter F Q such that W 2 F and such that F is D-generic. The proof is essentially the same as the proof that YG ¤ ;. Fix D and let X  hH.!3 /; h; 2i be the elementary substructure of elements which are definable in the structure with parameters from !1 [ ¹Dº. For each ˛ < !1 let X˛ D ¹f .s/ j f 2 X and s 2 ˛
|NS M .Pmax / ˛

hX˛ ; h; 2i Š hM ˛ ; h ˛ ; 2i; be the filter generated by the set, ¹p j  < ˛º. C D ¹X˛ \ !1 j ˛ < !1 º

and for each ˛ < ˇ < !1 , let ˛;ˇ W M ˛ ! M ˇ be the elementary embedding which corresponds to the inclusion map, X˛  Xˇ . Finally for each ˛ < !1 let ˛ be least such that (3.1) M ˛ is transitive, (3.2) h 2 M ˛ for all  < ˛ , (3.3) hM ˛ ; h ˛ ; 2i  ZFC n Powerset, M

(3.4) !1

˛

M

exists and !1

(3.5) H.!1 /

M

˛

M

(3.6) ˛ < !1

˛

˛

2 M ˛ , M

D ¹h./ j  < !1 .

˛

º,

|

NS 8.2 Pmax

563

Clearly C is a closed unbounded subset of !1 . The key point is that if  is a limit point of C then hM ˛ W ˛ <  i 2 M  and h˛;ˇ W ˛ < ˇ <  i 2 M  : For each ˛ < ˇ  !1 let and let

Y˛ D Y.p˛ ;0/ .ˇ / j˛;ˇ W M.p˛ ;0/ ! M.p ˛ ;0/

be the elementary embedding corresponding to the iteration which witnesses pˇ < p˛ . If ˇ D !1 then j˛;!1 D jp˛ ;G jM.p˛ ;0/ : Thus if hW˛ W ˛ <  i is any sequence such that: (4.1) hW˛ W ˛ <  i 2 M  ; (4.2) for each ˛ < ˇ <  , j˛;ˇ .W˛ /  Wˇ ; (4.3) for each ˛ <  , W˛ 2 Y˛ : Then there exists W 2 Y.p / such that j˛; .W˛ /  W for all ˛ <  . It is now straightforward to construct a sequence hW˛ W ˛ < !1 i such that: (5.1) for all ˛ < ˇ < !1 ,

W˛ 2 Y˛

and j˛;ˇ .W˛ /  Wˇ ; (5.2) the filter F  Q generated by the set ¹jp˛ ;G .W˛ / j ˛ < !1 º is D-generic. Clearly one can require that any given element of Q belong to F . Let 0 2 S be least such that !1 < 0 . Thus G 2 M 0 . Suppose that F0  Q is a filter which is M 0 -generic and let U0 2 YG be such that [F0  U0 : We prove that G n IU0 ;FG is dense in .P .!1 /G n IU0 ;FG ; /. This is an immediate consequence of the genericity of F0 . To see this suppose that p 2 G, W 2 jp;G .Y.p;0/ / and that W 2 F0 . The key point is that INS \ P .!1 /G D .\¹IU;FG j U 2 YG º/ \ P .!1 /G :

8 | principles for !1

564



Let A be the diagonal union of .IW;FG /M.p;0/ where  jp;G W hM.p;k/ W k < !i ! hM.p;k/ W k < !i:

The set A is unambiguously defined modulo a nonstationary set. By modifying the choice of A we can suppose that A 2 P .!1 /G . Suppose that U 2 YG . Then for some W1 2 jp;G .Y.p;0/ /, 

W1 D U \ P .!1 /M.p;0/ : If W 6 U then W ¤ W1 and it follows that for each U 2 YG either W  U or .!1 n A/ 2 IU;FG . Now suppose that B 2 P .!1 /G , B \ A D ;, and that B is stationary; i. e. that B … .\¹IU;FG j U 2 YG º/ \ P .!1 /G : Let U 2 YG be such that B … IU;FG . The ultrafilter U must exist since INS \ P .!1 /G D .\¹IU;FG j U 2 YG º/ \ P .!1 /G : Necessarily W  U . Therefore there exists q 2 G such that (6.1) q < p, 

(6.2) B 2 P .!1 /M.q;0/ , where  W k < !i: jq;G W hM.q;k/ W k < !i ! hM.q;k/  . Thus W1 2 jq;G .Y.q;0/ / and W  W1 . Suppose that Let W1 D U \ M.q;0/ F  Q is any M 0 -generic filter containing W1 and that U 2 YG is any ultrafilter such that [F  U . It follows that B … IU;FG . It follows by the M 0 -genericity of F0 that

G n IU0 ;FG is dense in .P .!1 /G n IU0 ;FG ; /. Thus Y0 satisfies the requirements of the lemma. u t Lemma 8.63 yields the following variation of Lemma 8.57. |

NS  H.!1 / defines a term for a dense Lemma 8.64 (ADL.R/ ). Suppose that   Pmax subset of P .!1 / n IG and that  2 L.R/:

|

NS Let A be the set of x 2 R which code an element of  . Then for each q0 2 Pmax there |NS exists p0 2 Pmax such that p0 < q0 and such that:

(1) hM.p0 ;k/ W k < !i is A-iterable; (2) hV!C1 \ M.p0 ;0/ ; A \ M.p0 ;0/ ; 2i  hV!C1 ; A; 2i;

|

NS 8.2 Pmax

565

|

NS \ M.p0 ;0/ such that g0 2 M.p0 ;0/ , such that (3) There exists a filter g0  Pmax

p0 < p for each p 2 g0 , and such that in M.p0 ;0/ ; a) g0 is semi-generic and F D Fg0 , b) for each U 2 Y.p0 ;0/ , . \ H.!1 /M.p0 ;0/ /g0 n IU;F is dense in .P .!1 /g0 n IU;F ; /, where F D F.p0 / . Proof. The proof is in essence identical to the proof of Lemma 8.57, using Lemma 8.63 in place of Lemma 8.55. Let B be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; A; 2i: Let B  be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; B; ¹q0 º; 2i: Thus B  2 L.R/. By Theorem 8.42 applied to B  , there exist a countable transitive model M and an ordinal ı 2 M such that the following hold. (1.1) M  ZFC. (1.2) ı is a Woodin cardinal in M . (1.3) B  \ M 2 M and hV!C1 \ M; B  \ M; 2i  hV!C1 ; B  ; 2i. (1.4) B  \ M is ı C -weakly homogeneously Suslin in M . (1.5) Suppose  is the least inaccessible cardinal of M . Then strong condensation holds for M in M . |

NS M / . (1.6) q0 2 .Pmax

By Lemma 8.63 there exist, in M , a filter |

NS M / g  .Pmax

and a set Y0  .Yg /M such that the following hold where for each p 2 g,  W k < !i jp;g W hM.p;k/ W k < !i ! hM.p;k/

is the iteration given by g. We let . /M D  \ H.!1 /M :

566

8 | principles for !1

(2.1) q0 2 g. (2.2) Suppose that p 2 g. For each W 2 jp;g .Y.p;k/ /, there exists U 2 Y0 such that  D W: U \ M.p;k/

(2.3) For each U 2 Y0 , the normal ideal IU;Fg is proper. (2.4) For each U 2 Y0 ,

../M /g n IU;Fg

is dense in .P .!1 /g n IU;Fg ; /. (2.5) Suppose that .p; k/ 2 g  !. For each U 2 Y0 , 

 D .IW;Fg /M.p;k/ , a) IU;Fg \ M.p;k/ 

 D .RW;Fg /M.p;k/ , b) RU;Fg \ M.p;k/  . where W D U \ M.p;k/

(2.6) Suppose that U0 2 Yg , U1 2 Y0 and that U0 \ P .!1 /g D U1 \ P .!1 /g : Then U0 2 Y0 . By (2.2)–(2.6), g and Y0 satisfy (in M ) the requirements of the hypothesis of Lemma 8.56 and so by Lemma 8.56 there exists Y 2 M such that Y  Y0 and such that in M , (3.1) for each U0 2 Y0 there exists U1 2 Y such that for all .p; k/ 2 g  !,   D U1 \ M.p;k/ ; U0 \ M.p;k/

(3.2) let I be the ideal dual to the filter F D \¹U j U 2 Y º; then \¹IU;Fg j U 2 Y º  I: Let a D .aI /M where I D ¹.IU;Fg /M j U 2 Y º: Let be the least strongly inaccessible cardinal of M above ı. By .1:3/, B  \ M 1 is not † 1 in M and so by (1.4), exists. and let X0  M

be an elementary substructure such that X0 2 M , X0 is countable in M and such that ¹B \ M; Y; gº 2 M:

|

NS 8.2 Pmax

567

Let M0 be the transitive collapse of X0 . Let .a0 ; Y0 ; F0 / be the image of .a; Y; Fg / under the collapsing map and let I0 be the image of .I
|

NS 8.2 Pmax

573

where MX  is the transitive collapse of X  . Therefore MX  is strongly iterable and so the set ¹X  N j X is countable and NX is strongly iterable º is stationary in P!1 .N /. Let X0  N be a countable elementary substructure such that (1.1) ¹q0 ; G; Aº  X0 , (1.2) N0 is strongly iterable, where N0 is the transitive collapse of X0 . Let Z0 D X0 \ H.!2 / and let M0 be the transitive collapse of Z0 . Thus M0 D .H.!2 //N0 : Let S D ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countable º where MX is the transitive collapse of X . By Lemma 8.66 there exists a function  W H.!2 /
D n IU;FG

is dense in .P .!1 / n INS ; /. Let J D ¹IU;FG j U 2 YFG º:

8 | principles for !1

574 Thus:

(2.1) INS D \¹I j I 2 Jº. (2.2) Suppose that I0 2 J, I1 2 J and that for some A  !1 , a) I0  ¹B  !1 j B \ A 2 I1 º, b) !1 n A … I1 . Then I0 D I1 . Let ¹G0 ; F0 ; F0 ; J0 ; Y0 º be the image of ¹G; FG ; FG ; J; YFG º under the transitive collapse of X0 . |NS The partial order Umax is !-closed and so ¹N0 ; F0 ; G0 ; F0 ; D0 ; Y0 º 2 L.R/: Further

|

h.N0 ; J0 /; G0 ; Y0 ; F0 i 2 M0 NS : We now work in L.R/. By Theorem 8.69 there exists a condition |

NS .h.MO k ; YOk / W k < !i; FO / 2 Pmax

and an iteration

j W .N0 ; J0 / ! .N0 ; J0 /

such that (3.1) hMO k W k < !i is A-iterable, (3.2) j 2 MO 0 , (3.3) j.F0 / D FO , (3.4) j0 .Y0 / D ¹U \ N0 j U 2 YO0 º, (3.5) for each U 2 YO0 ,



.RW;FO /N0 D RU;FO \ N0

and



.IW;FO /N0 D IU;FO \ N0 ;

where W D N0 \ U . Let

p0 D .h.MO k ; YOk / W k < !i; FO /

and let g0 D j.F0 /. We claim that p0 and g0 satisfy the requirements of the lemma. We verify (3(b)), the other requirements are immediate.

|

NS 8.2 Pmax

575

Suppose that U 2 YO0 and let W D U \ N0 : Since

|

h.N0 ; J0 /; G0 ; Y0 ; F0 i 2 M0 NS ; it follows that

|

h.N0 ; J0 /; G0 ; Y0 ; FO i 2 M0 NS

where hG0 ; Y0 i D j.hG0 ; Y0 i/. |

By the definition of M0 NS , in N0 , . \ N0 /g0 n IW;FO is dense in .P .!1 / n IW;FO ; /. Let 

D0 D ../N0 /g0 n IW;FO ; 



where as usual, . /N0 D  \ H.!1 /N0 . We work in MO 0 . .N0 ; J0 / is an iterate of .N0 ; J0 / and IU;FO is a normal uniform ideal on !1 such that IU;FO \ N0 2 J0 : Therefore by (2.2) and Lemma 4.10, !1 n .5D0 / 2 IU;FO : t u

This verifies (3b). |

NS is L.R/-generic. Then in Theorem 8.71. Assume AD L.R/ . Suppose G  Pmax L.R/ŒG,

(1) IG is !2 -saturated, (2) INS D IG . Proof. By Theorem 8.58, P .!1 / D P .!1 /G and IG D INS . Therefore it suffices to show that if D  P .!1 /G n IG is dense then there exists a set D0  D such that D0 is predense in .P .!1 /G n IG ; / and such that jD0 j  !1 . |NS Let   Pmax  H.!1 / be a set in L.R/ which defines a term for D. Let A  R be the set of x 2 R such that x codes an element of  .

8 | principles for !1

576

By Lemma 8.70 and genericity there exists p0 2 G such that: (1.1) hM.p0 ;k/ W k < !i is A-iterable; (1.2) hV!C1 \ M.p0 ;0/ ; A \ M.p0 ;0/ ; 2i  hV!C1 ; A; 2i; |

NS (1.3) There exists a filter g0  Pmax \ M.p0 ;0/ such that g0 2 M.p0 ;0/ , such that for each p 2 g0 , p0 < p;

and such that in M.p0 ;0/ ; a) g0 is semi-generic, b) F.p0 / D Fg0 , c) for each U 2 Y.p0 ;0/ , ../M.p0 ;0/ /g0 n IU;Fg0 is predense in .P .!1 / n IU;Fg0 ; /, where ./M.p0 ;0/ D  \ H.!1 /M.p0 ;0/ : A key point is that in M.p0 ;0/ ; for each U 2 Y.p0 ;0/ , j../M.p0 ;0/ /g0 j  !1 ; and so (1.3c) asserts that !1 n A 2 IU;Fg0 where Let

A D 5¹S j S 2 ../M.p0 ;0/ /g0 n IU;Fg0 º:  jp0 ;G W hM.p0 ;k/ W k < !i ! hM.p W k < !i 0 ;k/

be the iteration given by G. It follows that jp0 ;G .g0 /  G and that in L.R/ŒG, jp0 ;G .g0 / is a semi-generic filter. Let g0 D jp0 ;G .g0 / and let  D0 D g0 \ M.p : 0 ;0/

Thus D0  D and

jD0 j  !1 :

By (1.3) it follows that

5D0

contains the critical sequence of the iteration defining jp0 ;G and so D0 is necessarily predense in .P .!1 /G n IG ; / since .P .!1 /G n IG ; / D .P .!1 / n INS ; /:

t u

C

CC

8.3 The principles, |NS and |NS

577

As an immediate corollary of Theorem 8.71 we obtain the following. |

NS Corollary 8.72. Assume AD L.R/ . Suppose G  Pmax is L.R/-generic. Then in L.R/ŒG, YG D ;:

Proof. We note that the following must hold in L.R/ŒG. Suppose that S  !1 is stationary. Then there exists a set A  !1 such that both ¹˛ 2 S j A n F .˛/ is finiteº and ¹˛ 2 S j A \ F .˛/ is finiteº are stationary. Suppose YG ¤ ; and let U 2 YG . Thus, since P .!1 / D P .!1 /G ; it follows that IU;FG is a proper ideal. But INS is !2 -saturated and so for some stationary set S  !1 , IU;FG D INS jS D ¹T  !1 j T n S 2 INS º: t u

This contradicts the claim above. | PmaxNS

. This lemma will We end this section with one last lemma regarding the L.R/ be relevant to the absoluteness theorem we shall prove, see Theorem 8.99. |

NS Lemma 8.73. Assume AD L.R/ . Suppose G  Pmax is L.R/-generic. Let F D FG . Then in L.R/ŒG the following holds. There exists a co-stationary set S  !1 such that for all ultrafilters U  P .!1 /, if p 2 PU and

Zp;F … INS ; then Zp;F \ S … INS :

8.3

C

t u

CC

The principles, |NS and |NS |

|

NS NS The Umax -extension of L.R/ is a generic extension of the Pmax -extension. The relevant partial order is a product of a partial order PF which is defined in Definition 8.75. The definition of PF is closely related to two refinements of |NS one of which we |NS now define. These refinements in turn yield an absoluteness theorem for the Pmax extension. It is not clear if the version we prove is optimal and as we have indicated, more elegant versions are likely possible.

578

8 | principles for !1

We first fix some notation. Suppose that F W !1 ! Œ!1 ! is a function witnessing that |NS holds. For each S 2 P .!1 / n INS let FS;F denote the set of A  !1 such that there exists a club C  !1 such that for all ˛ 2 C \ S , F .˛/ n A is finite. Clearly FS;F is a filter on !1 which extends the club filter. The definition of C |NS involves Zh;F which is defined in Definition 8.26. C

Definition 8.74. |NS : There is a function F W !1 ! Œ!1 ! such that the following hold. (1) F witnesses |NS . (2) Suppose X  P .!1 / has cardinality !1 and that S  !1 is stationary. Then there exists a stationary set T  S and an ultrafilter U such that: a) FT;F \ X D U \ X . b) Suppose that h W Œ!1 
t u

Definition 8.75. Suppose that F W !1 ! Œ!1 ! is a function witnessing that |NS holds. Let PF be the partial order defined as follows. Conditions are sets X  P .!1 / such that jX j  !1 and such that X  FS;F for some S 2 P .!1 / n INS . Suppose X; Y 2 PF . Then X  Y if Y  X .

t u

The partial order PF is analogous to the partial order PNS which we defined in Section 6.1. There is however an interesting difference. It is not difficult to show that assuming ./, the partial order PNS is not !2 -cc. However if INS is !2 -saturated, which |NS

is the case in L.R/Pmax , then PF is trivially !2 -cc for any function F which witnesses |NS . More is actually true.

C

CC

8.3 The principles, |NS and |NS

579

Lemma 8.76. Suppose that F W !1 ! Œ!1 ! is a function which witnesses that |NS holds and that INS is !2 -saturated. Then there exists a complete boolean subalgebra B  P .!1 /=INS such that RO.PF / Š B: Proof. Define  W PF ! P .!1 /=INS as follows. Suppose X 2 PF . It follows from the !2 -saturation of INS that there exists a stationary set SX  !1 such that (1.1) X  FSX ;F , (1.2) for all S 2 P .!1 / n INS , if

X  FS;F

then S n SX 2 INS . Define .X / D b where b 2 P .!1 /=INS is the element given by SX . The element b is unambiguously defined. The function  induces the required isomorphism of RO.PF / with a complete t u boolean subalgebra of P .!1 /=INS . We also note the following reformulation of Corollary 8.72. |

NS Lemma 8.77. Assume AD L.R/ . Suppose G  Pmax is a semi-generic filter such that G is L.R/-generic and such that

P .!1 / D P .!1 /G : Then RO.PF / has no atoms where F D FG .

t u C

Lemma 8.76 and Lemma 8.77 suggest the following refinement of |NS . CC

Definition 8.78. |NS : There is a function F W !1 ! Œ!1 ! such that the following hold. C

(1) F witnesses |NS . (2) PF is !2 -cc. (3) RO.PF / has no atoms.

t u

580

8 | principles for !1

Remark 8.79. As we have already remarked, the most elegant manifestation of |NS would be to have for some ultrafilter U on !1 , (1) U extends the club filter, (2) the boolean algebra RO.PU / is isomorphic to a complete boolean subalgebra of P .!1 /=INS . Any function F W !1 ! Œ!1 ! CC

inducing the isomorphism for (2), witnesses that |NS holds. However by Lemma 8.25, CC

(1) and (2) cannot both hold for any ultrafilter U . |NS in some sense gives the best possible approximation to (1) and (2); cf. Corollary 8.88. t u There is an interesting question.  Suppose that

F W !1 ! Œ!1 ! C

is a function which witnesses that |NS holds. Can the boolean algebra, RO.PF /; be atomic? |

NS -extension easily yields, The basic analysis of the Pmax

|

NS Theorem 8.80. Assume AD L.R/ . Suppose G  Pmax is L.R/-generic. Then CC

L.R/ŒG  |NS : |

NS , the function FG Proof. By Theorem 8.58 and the definition of the order on Pmax witnesses that L.R/ŒG  |NS

and in L.R/ŒG; P .!1 / D P .!1 /G : |

NS that the function FG witnesses that It follows from the definition of Pmax C

L.R/ŒG  |NS : By Theorem 8.71, the nonstationary ideal is !2 -saturated in L.R/ŒG and so by Lemma 8.76, the function FG witnesses that CC

L.R/ŒG  |NS : |

t u |

NS NS -extension of L.R/, identifying the Umax We continue our analysis of the Pmax |NS extension of L.R/ as a generic extension of the Pmax -extension of L.R/. The relevant partial order, as we have indicated, is simply a product of PF .

C

CC

8.3 The principles, |NS and |NS

581

Definition 8.81. Suppose that F W !1 ! Œ!1 ! is a function witnessing that |NS holds. Let QF be the product partial order: Conditions are functions p W ˛ ! PF such that ˛ < !2 . The order is defined pointwise: Suppose that p1 and p2 are conditions in QF . Then p2  p1 if (1) dom.p1 /  dom.p2 /, (2) for all ˇ 2 dom.p1 /,

p2 .ˇ/  p1 .ˇ/ t u

in PF . Lemma 8.82. Suppose that F W !1 ! Œ!1 ! CC

is a function which witnesses that |NS holds. Then the partial order PF is .!1 ; 1/distributive. Proof. Suppose that g  Coll.!2 ; P .!1 // is V -generic. Since g is V -generic for a partial order which is .< !2 /-closed in V , it follows that in V Œg, F witnesses that CC |NS holds. Therefore we may assume without loss of generality that 2@1 D @2 . Suppose that G  PF is V -generic. Then, by reorganizing G as a subset of !2 , V ŒG D V ŒA where A  is a set such that A \ ˛ 2 V for all ˛ < !2V . This is a consequence of C the fact that F witnesses |NS in V ŒG, see Definition 8.74(2). Since PF is !2 -cc in V t u it follows that V is closed under !1 -sequences in V ŒG. !2V

|

|

NS NS The next four theorems detail the relationship between Pmax and Umax . We shall not need these theorems, we simply state them for completeness. The proofs are not difficult and we leave the details to the reader.

|

NS is L.R/-generic. Let Theorem 8.83. Assume AD L.R/ . Suppose that G  Umax F D FFG and let

H D ¹p 2 .QF /L.R/ŒFG  j for all ˇ 2 dom.p/; p.ˇ/  fG .ˇ/º: Then H is an L.R/ŒFG -generic filter and L.R/ŒG D L.R/ŒFG ŒH :

t u

8 | principles for !1

582

Theorem 8.84. Assume ADL.R/ and that V D L.R/Œg |NS Pmax

where g  is L.R/-generic. Let F D Fg be the function witnessing |NS given by g. Then PF is .!1 ; 1/-distributive and further suppose G  PF is V -generic. Then in V ŒG; (1) IU;F is a proper ideal, (2) INS is not saturated, (3) IU;F D sat.INS /, (4) IU;F is a saturated ideal, where U D [G.

t u

Theorem 8.84 combined with Theorem 8.58 yields the following theorem. Theorem 8.85. Assume ADL.R/ and that V D L.R/Œg |NS Pmax

where g  is L.R/-generic. Let F D Fg be the function witnessing |NS given by g. Suppose p 2 g and let  j W hM.p;k/ W k < !i ! hM.p;k/ W k < !i

of length !1 such that j.F.p/ / D F . Suppose that G  PF is V -generic and let U  P .!1 / be the ultrafilter, U D [G, given by G. Then in V ŒG the following hold where  : W D U \ M.p;0/ (1) W 2 j.Y.p;0/ /. 

 (2) .IW;F /M.p;0/ D IU;F \ M.p;0/ . 

 (3) .RW;F /M.p;0/ D RU;F \ M.p;0/ .

t u

Theorem 8.86. Assume ADL.R/ and that V D L.R/Œg |NS Pmax

where g  is L.R/-generic. Let F D Fg be the function witnessing |NS given by g. Then QF is .!1 ; 1/-distributive. Suppose G  QF is V -generic and for each ˛ < !2 let U˛ D [¹p.˛/ j ˛ 2 dom.p/ and p 2 Gº; and let Y D ¹U˛ j ˛ < !2 º:

C

CC

8.3 The principles, |NS and |NS

583

Then in V ŒG: (1) Y D Yg ; (2) For each U 2 Y , a) IU;F is a proper ideal, b) IU;F is a saturated ideal; (3) INS D \¹IU;F j U 2 Y º; (4) For each U 2 Y , INS \ U D ;.

t u CC

One corollary of Lemma 8.82 is that |NS cannot hold in L. More generally, strong CC

condensation for H.!2 / implies :|NS . CC

Corollary 8.87. Assume that strong condensation holds for H.!2 /. Then |NS fails. Proof. The proof is a modification of the proof of Lemma 8.25. We sketch the argument under the additional hypothesis that V D L. The proof from strong condensation for H.!2 / is essentially the same. Suppose that G  PF is V -generic and let U 2 V ŒG be the ultrafilter on !1 given by G; U D ¹X j X 2 Gº: Since F witnesses |NS in V it follows that U is a V -ultrafilter on !1V . However F CC witnesses |NS in V and so by Lemma 8.82, PF is .!1 ; 1/-distributive in V . This implies .P .!1 //V D .P .!1 //V ŒG and so U is an ultrafilter on !1 in V ŒG. Let  < !2 be least such that (1.1) F 2 L , (1.2) L  ZC, CC

(1.3) F witnesses |NS in L . The key point is that G \ L is L -generic for .PF /.L / . This implies that U \ L is generic over L . Let C D ¹X \ !1 j X  L ; F 2 X and jX j D !º and for each ˛ 2 C let

X ˛  L

be the (unique) elementary substructure X  L such that F 2 X and ˛ D X \ !1 .

584

8 | principles for !1

For each ˛ 2 C let ˛ be the image of  under the transitive collapse of X˛ . Therefore, since U extends the club filter on !1 , for each formula .x0 ; x1 / and for each ˇ < !1 , L  ŒF; ˇ if and only if ¹˛ j ˇ < ˛ and L˛  ŒF j˛; ˇº 2 U: Finally by the definition of  , every element of L is definable in L from paramt u eters in !1 [ ¹F º. But this contradicts that U \ L is generic over L . A second corollary of Lemma 8.82 is the following improvement of Lemma 8.76. The proof, which we leave to the reader, is an easy consequence of the definitions, cf. the proof of Lemma 8.76. Corollary 8.88. Suppose that F W !1 ! Œ!1 ! CC

is a function which witnesses that |NS holds and that INS is !2 -saturated. Then there exists a complete boolean subalgebra B  P .!1 /=INS such that RO.PF  PU / Š B; where U 2 V PF is the ultrafilter on !1 given by the generic filter for PF .

t u

|

NS We now come to the absoluteness theorem for the Pmax -extension. We first prove |NS a strong version of the homogeneity of Pmax . This is a corollary of the following iteration lemma.

Lemma 8.89. Suppose that |

h.M0 ; I0 /; g0 ; Y0 ; F0 i 2 M0 NS ; |

h.M1 ; I1 /; g1 ; Y1 ; F1 i 2 M0 NS ; and that strong condensation holds for H.!3 /. Then there exist iterations j0 W .M0 ; I0 / ! .M0 ; I0 / and j1 W .M1 ; I1 / ! .M1 ; I1 / of length !1 and a bijection  W j0 .Y0 / ! j1 .Y1 / such that: (1) !1 n ¹˛ < !1 j j0 .F0 /.˛/ D j1 .F1 /.˛/º 2 INS . (2) Suppose that W0 2 j0 .Y0 / and W1 D .W0 /. Then for all A0 2 W0 and for all A1 2 W1 , A0 \ A1 … INS :

C

CC

8.3 The principles, |NS and |NS

585

(3) Suppose that U  P .!1 / is an ultrafilter such that U \ M0 2 j0 .Y0 / and such that U \ M1 2 j1 .Y1 /: a) The ideal IU;F is proper. b) For each i 2 ¹0;1º, 

.RW;F /Mi D RU;F \ Mi ; and



.IW;F /Mi D IU;F \ Mi ;

where W D Mi \ U and where F D ji .Fi /. Proof. The proof is quite similar to that of Lemma 8.40 except we do not need to  . enforce AC Fix a function h W !3 ! H.!3 / which witnesses strong condensation for H.!3 /. For each  < !3 let M D ¹h.ˇ/ j ˇ < º and let h D hj: Let S be the set of  < !3 such that (1.1) M is transitive, (1.2) hˇ 2 M for all ˇ < , (1.3) hM ; h ; 2i  ZFC n Powerset, M

(1.4) !2

M

exists and !2

2 M ,

(1.5) M0# 2 H.!1 /M . We construct j0 as the limit of an iteration 0 W ˛ < ˇ  !1 i h.M0˛ ; G˛0 /; j˛;ˇ

and j1 as the limit of an iteration 1 W ˛ < ˇ  !1 i: h.M1˛ ; G˛1 /; j˛;ˇ

Simultaneously we construct a sequence h˛ W ˛  !1 i of bijections 0 1 .Y0 / ! j0;˛ .Y1 / ˛ W j0;˛ 0 such that for all ˛ < ˇ  !1 , and for all W 2 j0;˛ .Y0 /, 0 1 ˇ .j˛;ˇ .W // D j˛;ˇ .˛ .W //:

586

8 | principles for !1

Thus everything is completely determined by h.G˛0 ; G˛1 ; ˛ / W ˛ < !1 i. We say that this sequence satisfies the conditions of the lemma if the corresponding iterations 0 1 and j0;! ) together with the map !1 satisfy the requirements of the lemma. (j0;! 1 1 Similarly if  2 S then h.G˛0 ; G˛1 ; ˛ / W ˛ < .!1 /M i 0 1 satisfies the requirements of the lemma in M if both .j0; ; j0; ;  / 2 M and 0 1 .j0; ; j0; ;  / satisfies the requirements of the lemma interpreted in M where  D .!1 /M . We construct h.G˛0 ; G˛1 ; ˛ / W ˛ < ˇi by induction on ˇ, following the proof of Lemma 8.40, eliminating potential counterexamples. The construction is uniform and so for each  2 S, h.G˛0 ; G˛1 ; ˛ / W ˛ < .!1 /M i 2 M :

Suppose that h.G˛0 ; G˛1 ; ˛ / W ˛ < ˛0 i is given. If ˛0 is a successor ordinal then .G˛00 ; G˛10 ; ˛0 / D h.0 / where 0 is least such that h.0 / satisfies the minimum necessary conditions. Thus we may suppose that ˛0 is a (nonzero) limit ordinal. The function ˛0 is uniquely specified. We must define G˛00 and G˛10 . This we do by cases. The first case is that for all  2 S, either ˛0 ¤ .!1 /M ; 0 1 ; j0;˛ ; ˛0 / satisfies the requirements of the lemma interpreted in M . or .j0;˛ 0 0 There are two subcases. If for all  2 S,

˛0 ¤ .!1 /M ; then let 0 be least such that h.0 / D .g0 ; g1 / and 0 .I0 /, (2.1) for some I 2 j0;˛ 0

g0  .P .˛0 / \ M0˛0 n I / and g0 is M0˛0 -generic, 1 .I1 /, (2.2) for some I 2 j0;˛ 0

g1  .P .˛0 / \ M1˛0 n I / and g1 is M1˛0 -generic. Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: Otherwise let 0 2 S be least such that ˛0 D .!1 /M0 : Let 0 be least such that h.0 / D .g0 ; g1 / and

C

CC

8.3 The principles, |NS and |NS

587

0 (3.1) for some I 2 j0;˛ .I0 /, 0

g0  .P .˛0 / \ M0˛0 n I / and g0 is M0˛0 -generic, 1 .I1 /, (3.2) for some I 2 j0;˛ 0

g1  .P .˛0 / \ M1˛0 n I / and g1 is M1˛0 -generic, 0 1 .F0 /.˛0 / D j0;˛ .F1 /.˛0 /. (3.3) j0;˛ 0 C1 0 C1 0 1 Since .j0;˛ ; j0;˛ ; ˛0 / satisfies the requirements of the lemma interpreted in M 0 , 0 0 0 exists. Let

.G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: Finally let 0 2 S be least such that ˛0 D .!1 /M0 ; 0 1 and .j0;˛ ; j0;˛ ; ˛0 / fails to satisfy the requirements of the lemma interpreted in 0 0 M 0 . As in the analogous stage of the proof of Lemma 8.40, we shall extend the iterations, defining .G˛00 ; G˛10 /, attempting to eliminate the least counterexample, ignoring the requirement (1). We first suppose (2) fails. There are two subcases. First suppose that there exists .W0 ; W1 / 2 ˛0 such that for some A0 2 W0 and for some A1 2 W1 ,

A0 \ A1 D ;: Let 0 be least such that h.0 / is such a pair .W0 ; W1 / 2 ˛0 and let 1 be least such that h.1 / D .A0 ; A1 / with A0 2 W0 , A1 2 W1 and A0 \ A1 D ;: Let 0 be least such that h.0 / D .g0 ; g1 / and (4.1) (2.1)–(2.2) hold, (4.2) A0 2 g0 and A1 2 g1 . Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /:

588

8 | principles for !1

Otherwise let 0 be least such that for some .W0 ; W1 / 2 ˛0 , (5.1) h.0 / D .W0 ; W1 /, (5.2) there exist A0 2 W0 , A1 2 W1 , such that A0 \ A1 … .INS /M0 : Let 1 be least such that h.1 / D .A0 ; A1 / witnessing (5.2). Let 0 be least such that h.0 / D .g0 ; g1 / and (6.1) (2.1)–(2.2) hold, 0 1 .F0 /.˛0 / D j0;˛ .F1 /.˛0 /, (6.2) j0;˛ 0 C1 0 C1

(6.3) A0 2 g0 and A1 2 g1 . We can ensure (6.2) holds because in M 0 , W0 [ W1 can be extended to an ultrafilter. Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: 0 1 Finally we suppose that in M 0 , (3) fails for .j0;˛ ; j0;˛ ; ˛0 /. 0 0 Let 0 be least such that:

(7.1) h.0 / 2 M 0 ; (7.2) M 0  “h.0 / is a uniform ultrafilter on !1 ”; 0 (7.3) h.0 / \ M0˛0 2 j0;˛ .Y0 /; 0 1 (7.4) h.0 / \ M1˛0 2 j0;˛ .Y1 /; 0

(7.5) Let U D h.0 /. Either a) .IU;F /M0 is not a proper ideal, or b) there exists

˛0

.p; S / 2 .RW0 ;F /M0 such that .p; S / … .RU;F /M0 where 0  F D j0;˛ .F0 /, 0

 W0 D h.0 / \ M0˛0 , c) there exists

˛0

.p; S / 2 .RW1 ;F /M1 such that .p; S / … .RU;F /M0 where 1  F D j0;˛ .F1 /, 0

 W1 D h.0 / \ M1˛0 .

C

CC

8.3 The principles, |NS and |NS

589

Let (8.1) U D h.0 /, (8.2) W0 D U \ M0˛0 , ˛0

0 (8.3) I0 D .IW0 ;F /M0 where F D j0;˛ .F0 /, 0

(8.4) W1 D U \ M1˛0 , ˛0

0 (8.5) I1 D .IW1 ;F /M1 where F D j0;˛ .F1 /. 0

There are several subcases. First suppose that (7.5(a)) holds. Let 0 be least such that h.0 / D .g0 ; g1 / and (9.1) g0  .P .˛0 / \ M0˛0 n I0 / and g0 is M0˛0 -generic, (9.2) g1  .P .˛0 / \ M1˛0 n I1 / and g1 is M1˛0 -generic, 0 1 (9.3) j0;˛ .F0 /.˛0 / D j0;˛ .F1 /.˛0 /, 0 C1 0 C1 0 .F0 /.˛0 / is M 0 -generic for .PU /M0 . (9.4) j0;˛ 0 C1

Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: Otherwise (7.5(a)) fails. Hence either (7.5(b)) holds or (7.5(c)) holds. We next suppose that (7.5(b)) holds. Let 1 be least such that h.1 / D .p; S / witnessing (7.5(b)). Let 2 be least such that h.2 / D q and (10.1) q 2 .PU /M0 , (10.2) q  p, (10.3) .Zq;F /M0 \ S 2 .IU;F /M0 , 0 where F D j0;˛ .F0 /. 0 Let 0 be least such that h.0 / D .g0 ; g1 / and

(11.1) (9.1)–(9.4) hold, (11.2) S 2 g0 , (11.3) q belongs to the M 0 -generic filter for .PU /M0 given by 0 , where 0 1 0 D j0;˛ .F0 /.˛0 / D j0;˛ .F1 /.˛0 /: 0 C1 0 C1

Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: The final case is that both (7.5(a)) and (7.5(b)) fail. In which case (7.5(c)) holds. This is essentially the same as the case that (7.5(b)) holds: Let 1 be least such that h.1 / D .p; S / witnessing (7.5(c)). Let 2 be least such that h.2 / D q and

8 | principles for !1

590

(12.1) q 2 .PU /M0 , (12.2) q  p, (12.3) .Zq;F /M0 \ S 2 .IU;F /M0 , 1 where F D j0;˛ .F1 /. 0 Let 0 be least such that h.0 / D .g0 ; g1 / and

(13.1) (9.1)–(9.4) hold, (13.2) S 2 g0 , (13.3) q belongs to the M 0 -generic filter for .PU /M0 given by 0 , where 0 1 0 D j0;˛ .F0 /.˛0 / D j0;˛ .F1 /.˛0 /: 0 C1 0 C1

Let .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: This completes the inductive definition of h.G˛0 ; G˛1 ; ˛ / W ˛ < !1 i. Let .j0 ; j0 ; / D .j00;!1 ; j10;!1 ; !1 /: We claim that .j0 ; j0 ; / satisfies the requirements of the lemma. We prove (1) holds. For this we first prove that for all .W0 ; W1 / 2 , if A0 2 W0 and if A1 2 W1 then A0 \ A1 ¤ ;: Suppose this fails. Let 0 be least such that h.0 / is such a pair .W0 ; W1 / 2  and let 1 be least such that h.1 / D .A0 ; A1 / with A0 2 W0 , A1 2 W1 and A0 \ A1 D ;: Suppose X  hH.!3 /; h; 2i is a countable elementary substructure with  2 X . Let ˛0 D X \ !1 and let MX be the transitive collapse of X . Thus MX D M where  D MX \ Ord and so  2 S. Let .0X ; 1X / be the image of .0 ; 1 / under the collapsing map. Thus h.1X / D .X \ A0 ; X \ A1 /: It follows that .G˛00 ; G˛10 / was defined at stage ˛0 using .0X ; 1X / choosing 0 least such that h.0 / D .g0 ; g1 / and (14.1) (2.1)–(2.2) hold, (14.2) A0 \ X 2 g0 and A1 \ X 2 g1 ,

C

CC

8.3 The principles, |NS and |NS

591

and defining .G˛00 ; G˛10 / D h.0 / D .g0 ; g1 /: Thus A0 \ X 2 G˛00 and A1 \ X 2 G˛10 . But this implies ˛0 2 A0 \ A1 which is a contradiction. This proves that for all .W0 ; W1 / 2 , W0 [ W1 has the finite intersection property. Therefore there is a closed unbounded set C  !1 to which this reflects; if ˛0 2 C then for all .W0 ; W1 / 2 ˛0 , W0 [ W1 has the finite intersection property. Therefore, by inspection of the inductive construction, for all ˛0 2 C , if there exists  2 S such that ˛0 D .!1 /M ; then 0 1 j0;˛ .F0 /.˛0 / D j0;˛ .F1 /.˛0 /: 0 C1 0 C1

This proves (1). The verification that (2) and (3) hold is by similar reflection arguments. These arguments are essentially identical to arguments for the analogous claims given at the end of the proof of Lemma 8.40. t u Lemma 8.90. Suppose that |

h.M0 ; I0 /; g0 ; Y0 ; F0 i 2 M0 NS ; |

h.M1 ; I1 /; g1 ; Y1 ; F1 i 2 M0 NS ; and that strong condensation holds for H.!3 /. Then there exist iterations j0 W .M0 ; I0 / ! .M0 ; I0 / and

j1 W .M1 ; I1 / ! .M1 ; I1 /

of length !1 and a set Y  ¹U  P .!1 / j U is a uniform ultrafilter on !1 º such that !1 n ¹˛ < !1 j j0 .F0 /.˛/ D j1 .F1 /.˛/º 2 INS and such that for each i 2 ¹0;1º, the following hold. (1) For each U 2 Y , U \ Mi 2 j.Yi /. (2) For each U 2 Y , the ideal IU;ji .Fi / is proper, 

.RW;ji .Fi / /Mi D RU;ji .Fi / \ Mi ; and where W D Mi \ U .



.IW;ji .Fi / /Mi D IU;ji .Fi / \ Mi ;

592

8 | principles for !1

(3) Let I be the ideal on !1 which is dual to the filter, F D \¹U j U 2 Y º; then \¹IU;ji .Fi / j U 2 Y º  I: (4) j.Yi / D ¹U \ Mi j U 2 Y º. Proof. Let

j0 W .M0 ; I0 / ! .M0 ; I0 /

and

j1 W .M1 ; I1 / ! .M1 ; I1 /

be iterations of length !1 , and let  W j0 .Y0 / ! j1 .Y1 / be a bijection such that : (1.1) !1 n ¹˛ < !1 j j0 .F0 /.˛/ D j1 .F1 /.˛/º 2 INS . (1.2) Suppose that W0 2 j0 .Y0 / and W1 D .W0 /. Then for all A0 2 W0 and for all A1 2 W1 , A0 \ A1 … INS : (1.3) Suppose that U  P .!1 / is an ultrafilter such that

U \ M0 2 j0 .Y0 /

and such that

U \ M1 2 j1 .Y1 /:

a) The ideal IU;F is proper. b) For each i 2 ¹0;1º, 

.RW;F /Mi D RU;F \ Mi ; and



.IW;F /Mi D IU;F \ Mi ;

where W D Mi \ U . By Lemma 8.89, .j0 ; j1 ; / exists. Let F D j0 .F0 /. The desired set of ultrafilters Y is obtained just as in the proof of Lemma 8.41. Let Z be the set of uniform ultrafilters U on !1 such that U \ M0 2 j0 .Y0 /

C

CC

8.3 The principles, |NS and |NS

and such that

593

U \ M1 2 j1 .Y1 /:

We define by induction on ˛ a normal ideal J˛ as follows: J0 D \¹IU;F j U 2 Zº and for all ˛ > 0, J˛ D \¹IU;F j U 2 Z and for all  < ˛, J \ U D ;º: It follows easily by induction that if ˛1 < ˛2 then J˛1  J˛2 : Thus for each ˛, J˛ is unambiguously defined as the intersection of a nonempty set of uniform normal ideals on !1 . The sequence of ideals is necessarily eventually constant. Let ˛ be least such that J˛ D J˛C1 and let J D J˛ : Thus J is a uniform normal ideal on !1 . Let Y be the set of U 2 Z such that U \ J D ; and let I be the ideal dual to the filter F D \¹U j U 2 Y º: Then it follows that \¹IU;F j U 2 Y º  I: Similarly and

j0 .Y0 / D ¹U \ M0 j U 2 Y º j1 .Y1 / D ¹U \ M1 j U 2 Y º: |

t u

NS is an immediate corollary. We isolate the relevant fact The homogeneity of Pmax in the following lemma.

Lemma 8.91. Suppose that for each x 2 R, there exists |

h.M; I/; g; Y; F i 2 M0 NS |

|

NS NS such that x 2 M. Suppose that p0 2 Pmax and p1 2 Pmax . There exist

|

NS .h.Mk ; Yk / W k < !i; F / 2 Pmax

and functions F0 , F1 such that |

NS (1) .h.Mk ; Yk / W k < !i; F0 / 2 Pmax and .h.Mk ; Yk / W k < !i; F0 / < p0 ,

|

NS (2) .h.Mk ; Yk / W k < !i; F1 / 2 Pmax and .h.Mk ; Yk / W k < !i; F1 / < p1 ,

(3) ¹˛ < !1M0 j F0 .˛/ ¤ F1 .˛/º 2 .INS /M0 .

594

8 | principles for !1

Proof. Let x 2 R code the pair .p0 ; p0 / and let |

h.M; I/; g; Y; F i 2 M0 NS be such that x 2 M. Thus

|

NS M / : ¹p0 ; p1 º  .Pmax

Let N D .L.R//M and for i 2 ¹0;1º let |

NS N / gi  .Umax

be N -generic with pi 2 Fgi where |

NS N / Fgi  .Pmax

|

NS N is the induced N -generic filter on .Pmax / . Let hYi ; Fi i D hYgi ; Fgi iN Œgi 

and let Ii D ¹.IU;Fi /N Œgi  j U 2 Yi º: A key point is that since .M; I/ is iterable it follows by Lemma 8.66 and Theorem 3.46, that for each i 2 ¹0;1º, the structure .N Œgi ; Ii / is also iterable and so |

h.Mi ; Ii /; gi ; Yi ; Fi i 2 M0 NS ; where Mi D N Œgi . Strictly speaking Lemma 8.66 and Theorem 3.46 cannot be applied in N Œg since we have only N Œg  ZFC C ZC C †1 -Replacement; but both are easily seen to hold in this case. Let y 2 R code .N ; g0 ; g1 / and let O I; O ı; / 2 H.!1 / .M; be such that O (1.1) x 2 M, (1.2) MO is transitive and MO  ZFC C “ı is a Woodin cardinal”, O (1.3) IO D .I
9.4 Chang’s Conjecture

9.4

637

Chang’s Conjecture

There is a curious metamathematical possibility. Perhaps there is an interesting combinatorial statement whose truth in L.R/Pmax cannot be proved just assuming L.R/  AD; but can be proved from a stronger hypothesis. We recall the statement of Chang’s Conjecture. Definition 9.56. Chang’s Conjecture: The set ¹X  !2 j ordertype.X / D !1 º t u

is stationary in P .!2 /. D. Seabold has proved the following theorem.

Theorem 9.57 (Seabold). Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Suppose G  Pmax is L.; R/-generic. Then L.; R/ŒG  Chang’s Conjecture:

t u

A corollary of Seabold’s theorem is that from suitable determinacy hypotheses one obtains that L.R/Pmax  Chang’s Conjecture: The proof can be refined to establish the following theorem which identifies a sufficient condition which is first order in L.R/. Theorem 9.58. Suppose L.R/  AD and that there exists a countable set  R such that HODL.R/ . / \ R D and HODL.R/ . /  AD C DC: Suppose G  Pmax is L.R/-generic. Then L.R/ŒG  Chang’s Conjecture: The proofs adapt to prove the following improvement of Corollary 9.40.

t u

638

9 Extensions of L.; R/

Theorem 9.59. Suppose L.R/  AD and that there exists a countable set  R such that HODL.R/ . / \ R D and HODL.R/ . /  AD C DC: Suppose G  Pmax is L.R/-generic. Then L.R/ŒG  ZFC C Martin’s Maximum ZF .c/:

t u

Our goal in this section is to sketch the proof of the generalization of Theorem 9.58 to the Qmax -extension; Theorem 9.60. Suppose L.R/  AD and that there exists a countable set  R such that HODL.R/ . / \ R D and HODL.R/ . /  AD C DC: Suppose G  Qmax is L.R/-generic. Then L.R/ŒG  Chang’s Conjecture:

t u

The following theorem, in conjunction with Theorem 6.149, shows that L.R/Qmax  Chang’s Conjecture cannot be proved just assuming L.R/  AD: Pmax

The analogous question for L.R/

is open.

Theorem 9.61. Suppose ZFC C “ There is a normal, uniform, ideal on !1 which is !1 -dense” C Chang’s Conjecture is consistent. Then ZFC C “ There is a normal, uniform, ideal on !1 which is !1 -dense” C “ There are infinitely many Woodin cardinals” is consistent.

t u

Steel has generalized the analysis of scales in L.R/ to iterable Mitchell–Steel modQ i. e. Mitchell–Steel models relativized to R. With this machinels of the form L.R; E/; ery the method of the core model induction used to prove Theorem 6.149 on page 425, generalizes to prove the following theorem.

9.4 Chang’s Conjecture

639

Theorem 9.62. Suppose there is a normal, uniform, ideal on !1 which is !1 -dense and that Chang’s Conjecture holds. Then there exists a countable set  R such that HODL.R/ . /  AD C DC:

t u

Corollary 9.63. Suppose there is a normal, uniform, ideal on !1 which is !1 -dense and that Chang’s Conjecture holds. Then L.R/Qmax  Chang’s Conjecture: Remark 9.64.

t u

(1) The hypothesis of Theorem 9.58 is equiconsistent with ZFC C “There are ! C ! many Woodin cardinals”:

Thus Theorem 9.62 implies that ZFC C “There are ! C ! many Woodin cardinals” is equiconsistent with ZFC C “There is a normal, uniform, ideal on !1 which is !1 -dense” C Chang’s Conjecture: (2) We do not know if Theorem 9.62 can be generalized to ./. More precisely:  Suppose that ./ and Chang’s Conjecture hold. Is there a countable set  R such that HODL.R/ . /  AD C DC‹ u t We need a technical lemma which is a variant of Theorem 9.52. Lemma 9.65. Suppose A  R and that L.A; R/  ADC : Suppose G  Qmax is L.A; R/-generic. Then in L.A; R/ŒG the following holds. Suppose  > !2 , L .A; R/ŒG  ZFC ; and that L .A; R/ †2 L.A; R/: Then for each a 2 L .A; R/ŒG there exists a countable elementary substructure, X  L .A; R/ŒG such that ¹fG ; a; Aº  X and such that the following hold: (1) h.MX ; IX /; fX i 2 G; (2) for all ˛ 2 C , fG .˛/ is L.MX ; fG j˛/-generic for Coll.!; ˛/;

640

9 Extensions of L.; R/

where MX is the transitive collapse of X , fX is the image of fG under the collapsing map, IX D .INS /MX ; and where C is the critical sequence of the iteration j W .MX ; IX / ! .MX ; IX / such that j.fX / D fG . Proof. Fix p0 2 G and fix a term  2 L .A; R/ for a. We work in L.A; R/. Fix  W R ! L .A; R/ such that ŒR  L .A; R/ and such that ¹; Aº  ŒR. Let x0 2 R be such that .x0 / D  and let x1 be such that .x1 / D A. Let B  R code the set of pairs .ha0 ; : : : ; an i; .z0 ; : : : ; zn // such that ha0 ; : : : ; an i 2 R


,  is a formula and

L .A; R/  Œ.a0 /; : : : ; .an /: Let T be the theory of L .A; R/; i. e. a reasonable fragment of ZF C AD C DC C “V D L.A; R/” 

containing ZFC . By Lemma 9.52 there exists a countable transitive set N and a filter H  QN max such that (1.1) N  T, (1.2) ¹p0 ; x0 ; x1 º  N , (1.3) p0 2 H , (1.4) H is N -generic, (1.5) B \ N 2 N and hH.!1 /N ; B \ N; 2i  hH.!1 /; B; 2i, (1.6) .N ŒH ; .INS /N ŒH  / is B-iterable. Thus N ŒH  /; f i 2 Qmax h.N ŒH ; INS

and N ŒH  /; f i < p0 h.N ŒH ; INS

9.4 Chang’s Conjecture

where

641

f D fHN ŒH  D [¹f1 j h.M1 ; I1 /; f1 i 2 H º:

By genericity we can suppose that N ŒH  h.N ŒH ; INS /; f i 2 G

and that for all ˛ 2 D, fG .˛/ is L.N ŒH ; fG j˛/-generic for Coll.!; ˛/ where D is the critical sequence of the iteration N ŒH  j  W .N ŒH ; INS / ! .N  ; I  /

such that j  .f / D fG . N ŒH  Since h.N ŒH ; INS /; f i 2 G, H  G. Let X  L .A; R/ŒG be the set of b 2 L .A; R/ŒG such that b is definable in L .A; R/ŒG from parameters in ŒR \ N  [ ¹fG º. Thus X  L .A; R/ŒG. Further a 2 X since x0 2 N . The key points are that H G and hH.!1 /N ; B \ N; 2i  hH.!1 /; B; 2i; for these imply that X \ L .A; R/ D ŒR \ N : Let MX be the transitive collapse of X , let FX be the image of fG under the collapsing map and let IX D .INS /MX . Thus (2.1) .H.!2 //MX D H.!2 /N ŒH  , (2.2) IX D .INS /N ŒH  , (2.3) f D fX . Therefore by Theorem 9.52, .MX ; IX / is iterable, and so h.MX ; IX /; fX i 2 Qmax : However, N ŒH  /; f i 2 G: h.N ŒH ; INS N ŒH  /; f i 2 G induces an iteration witTherefore the iteration witnessing h.N ŒH ; INS nessing h.MX ; IX /; fX i 2 G

and this iteration has the same critical sequence, D. Therefore since MX 2 N ŒH , it follows that for all ˛ 2 D, fG .˛/ is t u L.MX ; fG j˛/-generic for Coll.!; ˛/.

642

9 Extensions of L.; R/

Lemma 9.66. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC C “ ‚ is regular ”: Suppose G  Qmax is L.; R/-generic. Then in L.; R/ŒG the following holds. Suppose N is a transitive set such that N  ZFC and such that N !1  N: There exists a function h W !1 ! N such that for all limit ordinals 0 <  < !1 , (1) hŒ  N , (2)   hŒ, (3) fG ./  fG . / is N -generic for Coll.!; fG .//  Coll.!; fG . // N

where N is the transitive collapse of hŒ and  D !2  . Proof. By Theorem 9.51, L.; R/ŒG  !2 -DC: Therefore there exists ZN such that Z  Z and such that jZj D !2 . Let NZ be the transitive collapse of Z. Thus we can suppose, by replacing N by NZ if necessary, that jN j D !2 . Fix a bijection F W !2 ! N: !1

Thus for some A  R with A 2 L.; R/, .F; N / 2 L.A; R/ŒG: Fix  2 Ord such that

L .A; R/  ZFC

and such that L .A; R/ †2 L.A; R/: By Lemma 9.65 there exists a countable elementary substructure X  L .A; R/ŒG such that ¹A; F; N; fG º  X and such that the following hold:

9.4 Chang’s Conjecture

643

(1.1) h.MX ; IX /; fX i 2 G; (1.2) for all ˛ 2 C , fG .˛/ is L.MX ; fG j˛/-generic for Coll.!; ˛/; where MX is the transitive collapse of X , fX is the image of fG under the collapsing map, IX D .INS /MX ; and where C is the critical sequence of the iteration j W .MX ; IX / ! .MX ; IX / such that j.fX / D fG . Let hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i be the iteration of .MX ; IX / such that j0;!1 .fX / D fG . Define a sequence hX˛ W ˛  !1 i of countable elementary substructures by induction on ˛ such that (2.1) X0 D X , (2.2) X˛C1 D ¹f .X˛ \ !1 / j f 2 X˛ º, (2.3) if ˇ  !1 is a limit ordinal then Xˇ D [¹X˛ j ˛ < ˇº: For each ˛  !1 let M˛ be the transitive collapse of X˛ , let ˛ W M˛ ! X˛ be the inverse of the collapsing map, and let 

G˛ D ¹a 2 .P .!1 //M˛ j X˛ \ !1 2 ˛ .a/º: Thus G˛ is simply the image of ¹S 2 P .!1 / \ X˛ j X˛ \ !1 2 S º in the transitive collapse of X˛ . For each ˛ < ˇ  !1 let

 j˛;ˇ W M˛ ! Mˇ

be the elementary embedding such that for all a 2 M˛ ,  .a//: ˛ .a/ D ˇ .j˛;ˇ

Thus

 hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i

 is an iteration of .MX ; IX / such that j0;! .fX / D fG . 1 Therefore  hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i D hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i:

9 Extensions of L.; R/

644

For each   !1 let

NQ D X \ N and let N be the transitive collapse of NQ . Thus N 2 M and NQ D  .N /: Further for each   !1 ,

N D j0; .N0 /: M

M

Thus for each  < !1 , fG .!1  / is L.M0 ; fG j!1  /-generic. However for each  < !1 , M

hM˛ ; G˛ ; j˛;ˇ W ˛ < ˇ  i 2 L.M0 ; fG j!1  / MC1

and so for each  < !1 , fG .!1

/ is L.N /-generic. Further

MC1

!1

M

D !2

N

D !2  :

Let h W !1 ! N be such that for all limit ordinals  < !1 , hΠD NQ : t u

The function h is as desired. The application of Lemma 9.66 requires the following additional lemma. Lemma 9.67. Suppose L.R/  AD and that  R is a countable set such that HODL.R/ . / \ R D and HODL.R/ . /  ZF C AD C DC Then

HODL.R/ . /  ADC :

Theorem 9.68. Suppose L.R/  AD and that there exists a countable set  R such that HODL.R/ . / \ R D and HODL.R/ . /  AD C DC: Suppose G  Qmax is L.R/-generic. Then L.R/ŒG  Chang’s Conjecture:

t u

9.4 Chang’s Conjecture

645

Proof. We work in L.R/ŒG. Fix F W !2
!2
˛ < !2 and in L.R/, p Qmax  .s/ D ˛:

If Chang’s Conjecture fails in L.R/ŒG then there is a function F and a corresponding term  such that F is a counterexample and such that A is 21 in L.R/. This follows by the usual reflection arguments and the fact that the pointclass .†21 /L.R/ has the scale property, Theorem 2.3. Again by the scale property of .†21 /L.R/ there must exist a condition p0 2 Qmax \ HODL.R/ such that p0 forces that  is a term for a counterexample to Chang’s Conjecture. Fix a countable set  R such that HODL.R/ . / \ R D

9 Extensions of L.; R/

646 and

HODL.R/ . /  ZF C AD C DC: By Lemma 9.67, we can suppose that HODL.R/ . /  ADC : Let S  Ord be a set such that LŒS  D HODL.R/ : It is easy to see that such a set S exists, essentially by Vopenka’s argument. In fact one can choose S to be a subset of ‚L.R/ . Let N D L.S; / and let  D P . / \ N: Thus

L.; /  ADC C “ ‚ is regular”

and D R \ L.; /: Let g0  Qmax \ L.; / be L.; /-generic such that p0 2 g0 and let f0 D [¹f j h.M; I /; f i 2 g0 º: Thus g0 is N -generic and P . / \ N Œg0  D P . / \ L.; /Œg0 : Therefore by Theorem 6.81 and Theorem 9.51, the following hold in N Œg0 . (1.1) AC . (1.2) The nonstationary ideal on !1 is !1 -dense. (1.3) f0 witnesses ˘++ .!1
9.4 Chang’s Conjecture

647

By Theorem 5.35, L.R/ HOD¹a D HODL.R/ Œa0  D LŒS; a0  D N Œg0 : 0º

By Theorem 5.34, there exists x0 2 R such that for all x 2 R, if x0 2 LŒS; a0 ; x then

!2LŒS;a0 ;x

is a Woodin cardinal in

0 ;x : HODLŒS;a ¹S;a0 º

However for each  < !1 , there exists x1 2 R such that for all x 2 R, if x1 2 LŒS; a0 ; x then 0 ;x  HODL.R/  LŒS; a0 ; P ./ \ HODLŒS;a ¹S;a0 º ¹S;a0 º

since HODL.R/ D LŒS . Therefore there exist a transitive inner model M , containing the ordinals, and ı0 < !1 such that M  ZFC; ¹S; a0 º  M ,

P .!3N Œg0  / \ N Œg0  D P .!3N Œg0  / \ M;

and such that ı0 is a Woodin cardinal in M . Thus (1.1)–(1.4) hold in M . Let M0 D M \ V where  is the least ordinal such that  > ı0 and such that  is strongly inaccessible in M . Since M  ZFC and since M  L.R/,  exists. M0 . Since Ord  M , the structure .M0 ; I0 / is iterable where I0 D INS Since (1.1)–(1.4) hold in M , (1.1)–(1.4) hold in M0 . Therefore h.M0 ; I0 /; f0 i 2 Qmax and it follows that h.M0 ; g0 /; f0 i < p for all p 2 g0 . By Lemma 6.23 there exists an iteration j W .M0 ; I0 / ! .M0 ; I0 / such that ¹˛ < !1 j j.f0 /.˛/ D fG .˛/º contains a club in !1 . By Theorem 6.34 there is an L.R/-generic filter G   Qmax such that fG  D fG and such that

L.R/ŒG   D L.R/ŒG:

9 Extensions of L.; R/

648

Thus it follows that

h.M0 ; I0 /; f0 i 2 G  :

We now come the key claim, which is a consequence of Lemma 9.66. Let P0 be the partial order given by the stationary tower .P ¹˛ < !1 j k0 .f0 /.˛/ D fG  .˛/º contains a club in !1 . By Theorem 6.34, there is an L.R/-generic filter G   Qmax , such that L.R/ŒG   D L.R/ŒG   D L.R/ŒG; and such that k0 .f0 / D fG  . We now come the second point. Note f0 D [¹f j h.M; I /; f i 2 j0 .g0 /º since f0 D [¹f j h.M; I /; f i 2 g0 º and so k0 .f0 / D [¹f j h.M; I /; f i 2 k0 ı j0 .g0 /º: Therefore, this is the second point, g0  j0 .g0 /  k0 ı j0 .g0 /  G  : Let

F  W !2
650

9 Extensions of L.; R/

be the interpretation of  by G  . Since g0  G  , p0 2 G  . Therefore F  is a counterexample to Chang’s Conjecture. Let 0 be the interpretation of  in M0 . Define F0 W !2M0 ! !2M0 by F0 .s/ D ˛ if for some x 2 R \ M0 , 0 .x/ D .p; s; ˛/ for some p 2 g0 . By the definition of M0 , R \ M0 D D R \ HODL.R/ Œ : Therefore since A is 21 , A \ M0 2 M0 and further hV!C1 \ L.; /; A \ ; 2i  hV!C1 ; A ; 2i: Therefore since g0 is L.; /-generic, this definition of F0 does yield a function. Let Q Q Z D k0 Œ!2M0  D ¹k0 .˛/ j ˛ < !2M0 º: Q

Since !2M0 D !1 , Z has ordertype !1 . Further by elementarity, k0 .j0 .F0 //ŒZ
By the choice of  and since !1 D !1 0 , k0 ı j0 .0 / D j.R \ MQ 0 /: The final point is that k0 ı j0 .A \ M0 / D A \ MQ 0 and so since k0 ı j0 .g0 /  G  , k0 ı j0 .F0 /  F  . But then Z witnesses that F  is not a counterexample to Chang’s Conjecture, a contradiction. We verify this final point which amounts to a certain form of A -iterability. Since M0 D V \ M it follows that the elementary embedding j0 lifts to define an elementary embedding j W M ! MQ  L.R/: It follows that

MQ 0 D j.M0 / D Vj. / \ MQ :

Therefore the elementary embedding k0 lifts to define an elementary embedding k W MQ ! MQ   MQ Œh0 : We must show that k ı j.A \ M / D A \ MQ  . The set A is 21 is L.R/. Therefore there exist trees T0 2 HODL.R/ and T1 2 HODL.R/ such that A D pŒT0  and R n A D pŒT1 :

9.5 Weak and Strong Reflection Principles

651

Since HODL.R/  M , T0 2 M and T1 2 M . Thus k ı j.A \ M / D pŒk ı j.T0 / \ MQ  : Clearly pŒT0   pŒk ı j.T0 / and pŒT1   pŒk ı j.T1 /: However by absoluteness pŒk ı j.T0 / \ pŒk ı j.T1 / D ;: Therefore pŒT0  D pŒk ı j.T0 / and so k ı j.A \ M / D A \ MQ  as desired.

9.5

t u

Weak and Strong Reflection Principles

A natural question is the following. Suppose   P .R/ is a pointclass, closed under continuous preimages, such that L.; R/  ADC C “‚ is regular”: Suppose that L.; R/Pmax  Martin’s Maximum.c/: Must L.; R/  ADR ‹ As we have previously noted (Theorem 9.41), the answer to this question is yes. One goal of this section is to sketch a proof of a stronger theorem, Theorem 9.87. The following theorems show that some condition on  is necessary. Theorem 9.69. Assume Martin’s Maximum.c/. Then for each set A  !2 , A# exists. t u Theorem 9.70. Suppose that Martin’s Maximum.c/ holds and that P is a partial order of cardinality !2 . Suppose G  P is V -generic. Then V ŒG  PD: t u We shall obtain Theorem 9.69 as a corollary of a slightly stronger theorem, Theorem 9.75, that involves a specific consequence of Martin’s Maximum.c/. This consequence is a reflection principle for stationary subsets of P!1 .!2 / which is a special case of the reflection principle WRP of .Foreman, Magidor, and Shelah 1988/. This and a generalization formulated in .Todorcevic 1984/ are discussed briefly in Section 2.6, see Definition 2.54. The special cases of interest to us here are given in the following definition. The actual formulation of Definition 9.71(3) is taken from .Feng and Jech 1998/. This formulation is more elegant than the original.

652

9 Extensions of L.; R/

(1) (Foreman–Magidor–Shelah) WRP.!2 /: Suppose that S  P!1 .!2 / is stationary in P!1 .!2 /. Then there exists !1 < ˛ < !2 such that S \ P!1 .˛/ is stationary in P!1 .˛/.

Definition 9.71.

(2) (Foreman–Magidor–Shelah) WRP.2/ .!2 /: Suppose that S1  P!1 .!2 / and S2  P!1 .!2 / are each stationary in P!1 .!2 /. Then there exists !1 < ˛ < !2 such that S1 \ P!1 .˛/ and S2 \ P!1 .˛/ are each stationary in P!1 .˛/. (3) (Todorcevic) SRP.!2 /: Suppose that S  P!1 .!2 / and that for each stationary set T  !1 , the set ¹ 2 S j \ !1 2 T º is stationary in P!1 .!2 /. Then there exists !1 < ˛ < !2 such that S \ P!1 .˛/ contains a closed, unbounded, subset of P!1 .˛/. Lemma 9.72 (Todorcevic). SRP.!2 / holds.

(1) Assume that Martin’s Maximum.c/ is true. Then

(2) Assume SRP.!2 /. Then WRP.2/ .!2 / holds. Lemma 9.73 (Todorcevic).

t u

t u

(1) Assume WRP.!2 /. Then 2@0  @2 .

(2) Assume SRP.!2 /. Then 2@1 D @2 .

t u

Remark 9.74. (1) It is not difficult to show that WRP.!2 / is consistent with 2@1 > @2 and WRP.!2 / is consistent with CH. Thus Lemma 9.73(1) cannot really be improved. (2) We shall prove that SRP.!2 / implies ı12 D !2 and so SRP.!2 / implies c D @2 , see Theorem 9.79. (3) In fact, SRP.!2 / implies AC , as we shall note below. This gives a different t u proof that SRP.!2 / implies c D @2 . At the heart of Theorem 9.69 is the following theorem from which one can obtain Theorem 9.69 as a corollary.

9.5 Weak and Strong Reflection Principles

653

Theorem 9.75. Assume WRP.2/ .!2 / and that for each set A  !1 , A# exists. Then for each set A  !2 , A# exists. Proof. Fix a set A  !2 . We must prove that A# exists. Clearly we may suppose that A is cofinal in !2 . For each countable set  !2 let ı be the ordertype of and let A  ı be the image of A under the transitive collapse of . Let  W ! ı be the collapsing map. For each i < ! let i D !2Ci . Thus for each bounded set b  !2 , i is a Silver indiscernible of LŒb. For each formula .x0 ; y0 ; z0 / and for each pair .s; t / 2 Œ!2 
654

9 Extensions of L.; R/

then S.:;s0 ;t0 / \ P!1 .˛/ is not stationary in P!1 .˛/. Similarly if LŒA \ ˛  :ŒA \ ˛; s0 ; t0 : then S.;s0 ;t0 / \ P!1 .˛/ is not stationary in P!1 .˛/. This contradicts the choice of ˛ and so proves our claim. Let T be the set of .; s; t / such that S.;s;t/ contains a closed unbounded subset of P!1 .!2 /. T is naturally interpreted as a complete theory in the language with additional constants for A, the ordinals less than !2 , and for the i . It follows easily that this theory is A# since every countable subset of this theory can be embedded into A# for almost all (in the sense of the filter generated by the closed unbounded subsets t u of P!1 .!2 /). Magidor has noted the following:  Suppose that is weakly compact and that G  Coll.!1 ; < / is V -generic. Then (by reflection) V ŒG  WRP.2/ .!2 /: Thus WRP.2/ .!2 / does not imply, for example, that 0# exists. Lemma 9.76 (Todorcevic). .SRP.!2 // INS is !2 -saturated.

t u

One corollary of Lemma 9.76 and the proof of Theorem 5.13 is that SRP.!2 / implies AC . This result, obtained independently by P. Larson, gives yet another proof that SRP.!2 / implies c D @2 . Corollary 9.77 (SRP.!2 /).

AC

holds.

t u

Theorem 9.69 is an immediate consequence of the following corollary of both Lemma 9.76 and Theorem 9.75. Corollary 9.78. Assume SRP.!2 /. Then for each set A  !2 , A# exists. Proof. By Lemma 9.72(2), WRP.2/ .!2 / holds. By Lemma 9.76, for every set A  !1 , t u A# exists. Therefore by Theorem 9.75, for every set A  !2 , A# exists. Another corollary is the following theorem which shows that SRP.!2 / implies that ı12 D !2 . Theorem 9.79. Assume SRP.!2 /. Then ı12 D !2 . Proof. We have by Lemma 9.76 and Corollary 9.78 the following. (1.1) INS is !2 -saturated and that for each A  !2 , A# exists.

9.5 Weak and Strong Reflection Principles

655

We claim that (1.1) implies that  ı 12 D !2 . This is an immediate corollary of Theorem 3.17 if one assumes in addition that 2@1 D @2 which in fact is a consequence of SRP.!2 /. However this additional assumption is unnecessary. Choose A  !2 such that (2.1) .!2 /LŒA D !2 , (2.2) .H.!2 //LŒA  H.!2 /. This is easily done, it is theorem of ZFC that such a set A exists. Thus (by (2.2)) for all ˛ < !2 , .A \ ˛/# 2 LŒA and so P .!1 / \ LŒA D P .!1 / \ LŒA# : Therefore

#

.H.!2 //LŒA   H.!2 /; which implies that LŒA#   “INS is !2 -saturated” #

since !2 D .!2 /LŒA  . But LŒA#   “P .!1 /# exists”. Thus by Theorem 3.17, which can be applied in the inner model LŒA# , LŒA#   “ı12 D !2 ” and so  ı 12 D !2 .

t u

Corollary 9.78 can be strengthened considerably, for example if SRP.!2 / holds then for every set A  !2 , A exists. Another generalization of Theorem 9.75 is given in the following theorem whose proof is closely related to the proof of Theorem 9.87. Theorem 9.80. Suppose that WRP.2/ .!2 / holds and that if g  Coll.!; !1 / is V -generic then in V Œg: (i) L.R/  AD; (ii) R# exists. Suppose that G  Coll.!; !2 / is V -generic. Then in V ŒG: (1) L.R/  AD; (2) R# exists.

t u

656

9 Extensions of L.; R/

An easier version of Theorem 9.80 is the following theorem. Recall that PD is the assertion that all projective sets are determined. Theorem 9.81. Suppose that WRP.2/ .!2 / holds and that if g  Coll.!; !1 / is V -generic then V Œg  PD: Suppose that G  Coll.!; !2 / is V -generic. Then V ŒG  PD:

t u

The method of proving Theorem 9.81 amplified by some of the machinery behind the proof of Theorem 5.104 yields the following improvements of Theorem 9.81. Theorem 9.82. Suppose that WRP.2/ .!2 / holds and that INS is !2 -saturated. Suppose that G  Coll.!; !2 / is V -generic. Then V ŒG  PD:

t u

Corollary 9.83. Suppose that SRP.!2 / holds and that P is a partial order of cardinality !2 . Suppose G  P is V -generic. Then V ŒG  PD:

t u

Remark 9.84. Theorem 9.82, and therefore Corollary 9.83, can be be strengthened to obtain more determinacy. The main results of Steel and Zoble .2008/ improve the results by obtaining ADL.R/ . The proof of Theorem 9.82 can be implemented using a weakened version of SRP.!2 /, see Theorem 9.95. This version is defined in Definition 9.88(2). Theorem 9.99 shows that this weakened version together with the assertion that INS is !2 t u saturated cannot imply significantly determinacy significantly past ADL.R/ . WRP.!2 / implies a weak variation of Chang’s Conjecture. Lemma 9.85 (WRP.!2 /). Suppose that F W !2
9.5 Weak and Strong Reflection Principles

657

Proof. Let S D P!1 .!2 / n CF : Assume toward a contradiction that S is stationary in P!1 .!2 /. Thus by WRP.!2 / there exists !1 < ˛ < !2 such that S \ P!1 .˛/ is stationary in P!1 .˛/. Let Z  H.!3 / be a countable elementary substructure such that (1.1) F 2 Z, (1.2) ˛ 2 Z, (1.3) Z \ ˛ 2 S . The requirement (1.3) is easily arranged since S \ P!1 .˛/ is stationary in P!1 .˛/. By (1.2) Z \ ˛ ¨ Z \ !2 : But Z \ !2 is closed under F and so Z \ !2 witnesses that Z \ ˛ 2 CF which contradicts that Z \ ˛ 2 S . t u An immediate corollary of the next lemma is that WRP.!2 / must fail in L.A; R/Pmax where A  R is such that L.A; R/  ADC : Lemma 9.86. Suppose that V D LŒA for some set A  !2 and that for each set B  !1 , B # exists. Then WRP.!2 / fails. Proof. Consider the structure h!2 ; A; 2i: For each countable elementary substructure X  h!2 ; A; 2i let AX be the image of A under the transitive collapse. Let S be the set of X 2 P!1 .!2 / such that !1 \ X is countable in LŒAX . We claim that that S is stationary in P!1 .!2 /. If not let Y0 be the set of a 2 H.!3 / such that a is definable in the structure hH.!3 /; A; 2i: Thus Y0  H.!3 / and so S 2 Y0 . Since S is not stationary it follows that Y0 \ !2 … S: Let M0 be the transitive collapse of Y0 and let A0 be the image of A under the transitive collapse. Thus every element of M0 is definable in the structure hM0 ; A0 ; 2i:

658

9 Extensions of L.; R/

However M0  “V D LŒA0 ” and so M0 2 LŒA0 . Therefore M0 is countable in LŒA0  and so Y0 \ !2 2 S , a contradiction. Thus S is stationary in P!1 .!2 /. We now assume toward a contradiction that WRP.!2 / holds. Fix !1 < ˛ < !2 such that S \ P!1 .˛/ is stationary in P!1 .˛/. Thus there exists Z  H.!3 / such that (1.1) Z \ ˛ 2 S , (1.2) A \ ˛ 2 Z. However .A \ ˛/# exists and so .A \ ˛/# 2 Z. Let Z0 D Z \ ˛. Let MZ be the transitive collapse of Z and let AZ be the image of A under the transitive collapse. Let AZ0 be the image of A \ ˛ under the transitive collapse of Z0 . Trivially AZ0 is the image of A \ ˛ under the transitive collapse of Z. However Z \ !1 D Z0 \ !1 and so since Z0 2 S ,

V!C1 \ LŒAZ0  6 MZ :

The key point is that since .A \ ˛/# 2 Z; it follows that .AZ0 /# 2 MZ . This in turn implies that V!C1 \ LŒAZ0   MZ ; t u

which is a contradiction.

By Lemma 9.72, Martin’s Maximum.c/ implies WRP.!2 / and so Theorem 9.41 is an immediate corollary of the next theorem. The proof requires more of the descriptive set theory associated to ADC and so we shall only sketch the argument. The proof is in essence a generalization of the proof of Lemma 9.86. Theorem 9.87. Suppose that   P .R/ is a pointclass, closed under continuous preimages, such that L.; R/  ADC C “ ‚ is regular”: and such that

L.; R/Pmax  WRP.!2 /:

Then L.; R/  AD R :

9.5 Weak and Strong Reflection Principles

659

Proof. We assume toward a contradiction that L.; R/ 6 AD R : Fix G  Pmax such that G is L.; R/-generic. We work in L.; R/ŒG. By Theorem 9.22, since L.; R/ 6 AD R ; there exists a set S  Ord such that L.; R/ D L.S ; R/: Thus, since L.R/ŒG  ZFC, L.; R/ŒG  ZFC: Let .0 ; ˛0 / be the least (lexicographical order) such that (1.1) 0   is a boolean pointclass closed under continuous preimages, (1.2) ˛0 2 Ord and L˛0 .0 ; R/  ZF n Replacement C †1 -Replacement; (1.3) L˛0 .0 ; R/  ADC C “ ‚ is regular”, (1.4) L˛0 .0 ; R/ŒG  WRP.!2 /, (1.5) L˛0 .0 ; R/ 6 AD R . It follows that 0 D P .R/ \ L˛0 .0 ; R/; for otherwise L˛0 .0 ; R/ŒG D L˛0 ŒA for some A  !2 which, by (1.4), contradicts Lemma 9.86. By (1.5) there is a largest Suslin cardinal (in L˛0 .0 ; R/), ı0 < .‚/L˛0 .0 ;R/ : Let 0 be the pointclass of sets A  R such that A is Suslin and co-Suslin in L˛0 .0 ; R/: It follows that there exists a tree T on !  ı0 such that (2.1) T 2 L˛0 .0 ; R/, (2.2) let 0 be the set of A  R such that A is †1 -definable in the structure hM 0 ; T; 2i with parameters from R, then T is the tree of 0 -scale on the universal 0 set, (2.3) 0  .HODT .R//L˛0 .0 ;R/ . The key consequence of ADC is the following.

660

9 Extensions of L.; R/

(3.1) Suppose that A 2 0 is definable in L˛0 .0 ; R/ from .T; x/ where x 2 R. Then there is a scale on A (in L.; R/) each norm of which is definable in L˛0 .0 ; R/ from .T; x/. In particular, and this is all we require, (4.1) every set in 0 is Suslin in L.; R/. Consider the structure hM 0 ŒG; T; 2i: For each countable elementary substructure X  hM 0 ŒG; T; 2i let NX be the transitive collapse of X . Let S be the set of countable X  hM 0 ŒG; T; 2i such that !1 \ X is countable in LŒT; NX . We claim that in L˛0 .0 ; R/ŒG, the set S is stationary in P!1 .M 0 ŒG/. If not let Y0 be the set of a 2 M0 ŒG such that a is definable in the structure hM0 ŒG; 0 ; T; 2i from G. Note that jM 0 ŒGjL˛0 .0 ;R/ŒG D !2 and .H.!3 //L˛0 .0 ;R/ŒG D M0 ŒG: Therefore, by (2.3), there is a wellordering of M0 which is definable in hM0 ŒG; 0 ; T; 2i from G. Thus Y0  M0 ŒG and so it follows that S 2 Y0 . Since S is not stationary it follows that Y0 \ M 0 ŒG … S: Let M0 be the transitive collapse of Y0 and let N0 be the image of hM 0 ŒG; T; 2i under the transitive collapse. The key point is that M0 2 LŒT; N0 : This is another consequence of ADC , it is closely related to the proof of (2.3). Thus N0 is countable in LŒT; N0  which contradicts that Y0 \ M 0 ŒG … S: Therefore S is stationary in P!1 .M 0 ŒG/. We show this contradicts that L˛0 .0 ; R/ŒG  WRP.!2 /:

9.5 Weak and Strong Reflection Principles

661

Fix a bijection  W !2 ! M 0 ŒG with  2 L˛0 .0 ; R/ŒG. This exists since jM 0 ŒGj D !2 in L˛0 .0 ; R/ŒG. Therefore, by WRP.!2 /, there exists  < !2 such that (5.1) Œ  M 0 ŒG, (5.2) !1  Œ, (5.3) ¹X 2 P!1 ./ j ŒX  2 S º is stationary in P!1 ./. The key point is that H.!2 /L.;R/ŒG D H.!2 /L˛0 .0 ;R/ŒG and so in L.; R/ŒG, the set ¹X 2 P!1 ./ j ŒX  2 S º is stationary in P!1 ./. Let B 2 0 be the set of x 2 R such that x codes a triple .˛; a; b/ such that (6.1) ˛ < !1 , (6.2) a  ˛, (6.3) b D P .˛/ \ LŒT; a. The set B is Suslin in L.; R/. Let TB 2 M be a tree such that B D pŒTB : Let

TB D ¹f W !1 ! TB j f 2 M º=

be the ultrapower computed in L.; R/ of TB by the measure on !1 generated by the closed unbounded subsets of !1 . Thus if g  .P .!1 / n INS ; /L.;R/ŒG is L.; R/ŒG-generic then

TB D j.TB /

where j W L.; R/ŒG ! N  L.; R/ŒGŒg is the associated generic elementary embedding. Since in L.; R/ŒG, the set ¹X 2 P!1 ./ j ŒX  2 S º is stationary in P!1 ./, there exists Z  hM ŒG; ; 2i such that (7.1) ¹T; TB ; 0 ; ; º 2 Z, (7.2) ŒZ \  2 S .

662

9 Extensions of L.; R/

Let M0 be the transitive collapse of Z and let N0 be the transitive collapse of ŒZ \ . By (5.2), .!1 /M0 D .!1 /N0 D Z \ !1 : Since TB 2 Z it follows that P .N0 / \ LŒT; N0   M0 : However ŒZ \  2 S and so this implies that Z \ !1 is countable in M0 , a contradiction. t u There are natural weakenings of the principles, WRP.!2 / and SRP.!2 /. We discuss these briefly and state some theorems. Our purpose is to illustrate how possibly subtle variations are stratified, in the context of Pmax -extensions, by the strength of the underlying model of ADC . Suppose that I  P .P!1 .!2 // is an ideal. Recall that the ideal I is normal if for all functions F W !2 ! I; SF 2 I where SF D ¹ 2 P!1 .!2 / j 2 F .˛/ for some ˛ 2 º: The ideal is fine if for each 2 P!1 .!2 /, ¹ 2 P!1 .!2 / j 6  º 2 I: (1) WRP .!2 /: There is a proper normal, fine, ideal I  P .P!1 .!2 // such that for all T 2 P .!1 / n INS , ¹X 2 P!1 .!2 / j X \ !1 2 T º … I and such that if S  P!1 .!2 /

Definition 9.88.

is I -positive then there exists !1 < ˛ < !2 such that S \ P!1 .˛/ is stationary in P!1 .˛/. (2) SRP .!2 /: There is a proper normal, fine, ideal I  P .P!1 .!2 // such that for all T 2 P .!1 / n INS , ¹X 2 P!1 .!2 / j X \ !1 2 T º … I; and such that if S  P!1 .!2 / is a set such that for each T 2 P .!1 / n INS , ¹X 2 S j X \ !1 2 T º … I; then there exists !1 < ˛ < !2 such that S \ P!1 .˛/ contains a closed, unbounded, subset of P!1 .˛/.

t u

9.5 Weak and Strong Reflection Principles

663

Remark 9.89. WRP .!2 / simply asserts that the set of counterexamples to WRP.!2 / t u generates a normal, fine, ideal which is proper on each stationary subset of !1 . One connection between these weakened versions is given in the following lemma. Lemma 9.90. Assume that INS is !2 -saturated and that SRP .!2 / holds. Let I  P .P!1 .!2 // be a normal ideal witnessing that SRP .!2 / holds. Suppose that S1  P!1 .!2 / and S2  P!1 .!2 / are each I -positive. Then there exists !1 < ˛ < !2 such that S1 \ P!1 .˛/ and S2 \ P!1 .˛/ are each stationary in P!1 .˛/. Proof. Let J1  P .!1 / be the set of A  !1 such that ¹X 2 S1 j X \ !1 2 Aº 2 I: It is easily verified that J1 is a normal (uniform) ideal and so since INS is !2 -saturated, there exists A1 2 P .!1 / n INS such that J1 D ¹A  !1 j A \ A1 2 INS º: Similarly there exists A2 2 P .!1 / n INS such that J2 D ¹A  !1 j A \ A2 2 INS º; where J2 is the set of A  !1 such that ¹X 2 S2 j X \ !1 2 Aº 2 I: Choose stationary sets B1  A1 and B2  A2 such that B1 \ B2 D ;. Define S  P!1 .!2 / to be the set of X such that; (1.1) X 2 S1 if X \ !1 2 B1 , (1.2) X 2 S2 if X \ !1 2 B2 . It follows that for each stationary set T  !1 , ¹X 2 S j X \ !1 2 T º … I: 

Thus since I witnesses SRP .!2 / there exists !1 < ˛ < !2 such that S \ P!1 .˛/ is closed, unbounded, in P!1 .˛/. This implies that both S1 \ P!1 .˛/ and S2 \ P!1 .˛/ t u are stationary in P!1 .˛/. The following lemmas show that while WRP .!2 / is a significant weakening of WRP.!2 /, it is plausible that SRP .!2 / is not as significant a weakening of SRP.!2 /.

9 Extensions of L.; R/

664

Lemma 9.91 (2@1 D @2 ). Assume WRP.!2 / and suppose that A  !2 is a set such that H.!2 /  LŒA: Then

LŒA  WRP .!2 /:

Proof. Let I be the normal ideal defined in LŒA, generated by sets S  P!1 .!2 / such that (1.1) S 2 LŒA, (1.2) for all !1 < ˛ < !2 , S \ P!1 .˛/ is not stationary in P!1 .˛/. Since WRP.!2 / holds in V , I is contained in the ideal of nonstationary subsets of P!1 .!2 /. Therefore I is a proper ideal in LŒA and so I witnesses WRP .!2 / in LŒA. t u The proof of Theorem 9.75 easily adapts, using Lemma 9.90 in place of WRP2 .!2 /, to prove the following variation of Theorem 9.75. Lemma 9.92. Assume SRP .!2 / and that INS is !2 -saturated. Then for each A  !2 , t u A# exists. As an immediate corollary we obtain a weak version of Theorem 9.87. Corollary 9.93. Suppose that   P .R/ is a pointclass, closed under continuous preimages, such that L.; R/  ADC C “ ‚ is regular” and such that

L.; R/Pmax  SRP .!2 /:

Then for each A 2 P .R/ \ L.; R/, A# 2 L.; R/.

t u

The situation for WRP .!2 / seems analogous to that for Chang’s Conjecture. Theorem 9.94. Suppose L.R/  AD and that there exists a countable set  R such that HODL.R/ . / \ R D and HODL.R/ . /  AD C DC: Suppose G  Pmax is L.R/-generic. Then L.R/ŒG  WRP .!2 /:

t u

9.5 Weak and Strong Reflection Principles

665

The proof of Theorem 9.82 actually proves the following theorem. Theorem 9.95. Suppose that SRP .!2 / holds and that INS is !2 -saturated. Suppose that G  Coll.!; !2 / is V -generic. Then V ŒG  PD:

t u

Some information about SRP .!2 / is provided by Theorem 9.99. This theorem places an upper bound on the consistency strength of the theory ZFC C SRP .!2 / C “INS is !2 -saturated” which is not far beyond the lower bound established by Theorem 9.95, and significantly below the known upper bounds for SRP.!2 /. Theorem 9.99 involves the following determinacy hypothesis:  (ZFC) Let F be the club filter on P!1 .R/. Then (1) F jL.R; F / is an ultrafilter, (2) L.R; F /  ADC . We note the following corollary of Theorem 9.14. Theorem 9.96. Let F be the club filter on P!1 .R/. Suppose that L.R; F /  AD: C

t u

Then L.R; F /  AD .

The proof of Theorem 9.99 is relatively straightforward using the following theorem. Theorem 9.97. Let F be the club filter on P!1 .R/. Suppose that F jL.R; F / is an ultrafilter and that L.R; F /  AD: Let  D

2 L.R;F / . . 1 /

Then:

(1) Suppose A  R! and that A 2 M . The real game corresponding to A is determined in L.R; F /. (2) hM ; F \ M ; 2i †1 hL.R; F /; F \ L.R; F /; 2i. Remark 9.98. Theorem 9.97 might seem to suggest that the hypothesis:  (ZFC) Let F be the club filter on P!1 .R/. Then (1) F jL.R; F / is an ultrafilter, (2) L.R; F /  ADC ;

t u

666

9 Extensions of L.; R/

is very strong, close in strength to ZF C ADR : However the hypothesis is in fact equiconsistent with ZFC C “There are ! 2 many Woodin cardinals” and ADR is considerably stronger; ADR implies there are inner models in which there are measurable cardinals which are limits of Woodin cardinals, and much more. t u Theorem 9.99. Let F be the club filter on P!1 .R/. Suppose that F jL.R; F / is an ultrafilter and that L.R; F /  AD: Suppose that G  Pmax is an L.R; F /-generic filter. Then L.R; F /ŒG  SRP .!2 /:

t u



We conjecture that Theorem 9.87 holds for SRP .!2 /. This conjecture is not refuted by Theorem 9.99. The explanation lies in the subtle, but important, distinction between models of ADC of the form L.; R/ versus models of the form L.S; ; R/ where S is a set of ordinals and  is a pointclass (closed under continuous preimages). We discuss below an example which illustrates this point. Let F be the club filter on P!1 .R/. Suppose that, as in Theorem 9.99, F jL.R; F / is an ultrafilter and that L.R; F /  AD: Thus L.R; F /  “There is a normal fine measure on P!1 .R/”: The basic theory of ADC applied to L.R; F / shows that L.R; F / D HODL.R;F / .R/: Thus L.R; F / is of the form L.S; ; R/ with  D ;. However the basic theory of ADC also yields the following theorem. Theorem 9.100. Suppose that   P .R/ is a pointclass, closed under continuous preimages, such that L.; R/  ADC and that L.; R/  “There is a normal fine measure on P!1 .R/”: Then L.; R/  ADR : t u Thus to obtain a model of ADC in which there is a normal fine measure on P!1 .R/, the distinction between models of the form L.; R/ and of the form L.S; ; R/ is an important one. We conjecture that the situation is similar for SRP .!2 /. Of course for the other principles (SRP.!2 /, WRP.!2 /, WRP.2/ .!2 /, and WRP .!2 /) the distinction is not important. The reason is simply that these other principles are absolute between V and L.P .!2 //.

9.6 Strong Chang’s Conjecture

9.6

667

Strong Chang’s Conjecture

We briefly discuss the following strengthenings of Chang’s Conjecture. One of these is Strong Chang’s Conjecture which is discussed in .Shelah 1998/. Definition 9.101 (ZF C DC).

(1) Chang’s ConjectureC : Suppose that F W !2
Then there exists G W !2
t u

Remark 9.102. In general we shall only consider Strong Chang’s Conjecture in the situation that L.P .!2 //  !2 -DC: u t Lemma 9.103 (ZFC). The following are equivalent: (1) Strong Chang’s Conjecture. (2) There exists a function F W H.!3 / ! H.!3 / such that if X  H.!3 / is countable and closed under F , then there exists Y  H.!3 / such that

668

9 Extensions of L.; R/

a) F ŒY   Y , b) X  Y , c) X \ !1 D Y \ !1 , d) X \ !2 ¤ Y \ !2 . (3) There exists a function F W H.!3 / ! H.!3 / such that if X  H.!3 / is countable and closed under F , then there exists Y  H.!3 / such that a) X  Y , b) X \ !1 D Y \ !1 , c) X \ !2 ¤ Y \ !2 . (4) There exists a transitive inner model N such that a) P .!2 /  N , b) N  ZF C DC C Strong Chang’s Conjecture. Proof. It is straightforward to show that (1) implies (2), the relevant observation is the following. Suppose that M is a transitive set such that M H.!3 /  M: Then there exist a countable elementary substructure X0  M and a function F0 W H.!3 / ! H.!3 / such that the following holds. Suppose that X  H.!3 / is countable, X0 \ H.!3 /  X; and X is closed under F0 . Then there exists Y M such that X0  Y and Y \ H.!3 / D X . Thus it suffices to prove that (4) implies (3) and that (3) implies (1). We first prove that (3) implies (1), noting that for this implication one only needs !2 -DC. Let M be a transitive set such that M H.!3 /  M and let X  M be a countable elementary substructure. Since H.!3 / is definable in M , there exists F W H.!3 / ! H.!3 /

9.6 Strong Chang’s Conjecture

669

such that F 2 X and such that F witnesses (3). Let Y  H.!3 / be a countable elementary substructure closed under F such that (1.1) X  Y , (1.2) X \ !1 D Y \ !1 , (1.3) X \ !2 ¤ Y \ !2 , and let Z D ¹f .a/ j f W !2 ! M; f 2 X; and a 2 ŒY \ !2  jH.!3 /j; and let M D N . Thus M H.!3 /  M in N . Let F W H.!3 / ! H.!3 / be a function (in V ) such that if X  H.!3 / is a countable set closed under F then there exists X  M such that X  \ H.!3 / D X . We claim that F witnesses (3). Assume toward a contradiction that this fails and let X  H.!3 / be a countable set, closed under F , which witnesses that F fails to satisfy (3). However X 2 N (since H.!3 /  N ) and so by absoluteness and the choice of F , there exists a countable elementary substructure X  M such that X  2 N and X  \ H.!3 / D X . Therefore since N  Strong Chang’s Conjecture; there exists a countable elementary substructure Y M such that

670

9 Extensions of L.; R/

(4.1) X  Y , (4.2) X \ !1 D Y \ !1 , (4.3) X \ !2 ¨ Y \ !2 . Finally Y \ H.!3 / contradicts the choice of X .

t u

The next lemma is an immediate consequence of the definitions. Lemma 9.104 (ZF C !2 -DC). (1) Assume Strong Chang’s Conjecture holds. Then Chang’s ConjectureC holds. (2) Assume Chang’s ConjectureC holds. Then Chang’s Conjecture holds. Proof. (2) is immediate, we prove (1). Fix a function F W !2
XZ0 \ !2 D X \ !2 GŒX
Finally since Strong Chang’s Conjecture holds, the function G is as required.

t u

9.6 Strong Chang’s Conjecture

671

The primary goal of this section is to sketch the construction of a model in which ./ holds and in which Strong Chang’s Conjecture holds. This improvement of Theorem 9.57 will require an even stronger determinacy hypothesis. The formulation involves the sequence h‚˛ W ˛ < i which is discussed at the end of Section 9.1. The proof of Theorem 9.114 requires the following theorems concerning models of ADC . Theorem 9.105. Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC : Let h‚˛ W ˛ < i be the ‚-sequence of L.; R/. Suppose that either (i)  is a limit ordinal, or (ii) if  D ˛ C 1 then ˛ < ı where ı D max¹ < ‚ j is a Suslin cardinal in L.; R/º: Then there is a surjection  W ‚! \ V‚ ! P .R/ \ L.; R/ such that  is †1 -definable in L.; R/ from ¹Rº. Remark 9.106.

t u

(1) We shall only use Theorem 9.105 when L.; R/  AD R C “ ‚ is regular”:

In this situation  D ‚ and so the hypothesis of Theorem 9.105 is satisfied. (2) Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC : We do not know if the conclusion of Theorem 9.105 must hold in general. However the assumption that both (i) and (ii) fail; i. e. that  D ı C 1 where ı is the largest Suslin cardinal in L.; R/, is far stronger than any determinacy hypothesis we shall require in this book. t u The second theorem we shall require generalizes Theorem 9.29. Note that as an immediate corollary one obtains, with notation from the statement of Theorem 9.107, that .HODa /L.;R/ D .HOD/L.;R/ .a/; for each a 2 P!1 .‚! / \ V‚ :

672

9 Extensions of L.; R/

Theorem 9.107. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC : Let ‚ D .‚/L.;R/ and suppose that A  ‚ ! \ V‚ is ordinal definable in L.; R/. Then there exist a formula .x; y/ and a set b 2 P .‚/ \ HOD such that for all a 2 ‚ ! \ V‚ ; a 2 A if and only if

LŒa; b  Œa; b:

t u

Theorem 9.107 easily yields the following corollary which is what we shall require. Corollary 9.108. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC : Let ‚ D .‚/L.;R/ and suppose that a 2 P!1 .‚! / \ V‚ : Suppose that P 2 HODL.;R/ .a/ is a partial order which is countable in V and that X is a comeager set of filters in P such that X is ordinal definable in L.; R/ with parameters from a [ ¹aº. Suppose that g  P is a filter which is HODL.;R/ .a/-generic. Then g 2 X . Proof. Fix  2 Ord such that a 2 V , jV j D  , and such that X is definable in L .; R/ with parameters from a [ ¹aº. Let Y be the set of all finite sequences ha0 ; b0 ; P0 ; 0 ; g0 i such that: (1.1) a0 2 P!1 .‚! / \ V‚ . (1.2) P0 is a partial order. (1.3) P0 2 H.!1 / \ HODL.;R/ .a0 /. (1.4) b0 2 a0
9.6 Strong Chang’s Conjecture

673

Thus Y is ordinal definable and nonempty. Fix a reasonable coding of elements of .P!1 .‚! / \ V‚ /  .‚! \ V‚ /

LŒB; s  ŒB; s:

and a formula  such that X D ¹g j L .; R/  Œa; b; gº:

Now suppose that g  P is a filter which is HOD.a/-generic. Since (3.1) B 2 HOD, (3.2) X is a comeager set of filters in P , it follows by the definability of forcing that there must exist a partial order Q 2 HOD.a/Œg \ H.!1 / such that if h  Q is HOD.a/Œg-generic then there exists s 2 HOD.a/ŒgŒh such that .s/ D ha; b; P ; ; gi: Thus X must contain all HOD.a/-generic filters.

t u

The next theorem which we shall require generalizes Theorem 9.7. Recall that if   P .R/ is a pointclass closed under continuous images, continuous preimages, and complements, then we have associated to  a transitive set M constructed from those sets, X , which are coded by an element of , see Definition 2.18.

674

9 Extensions of L.; R/

Theorem 9.109. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Let h‚˛ W ˛ < i be the ‚-sequence of L.; R/. For each ˛ <  let ı˛ D sup¹ < ‚˛ j is a Suslin cardinalº and let ˛ D ¹A  ! ! j w.A/ < ı˛ º. Then M ˛ †1 L.; R/.

t u

Remark 9.110. By Theorem 9.19, the Suslin cardinals are closed below ‚. Thus the essential content of Theorem 9.109 is in the case that ı˛ < ‚ ˛ : This is the case that ı˛ is the largest Suslin cardinal below ‚˛ . For example if ˛ D 0 ı 21 . u t then ı˛ D  We shall also need the following theorem concerning generic elementary embeddings. For this theorem it is useful to define in the context of DC, a partial embedding, jU , for each countably complete ultrafilter U . Definition 9.111 (DC). Suppose that X ¤ ; and that U  P .X / is a countably complete ultrafilter. Let jU W [¹LŒS  j S  Ordº ! V be defined as follows: Suppose that S  Ord. Then [¹jU .a/ j a 2 LŒS º is the transitive collapse of the ultrapower, ¹f W X ! LŒS  j f 2 V º=U; and jU jLŒS  W LŒS  ! LŒjU .S / is the associated (elementary) embedding.

t u

It is clear from the definition that jU .S / is unambiguously defined. Theorem 9.112. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Suppose that G  Coll.!; R/ is L.; R/-generic. Then there exists a generic elementary embedding jG W L.; R/ ! N  L.; R/ŒG such that: (1) N !  N in L.; R/ŒG; (2) for each set S  Ord with S 2 L.; R/, jG jLŒS  D j jLŒS  where 2 L.; R/ is the measure on P!1 .R/ generated by the closed unt u bounded subsets of P!1 .R/.

9.6 Strong Chang’s Conjecture

675

The last of the theorems which we shall need is in essence a corollary of Theorem 5.34. Theorem 9.113. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Let h‚˛ W ˛ < i be the ‚-sequence of L.; R/. Then for each ˛ < , HODL.;R/  “‚˛C1 is a Woodin cardinal”:

t u

Theorem 9.114(2) specifies conditions on a pointclass  which imply that L.; R/Pmax  Chang’s ConjectureC : We do not know if the hypothesis, L.; R/  ADR C “‚ is regular”; of Theorem 9.39 actually suffices. Nevertheless the requirements of Theorem 9.114(2) are implied by a number of much simpler assertions. For example the assertion, L.; R/  AD R C “ ‚ is Mahlo”; suffices. Theorem 9.114. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Let h‚˛ W ˛ < i be the ‚-sequence of L.; R/. For each ı <  let ı D ¹A  ! ! j w.A/ < ‚ı º and let Nı D HODL.;R/ .ı /: Let W be the set of ı <  such that (i) ı D ‚ı , (ii) Nı  “ı is regular”. Suppose G0  Pmax is L.; R/-generic. (1) Suppose that ı 2 W . Then Nı ŒG0   ZF C !2 -DC C Strong Chang’s Conjecture: (2) Suppose that W is cofinal in . Then L.; R/ŒG0   Chang’s ConjectureC :

676

9 Extensions of L.; R/

Proof. By Lemma 9.104, (2) is an immediate corollary of (1). We prove (1). Fix ‚ D .‚/L.;R/ : Let G1  Coll.!; R/ be L.; R/-generic and let j1 W L.; R/ ! L. 1 ; R1 /  L.; R/ŒG1  be the associated embedding as given by Theorem 9.112. Thus for each set S  Ord with S 2 L.; R/, j1 jLŒS  D j jLŒS  where 2 L.; R/ is the measure on P!1 .R/ generated by the closed unbounded subsets of P!1 .R/. It is convenient to work in L.; R/ŒG1 . Fix G0  Pmax such that G0 is L.; R/-generic and such that G0 2 L.; R/ŒG1 . We begin by observing that a very weak version of Chang’s ConjectureC does hold. Suppose that F W !2
9.6 Strong Chang’s Conjecture

677

such that: (2.1) h 2 L.; R/; (2.2) Suppose 2 P!1 .R/ and hŒ
678

9 Extensions of L.; R/

We let A be the code of  . Let † be the set of A 2 ı such that A D A for some term  . Fix a surjection  W ‚! !  such that  is †1 definable in L.; R/, such a function exists by Theorem 9.105. We now come to the first key point. Suppose that A 2 † and that  2 L.; R/Pmax is a term such that A D A . Suppose in addition that s 2 ı ! is such that both A and R n A have scales which are †11 .B/ where B D .s/. Let D R \ HODL.;R/ Œs: The key claim is that for every filter g  HODL.;R/ Œs \ Pmax ; if g is HODL.;R/ Œs-generic and if p0 2 Pmax is a condition such that p0 < q for each q 2 g, then there exist a condition p 2 Pmax and a countable set Z  !2 such that (4.1) Œ  \ !2 ¨ Z, (4.2) Œ  \ !1 D Z \ !1 , (4.3) p  “ŒZ
p  “ ŒZ
9.6 Strong Chang’s Conjecture

679

This too follows from Corollary 9.108, using Theorem 9.109 and Theorem 9.105. We now fix ı 2 W . We first apply this last claim in L. 1 ; R1 / where j1 W L.; R/ ! L. 1 ; R1 /  L.; R/ŒG1  is the generic elementary embedding associated to G1 . Let HOD1 D j1 .HODL.;R/ / and let

1 D ¹j1 .s/ j s 2 ı ! \ L.; R/º:

Let Y†  Mı be the set of all terms  2 Mı \ L.; R/Pmax such that 1  “ W !2
Nı ŒG0   !2 -DC:

Let M0 D ¹f W !2
T  P!1 .M0 / 2 P!1 .M0 /

such that there exists Z  !2 such that

9 Extensions of L.; R/

680

(7.1) \ !2 ¨ Z, (7.2) \ !1 D Z \ !1 , (7.3) f ŒZ
9.6 Strong Chang’s Conjecture

681

it follows that Nı Œg0   ZFC: Let jS W Nı ŒG0 ŒH0  ! N .S/  Nı ŒG0 ŒH0 ŒGS  be the associated generic elementary embedding. Thus since M0 D [S and since S 2 GS , jS ŒM0  2 jS .S/: From the definition of S it follows that the following must hold in jS ŒNı ŒG0 : (8.1) There exists p0 2 jS .G0 / such that p0 < p for all p 2 G0 and such that for all sets Z 2 jS .P!1 .!2 // if a) jS Œ!2L.;R/  ¨ Z, b) !1L.;R/ D Z \ jS .!1L.;R/ /, then there exist p 2 jS .G0 / and  2 Y† such that a) p  “jS . /ŒZ
682

9 Extensions of L.; R/

(10.1) B \ N .S/ 2 N .S/ , (10.2) hH.!1 /N

.S/

; B \ N .S/ ; 2i  hH.!1 /L.;R/ŒG1  ; B; 2i.

Finally, (6.1) is naturally expressible in the structure hH.!1 /L.;R/ŒG1  ; B; 2i by a formula in the language for this structure involving G0 . This is because G0 2 H.!1 /L.;R/ŒG1  : However (8.1) is expressible in the structure .S/ hH.!1 /N ; B \ N .S/ ; 2i; by the negation of this formula since .S/ G0 2 H.!1 /N : This contradicts (10.2). Thus, in Nı ŒG0 ŒH0 , the set S is not stationary in P!1 .M0 /. This proves that Nı ŒG0 ŒH0   ZFC C Strong Chang’s Conjecture; which is equivalent to (1).

t u

As an immediate corollary of Theorem 9.114, we obtain the following improvement of Theorem 9.39. Theorem 9.115. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: Let h‚˛ W ˛ < i be the ‚-sequence of L.; R/. For each ı <  let ı D ¹A  ! ! j w.A/ < ‚ı º and let Nı D HODL.;R/ .ı /: Let W be the set of ı <  such that (i) ı D ‚ı , (ii) Nı  “ı is regular”, (iii) cof.ı/ > !: Suppose that ı 2 W , G0  Pmax is Nı -generic and that H0  .Coll.!3 ; P .!2 ///Nı ŒG0  is Nı ŒG0 -generic. Then (1) Nı ŒG0 ŒH0   ZFC C Martin’s MaximumCC .c/, (2) Nı ŒG0 ŒH0   Strong Chang’s Conjecture. Proof. By Theorem 9.114, Nı ŒG0   ZF C Strong Chang’s Conjecture: Further P .!2 /Nı ŒG0  D P .!2 /Nı ŒG0 ŒH0  : There by Lemma 9.103, Nı ŒG0 ŒH0   Strong Chang’s Conjecture:

t u

9.7 Ideals on !2

9.7

683

Ideals on !2

Let JNS be the nonstationary ideal on !2 restricted to the ordinals of cofinality !. We shall consider several potential properties of JNS which approximate !3 -saturation. The first is the property of !-presaturation. This notion (for an arbitrary normal ideal) originates in Baumgartner and Taylor .1982/. Definition 9.116. The ideal JNS is !-presaturated if for all S 2 P .!2 / n JNS and for all sequences hDi W i < !i of subsets of P .!2 / n JNS such that for each i < !, Di is predense in .P .!2 / n JNS ; /I there exists a stationary set T  S such that for each i < !, j¹A 2 Di j A \ T … JNS ºj  !2 :

t u

For the definition of the second saturation property for JNS that we shall define it is convenient to define the notion of a canonical function. Definition 9.117. Suppose h W !2 ! !2 . Then h is a canonical function if there exists an ordinal ˛ < !3 and a surjection  W !2 ! ˛ such that !2 n ¹˛ < !2 j f .˛/ D ordertype.Œ˛/º t u

is not stationary in !2 . Definition 9.118. The ideal JNS is weakly presaturated if for every function f W !2 ! !2

and for every set S 2 P .!2 / n JNS , there exists a canonical function h W !2 ! !2 such that ¹˛ 2 S j f .˛/  h.˛/º … JNS : t u Remark 9.119. Suppose that JNS is weakly presaturated and that G  .P .!2 / n JNS ; / is V -generic. Let j W V ! .M; E/  V ŒG be the associated generic elementary embedding. Then j.!2V / D !3V .

t u

It is a theorem of Shelah that JNS is not presaturated. This is a corollary of the following lemma of .Shelah 1986/.

684

9 Extensions of L.; R/

Lemma 9.120 (Shelah). Suppose that is a regular cardinal and that P is a partial order such that V P  cof. / < cof.j j/: Then C is not a cardinal in V P .

t u

Theorem 9.121 (Shelah). JNS is not presaturated. Proof. Let P D .P .!2 / n JNS ; /: Assume toward a contradiction that JNS is presaturated and let G  P be V -generic. Let j W V ! M  V ŒG be the associated generic elementary embedding. Since JNS is presaturated in V , (1.1) j.!2V / D !3V , (1.2) !1V D !1V ŒG , (1.3) N !1  N in V ŒG. By (1.3), !3V D !2V ŒG : In V ŒG, N is the the V -ultrapower of V by a V -normal, V -ultrafilter disjoint from V JNS and so cof.!2V / D ! in N . Therefore ! D .cof.!2V //V ŒG < .cof.j!2V j//V ŒG D !1V ŒG : By Lemma 9.120, !3V is not a cardinal in V ŒG which is a contradiction.

t u

In contrast, by the results of .Foreman, Magidor, and Shelah 1988/ if is supercompact and if G  Coll.!2 ; < / is V -generic then in V ŒG, JNS is !-presaturated. The same theorem is true with only the assumption that is a Woodin cardinal. If GCH holds then JNS is not weakly presaturated. In fact if GCH holds then there is a single function f W !2 ! !2 such that if G  .P .!2 / n JNS ; / is V -generic then !3V < j.f /.!2V /. An even easier argument proves the following lemma which shows that Martin’s Maximum does not imply that JNS is weakly presaturated. Lemma 9.122. Assume that JNS is weakly presaturated. Suppose that N is a transitive inner model containing the ordinals such that (i) N  ZFC, (ii) !2V is inaccessible in N . Let D !2V . Then . C /N < !3 .

9.7 Ideals on !2

685

Proof. Define f W !2 ! !2 by f .˛/ D .j˛jC /N . Let D ..!2V /C /N . For each  < let  W !  be a surjection such that  2 N . Define f W !2 ! !2 by f .˛/ D ordertype. Œ˛/. Thus for each  < there exists a closed unbounded set C  !2 such that f .˛/ < f .˛/ for all ˛ 2 C . Suppose G  .P .!2 / n JNS ; / is V -generic and let j W V ! M  V ŒG be the associated generic elementary embedding. Thus for each  < ,  D j.f /.!2V / < j.f /.!2V /: Therefore

 j.f /.!2V / < j.!2V / D !3V :

t u

We define a natural strengthening of the notion that JNS is weakly presaturated. This we shall define for an arbitrary normal ideal I  P .!2 /, though we shall be primarily interested only in those ideals which extend JNS . This definition requires the obvious generalization of Definition 4.13. Definition 9.123. Suppose that U  .P .!2 //V is a uniform ultrafilter which is setgeneric over V . The ultrafilter U is V -normal if for all functions f W !2V ! !2V with f 2 V , either

¹˛ < !2V j f .˛/ ˛º 2 U

or there exists ˇ < !2V such that ¹˛ < !2V j f .˛/ D ˇº 2 U:

t u

We now generalize the notion of a semi-saturated ideal to ideals on !2 . Definition 9.124. Suppose that I  P .!2 / is a normal uniform ideal. The ideal I is semi-saturated if the following holds. Suppose that U is a V -normal ultrafilter which is set generic over V and such that U  P .!2 / n I: Then Ult.V; U / is wellfounded.

t u

686

9 Extensions of L.; R/

Remark 9.125. In light of Shelah’s theorem that no normal ideal extending JNS can be presaturated, semi-saturation (together with !-presaturation) is perhaps the strongest saturation property that such an ideal can have. It implies, for example, that every t u normal ideal which extends JNS is precipitous and much more. The next theorem, which is essentially an immediate consequence of the definitions, shows, in essence, that semi-saturated ideals on !2V in V correspond to semisaturated ideals on !1V Œg in V Œg where g  Coll.!; !1V / is V -generic. We state the theorem only for the nonstationary ideal, the general version is similar. Theorem 9.126. Suppose that g  Coll.!; !1 / is V -generic. The following are equivalent. (1) V  “The nonstationary ideal on !2 is semi-saturated”. (2) V Œg  “The nonstationary ideal on !1 is semi-saturated”.

t u

Theorem 9.127 and Lemma 9.128 correspond to Lemma 4.27 and Corollary 4.28 respectively. The proofs are similar, we leave the details to the reader. Theorem 9.127. Suppose that I  P .!2 / is a normal uniform ideal such that the ideal I is semi-saturated. Suppose that U is a V -normal ultrafilter which is set generic over V and such that U  P .!2 / n I and let j W V ! M  V ŒU  be the associated embedding. Then j.!2V / D !3V .

t u

The next lemma is an immediate corollary of Theorem 9.127. This lemma shows, for example, that if JNS is semi-saturated then every function f W !2 ! !2 is bounded by a canonical function modulo JNS . Thus ˘! .!2 / implies that JNS is not semi-saturated and so by Shelah’s theorem on ˘! .!2 / .Shelah 2008/, if 2!1 D !2 then JNS is not semi-saturated. Lemma 9.128. Suppose that I  P .!2 / is a normal uniform ideal such that the ideal I is semi-saturated. Suppose that f W !2 ! !2 : Then there exists a canonical function h W !2 ! !2 such that ¹˛ < !2 j h.˛/ < f .˛/º 2 I:

t u

9.7 Ideals on !2

687

Corollary 9.129. Assume 2!1 D !2 . Then JNS is not semi-saturated.

t u

If ./ holds then every (normal) semi-saturated ideal on !2 must properly extend JNS . Therefore we shall only be considering ideals which properly extend JNS but we note that there are several obvious questions concerning the general case of arbitrary normal ideals on !2 , with no restriction on 2!1 . (1) Can JNS be semi-saturated? (2) Is is possible for every function f W !2 ! !2 to be bounded by a canonical function pointwise on a closed unbounded set? (3) Can the nonstationary ideal on !2 be semi-saturated? (4) Let I be the nonstationary ideal on !2 restricted to the ordinals of cofinality !1 . Can the ideal I be semi-saturated? (5) Suppose that there exists a normal uniform ideal I  P .!2 / such that I is semi-saturated and contains JNS . Suppose that J  P .!2 / is a normal uniform semi-saturated ideal. Must JNS  J ‹ Remark 9.130. (1) It is plausible that if there is a huge cardinal, then in a generic extension of V one can arrange that every function f W !2 ! !2 is bounded pointwise on an !1 -club by a canonical function. Granting this, a negative answer to the first question would in effect be an interesting dichotomy theorem. (2) The likely answer to the second question is no. Theorem 9.131 shows that if P .!2 /# exists and if the nonstationary ideal on !2 is semi-saturated, then a generalization of Theorem 3.19(4) to !2 must hold. This seems impossible. (3) Let I be the nonstationary ideal on !2 restricted to the ordinals of cofinality !1 . It is not known whether the ideal I can be !3 -saturated. This is a well known problem. Question (3) is a weaker question, possibly significantly weaker as the t u results concerning JNS show. Theorem 9.131. Assume P .!2 /# exists. Let I be the nonstationary ideal on !2 . Suppose that I is semi-saturated and that C  !2 is closed and unbounded. Then there exists a set A  !1 such that ¹ j !1 <  < !2 and L ŒA is admissibleº  C:

688

9 Extensions of L.; R/

Proof. Fix C  !2 such that C is closed and unbounded in !2 . Suppose that g  Coll.!; !1 / is V -generic. Then by Theorem 9.126, V Œg  “INS is semi-saturated”: Further since P .!2 /# exists in V , V Œg  “P .!1 /# exists”: Thus by Theorem 3.19 and Theorem 4.29, there exists x 2 RV Œg , such that ¹˛ < !1 j L˛ Œx is admissibleº  C: Let A  !1 code a term for x. It follows that ¹ j !1 <  < !2 and L ŒA is admissibleº  C:

t u

The next theorem, which is a corollary of Theorem 9.126, shows that if the nonstationary ideal on !2 is semi-saturated then one formulation of the Effective Continuum Hypothesis must hold. Theorem 9.132. Let I be the nonstationary ideal on !2 . Suppose that I is semisaturated. Suppose that M is a transitive inner model containing the reals such that M  ZF C DC C AD and such that every set X 2 P .R/ \ M is weakly homogeneously Suslin in V . Then ‚M  !2 : Proof. Assume toward a contradiction that ‚M > !2 : Let A 2 P .R/ \ M be such that !2   ı 11 .A/ and let ˛ 2 Ord be least such that L˛ .A; R/  ZF C DC: We note that the existence of ˛ is immediate since (trivially) there must exist a measurable cardinal in V . By the choice of A, !2 < ˛: Fix a partial map  W R ! !2 such that: (1.1)  2 L˛ .A; R/; (1.2) ¹.t / j t 2 dom./º D !2 ; 1 (1.3) ¹.x; y/ j .x/  .y/º 2 † 1 .A/; 1 (1.4) Suppose Z  dom./ is † 1 , then ¹.t / j t 2 Zº is bounded in !2 .

Such a function  exists by Steel’s theorem, Theorem 3.40.

9.7 Ideals on !2

689

By the minimality of ˛, every element of L˛ .A; R/ is definable in L˛ .A; R/ with parameters from ¹Aº [ R. Let B be the set of x 2 R such that x codes a pair ..x0 ; x1 /; t / such that .x0 ; x1 / is a formula, t 2 R and such that L˛ .A; R/  ŒA; t : Thus B naturally codes L˛ .A; R/ and B 2 M . Let TA be a weakly homogeneous tree such that A D pŒTA  and let TB be a weakly homogeneous tree such that B D pŒTB . Let T be a weakly homogeneous tree such that dom./ D pŒT . Suppose that g  Coll.!; !1 / is V -generic. Then by Theorem 9.126, V Œg  “INS is semi-saturated”: Let (in V Œg), Ag D pŒTA  and Bg D pŒTB : Let ˛g be the least ordinal such that L˛g .Ag ; Rg /  ZF; where Rg D .R/V Œg . In V , every set which is projective in B is weakly homogeneously Suslin. Therefore by Lemma 2.28 it follows that in V Œg, Bg codes L˛g .Ag ; Rg / and that the natural map jg W L˛ .A; R/ ! L˛g .Ag ; Rg / is elementary. Let g D jg ./. Thus g W dom.g / ! jg .!2V / is a surjection and dom.g / D pŒT V Œg . Let X D ¹jg .ˇ/ j ˇ < !2V º: Thus in V Œg, jX j D !1 . However in V Œg: (2.1) L˛g .Ag ; Rg /  ZF C DC C AD; (2.2) X is a bounded subset of ‚L˛g .Ag ;Rg / ; (2.3) Every set D 2 P .Rg / \ L˛g .Ag ; Rg / is weakly homogeneously Suslin; (2.4) INS is semi-saturated. Therefore by Theorem 4.32 there exists a set Y 2 L˛g .Ag ; Rg / such that

9 Extensions of L.; R/

690

(3.1) X  Y  jg .!2V /, (3.2) jY j D !1 in L˛g .Ag ; Rg /. Let  D sup.X /: By (3.1) and (3.2),  is singular in L˛g .Ag ; Rg / and so since by the elementarity of jg , jg .!2V / is a regular cardinal in L˛g .Ag ; Rg /, it follows that  < jg .!2V /: Fix t 2 jg .dom.// such that Let  2 V in V ,

Coll.!;!1 /

g .t / D :

be a term for t . We may suppose without loss of generality that

1  “ 2 dom.g / and g ./ D sup¹jg .ˇ/ j ˇ < !2V º”; which implies that in V , 1  “ 2 pŒT ”: We now work in V . Fix 2 Ord such that V  ZFC and such that ¹; TB ; T º 2 V : Let Z0  V be a countable elementary substructure such that ¹; TB ; T º 2 Z0 : For each   !1 let Z D ZŒ D ¹f .s/ j f 2 Z0 and s 2 
1 † 1

Y D pŒT :

and Y D pŒT   pŒT  D dom./:

9.7 Ideals on !2

691

Thus by (1.4) there exists  < !2 such that .z/ <  for all z 2 Y . Therefore there exists t0 2 dom./ such that .z/ < .t0 / for all z 2 [¹Y j  < !1 º. However 1  “ 2 dom.g / and g ./ D sup¹jg .ˇ/ j ˇ < !2V º”; and so 1  “g .t0 / < g ./”: Note that !1  Z!1 and so

1  “ 2 pŒT!1 ”:

This is a contradiction for choose Z0  V such that Z0 is countable and such that ¹Z0 ; t0 º 2 Z0 . Thus hM W   !1 i 2 Z0 : Let M0 be the transitive collapse of Z0 and let T0 be the image of T!1 under the M

collapsing map and let 0 be the image of . Finally suppose that g   Coll.!; !1 0 / is M0 -generic and let tg  be the interpretation of 0 by g  . Thus tg  2 dom./ and by absoluteness, .t0 / < .tg  /: But tg  2 pŒT0  and T0 D T where M0

 D !1

D Z0 \ !1 :

Therefore tg  2 [¹Y j  < !1 º which contradicts the choice of t0 .

t u

There are three closely related results which improve slightly on the results of .Foreman and Magidor 1995/; these are stated as Theorem 9.134, Theorem 9.135 and Theorem 9.136 below. These theorems are straightforward corollaries of Theorem 10.62, Theorem 10.63, and Lemma 10.65. We leave the details to the interested reader. Remark 9.133. (1) The condition (iii) of Theorem 9.134 is trivially implied by, for example, the hypothesis that 2@2 < @! . (2) The condition (ii), that the ideal I be !-presaturated, is certainly easier to achieve than the condition that I be presaturated. If ı is a Woodin cardinal and G  Coll.!2 ; ı and is inaccessible then there exists a countable elementary substructure X  V such that Q 2 X, a) ¹ı; Eº b) hM; EQM i has a iteration scheme which is 1 -homogeneously Suslin, where M is the transitive collapse of X and EQM is the image of EQ under the collapsing map. u t WHIH holds in all of the current inner models in which there is a proper class of Woodin cardinals. The existence of 1 -weakly homogeneously Suslin iteration schemes for a countQ trivializes the question of what can happen in set generic extenable structure hM; Ei sions of M . If M elementarily embeds into a rank initial segment of V then similarly essentially anything can happen in some generic extension of V . Q is a countable structure where EQ is a weakly coherent Theorem 10.5. Suppose hM; Ei Doddage in M witnessing that ı is a Woodin cardinal for some ı 2 M . Suppose there Q which is 1 -weakly homogeneously Suslin. Then is a an iteration scheme for hM; Ei any sentence true in a rank initial segment of V is true in a rank initial segment of a set generic extension of M . t u Theorem 10.6. Suppose there are ! 2 many Woodin cardinals less than  . Suppose Q is a countable structure where EQ is a weakly coherent Doddage in M witnesshM; Ei ing that ı is a Woodin cardinal for some ı 2 M . Suppose there is a an iteration scheme Q which is < -weakly homogeneously Suslin. Then there is a set  R such for hM; Ei that M. / is a symmetric extension of M for set forcing and t u M. /  ADC :

10.1 Forcing notions and large cardinals

697

Since the symmetric extension M. / is a model of ADC , the analysis of both Pmax and Qmax can be carried out in M. /. This yields the following corollary. Theorem 10.7. Suppose there are ! 2 many Woodin cardinals less than  . Suppose Q is a countable structure where EQ is a weakly coherent Doddage in M witnesshM; Ei ing that ı is a Woodin cardinal for some ı 2 M . Suppose there is a an iteration scheme Q which is < -weakly homogeneously Suslin. Then: for hM; Ei (1) There is a set generic extension of M in which the axiom ./ holds. (2) There is a set generic extension of M in which the nonstationary ideal on !1 is t u !1 -dense. As a corollary we obtain, for example: Theorem 10.8 (WHIH). There exists a partial order P such that V P  ./:

t u

We now generalize the notion of an iteration scheme to allow the use of generic elementary embeddings in the construction of the iterations. Suppose M is a countable transitive model of ZFC. Suppose that ı0 ; ı1 are Woodin cardinals in M with ı0 < ı1 . Suppose hE˛ W ˛ < ı1 i is a weakly coherent Doddage of sets of extenders in Mı1 such that the sequence is in M and the sequence witnesses that Q is a function ı1 is a Woodin cardinal in M . A mixed iteration scheme for hM; ı0 ; Ei which assigns to each countable generic iteration hMˇ ; Gˇ ; j˛;ˇ W ˛ < ˇ   i an iteration scheme for .M ; EQ  / where: (1) M0 D M , G0 is M -generic for the stationary tower forcing P˛

This, together with the assertion that L.R/  AD C ; is expressible by a †2 sentence in hH.!2 /; 2i. There must exist a choice of  such that L.R/   and such that this †2 sentence cannot be realized in the structure P

hH.!2 /; 2iL.R/

for any partial order P 2 L.R/. Of course this is a trivial claim if P is .!; 1/distributive in L.R/. The general case, for arbitrary partial orders P 2 L.R/, is more subtle. It is a plausible conjecture that if L.R/  AD

10.2 Coding into L.P .!1 //

703

then there exists a partial order P 2 L.R/ such that L.R/P  ZFC C “R# exists”: It is not difficult to show that if INS is !2 -saturated and if B is any -centered boolean algebra, then INS is !2 -saturated in V B . Thus obtaining models in which INS is !2 -saturated and in which L.P .!1 // is large is straightforward. However if one requires that INS be !1 -dense then the problem appears to be far more subtle. One indication is provided by the theorem of Shelah (see Theorem 3.50); if INS is !1 -dense then 2@0 D 2@1 . Therefore if INS is !1 -dense then necessarily there exists a set A  R which is not !1 -borel. Nevertheless one can probably define a variation of Qmax via which one obtains extensions of, say L.R# /, in which INS is !1 -dense and in which R# is !1 -borel. Finally our particular approach to coding sets into L.P .!1 // is chosen with the .;/ .;;B/ particular kind of applications discussed above in mind, (involving Pmax and Pmax ). .;/ .X/ One can easily define the general version of Pmax obtaining Pmax corresponding to .X/ Q.X/ max . Using Pmax one can show that it is possible to realize all the …2 consequences of ./ for the structure hH.!2 /; Y; 2 W Y  R; Y 2 L.R/i and yet have R 2 L.P .!1 //. However suppose that #

L.R/  AD and that for each …2 -sentence , if Pmax

hH.!2 /; INS iL.R/



then hH.!2 /; 2i  . Then it is easily verified that R# is not !1 -borel.

10.2.1 Coding by sets, SQ We define our basic coding machinery. For this we recall Definition 5.2, which for each set S  !1 defines SQ to be the set of  < !2 such that (1) !1   , (2) if h W !1 !  is a surjection then ¹˛ j ordertype.hŒ˛/ … S º 2 INS : Suppose  < !2 , !1   and that A   . The set A is stationary in  if A\C ¤; for each C   such that C is closed and cofinal in  . Of course if A is stationary in  then  has cofinality !1 . We caution that this notion that A is stationary in  does not coincide with the notion that a is stationary in b as defined in Definition 2.33 unless  D !1 .

704

10 Further results

Suppose hSi W i < !i is a sequence of pairwise disjoint subsets of !1 . For each  < !2 let b D ¹i < ! j SQi \  is stationary in  º: Given X  P .!/, the most natural way to have X definable in H.!2 / would be to have X [ ¹;º D ¹b j  < !2 º for a suitable choice of hSi W i < !i. This we cannot quite arrange. For technical reasons we shall thin the sequence hb W  < !2 i and, in essence, obtain X [ ¹;º D ¹b j  < !2 º for a suitable choice of hSi W i < !i. This thinning will be definable in H.!2 / and more precisely X [ ¹;º D ¹c j  < !2 º where for each  < !2 ,

c D ¹i j 2iC1 2 b º:

Suppose Y  P .!/. An ordinal  is a Y -uniform indiscernible if  is an indiscernible of LŒa for each a 2 Y . We caution that the Y -uniform indiscernibles are not necessarily Y  Y -uniform indiscernibles. However the following lemma is easily proved. Lemma 10.21. Suppose that  R is a nonempty set such that for all a 2 !1 º D min¹ 2 S j  > !1 º where (i) !1 D sup¹.!1 /LŒx j x 2 º, (ii) S is the set of -uniform indiscernibles, (iii) S is the set of  -uniform indiscernibles. Then S n !1 D S .

t u

Definition 10.22. Suppose that hSi W i < !i is a sequence of pairwise disjoint stationary subsets of !1 and suppose that z  !. Let S D hSi W i < !i. We associate to the pair .S; z/ two subsets of P .!/; X(Code) .S; z/ D [¹X j  < ıº and Y(Code) .S; z/ D [¹Y j  < ıº where S(Code) .S; z/ D h.  ; X ; Y / W  < ıi is the maximal sequence generated from .S; z/ as follows.

10.2 Coding into L.P .!1 //

705

(i) Y0 D ¹zº, X0 D ¹;º, and 0 is the least indiscernible of LŒz above !1 . (ii) For all a 2 Y , a# exists and  is the least Y -uniform indiscernible, , such that ˛ <  for all ˛ <  . (iii) Suppose  is not the successor of an ordinal of cofinality !1 . Then X D [¹X˛ j ˛ <  º and Y D [¹Y˛ j ˛ <  º: (iv) Suppose  has cofinality !1 and let b D ¹i < ! j SQi is stationary in  º: iC1 2 bº and let d D ¹i < ! j 3iC1 2 bº. Let c D ¹i < ! j 2 a) YC1 D Y [ ¹d º, b) suppose that  <  where  is the least indiscernible of LŒd  above !1 , then XC1 D X [ ¹cº; otherwise XC1 D X . (v)  < !2 . Suppose M is a countable transitive model of ZFC, S 2 M, and that S D hSi W i < !i is a sequence of pairwise disjoint sets such that for all i < !, Si 2 .P .!1 / n INS /M : Suppose z  ! and z 2 M. Let S(Code) .M; S; z/ D .S(Code) .S; z//M ; let X(Code) .M; S; z/ D .X(Code) .S; z//M ; and let

Y(Code) .M; S; z/ D .Y(Code) .S; z//M :

t u

Remark 10.23. (1) It might seem more natural to define the set b  ! in (iv) using the first ! many uniform Y -uniform indiscernibles. This would allow the use of single stationary set S in the decoding process instead of a sequence hSi W i < !i of stationary sets. In fact such an approach is possible, the details are quite similar. One advantage is that with further modifications the coded set, X, is †1 definable, from parameters, in the structure hH.!2 /; 2i; instead of the expanded structure, hH.!2 /; INS ; 2i. For our applications this feature is at best more difficult to achieve; cf. Theorem 10.55, and there are more elegant ways to achieve this (by simply making the coded set !1 -borel).

706

10 Further results

However one can, by further refinements, arrange in the resulting extension that there exists A  !1 such that if N is any transitive set such that  A 2 N,  .INS /N D INS \ N ,  N  ZFC , then N  “2@0 D @2 ”. This yields a †2 sentence in the language of the structure, hH.!2 /; INS ; 2i; which if true implies c D !2 . (2) The sequence h  W  < ıi can be generated in a variety of ways rather just using the Y -uniform indiscernibles. Similarly the condition (iv)(b) can be modified to further thin the sequence. This in effect we shall do in Section 10.2.4, see Definition 10.72 and and the subsequent Remark 10.73. (3) If for every x 2 R, x # exists, then S(Code) .S; z/ has length !2 . (4) The requirement that X0 D ¹;º, rather than X0 D ;, is just for convenience. u t We extend these notions to sequences of models. Definition 10.24. Suppose hMk W k < !i is a sequence such that for all k < !, (1) for each t 2 R \ Mk , t # 2 MkC1 , (2) Mk is countable, transitive and Mk  ZFC; (3) Mk  MkC1 and

.!1 /Mk D .!1 /MkC1 ;

(4) .INS /MkC1 \ Mk D .INS /MkC2 \ Mk . Suppose S 2 M0 ,

S D hSi W i < !i

and that S is a sequence of pairwise disjoint sets such that for all i < !, Si 2 .P .!1 / n INS /M1 : Suppose z  ! and z 2 M0 . Let S(Code) .hMk W k < !i; S; z/ D h.  ; X ; Y / W  < ıi;

10.2 Coding into L.P .!1 //

707

let X(Code) .hMk W k < !i; S; z/ D [¹X j  < ıº; and let Y(Code) .hMk W k < !i; S; z/ D [¹Y j  < ıº; where h.  ; X ; Y / W  < ıi is the maximal sequence such that for all  < ı, there exists n 2 ! with h. ˛ ; X˛ ; Y˛ / W ˛ <  i D S(Code) .Mk ; S; z/ j  t u

for all k > n.

Suppose that hMk W k < !i is a sequence of countable transitive sets satisfying the conditions in Definition 10.24. We note that since for all k < !, .INS /MkC1 \ Mk D .INS /MkC2 \ Mk M

the following holds. Suppose k < !, S 2 .P .!1 //M0 and that ˛ < !2 k . Then for all i > k, Q MkC1 : .SQ /Mi \ ˛ D .S/ This observation yields the following important corollary which concerns the behavior of X(Code) .hMk W k < !i; S; z/ under iterations. Suppose that j W hMk W k < !i ! hMk W k < !i is an iteration. Then j ŒX(Code) .hMk W k < !i; S; z/  X(Code) .hMk W k < !i; j.S/; z/ and

j ŒY(Code) .hMk W k < !i; S; z/  Y(Code) .hMk W k < !i; j.S/; z/:

In many situations if one defines j.S(Code) .hMk W k < !i; S; z// [¹j.S(Code) .hMk W k < !i; S; z/j / j  < ıº: then one actually obtains j.X(Code) .hMk W k < !i; S; z// D X(Code) .hMk W k < !i; j.S/; z/ and

j.Y(Code) .hMk W k < !i; S; z// D Y(Code) .hMk W k < !i; j.S/; z/:

These claims are easily verified using the properties (1)–(3) of Definition 10.24. It certainly can happen that X(Code) .hMk W k < !i; S; z/ 2 M0 : Thus if

j W hMk W k < !i ! hMk W k < !i

is an iteration it may be that j.X(Code) .hMk W k < !i; S; z// does not coincide with the definition given above. In the cases we shall be interested in, X(Code) .hMk W k < !i; S; z/ 2 M0 and the two possible definitions of j.X(Code) .hMk W k < !i; S; z// coincide, cf. Remark 10.27(5).

708

10 Further results

10.2.2 Q.X/ max Suppose the nonstationary ideal on !1 is !1 -dense and there are infinitely many Woodin cardinals with a measurable cardinal above. The covering theorems show that L.P .!1 // is close to L.R/ below ‚L.R/ . A natural question is whether it must necessarily be the case that L.P .!1 // is a generic extension of L.R/ or whether covering must hold between L.P .!1 // and L.R/. We note that assuming AD L.R/ D L.P .!1 // and so covering trivially holds between these inner models in this case. We focus on the case when the nonstationary ideal is !1 -dense since this case is the most restrictive. It eliminates the possibility that sets appear in L.P .!1 // because they are coded into the structure of the boolean algebra P .!1 /=INS : Suppose that X  P .!/ is a set such that L.X; R/  ADC : .X/ We define a variation, Q.X/ max , of Qmax such that if G  Qmax is L.X; R/-generic then in L.X; R/ŒG, the nonstationary ideal is !1 -dense and X 2 L.P .!1 //. In fact in L.X; R/ŒG, X is a definable subset of H.!2 /. Thus, for example, if X codes R# then in L.X; R/ŒG, R# 2 L.P .!1 //:  Before defining Q.X/ max it is convenient to define a refinement of Qmax .

Definition 10.25. Q max is the set of .hMk W k < !i; f / 2 Qmax for which the following holds. For all k < ! there exists x 2 R \ MkC1 such that for all C  !1M0 if C 2 Mk and if C is closed and unbounded in !1M0 , then D \ .!1M0 n C / is bounded in !1M0 where D  !1M0 is the set of  < !1M0 such that  is an indiscernible of LŒx. t u Suppose f W !1 ! H.!1 / be a function such that for all ˛ < !1 , f .˛/ is a filter in Coll.!; ˛/. Let Sf be the sequence hSi W i < !i where for each i < !, Si D ¹˛ < !1 j ¹.0; i /º 2 f .˛/º:

10.2 Coding into L.P .!1 //

709

Definition 10.26. Suppose that X  P .!/: Q.X/ max

is the set of triples .hMk W k < !i; f; z/

such that the following hold. (1) .hMk W k < !i; f / 2 Q max . (2) For all k < !, Mk  ZFC C CH. (3) z  ! and z # 2 M0 . (4) Let h.  ; X ; Y / W  < ıi D S(Code) .hMk W k < !i; Sf ; z/ be the associated sequence. Then there exist sequences hıi W i < !i and hxi W i < !i such that ı D sup¹ıi j i < !º and for all i < ! a) ı0 < !2M0 and ıiC1 D ıi C !1M0 , b) xi  !, xi 2 Mi and xi# … Mi , c) S(Code) .hMk W k < !i; Sf ; z/jıi D S(Code) .Mi ; Sf ; z/jıi , d) Ord \ Mi <  where  is the least indiscernible of LŒxi  above !1M0 , e) Xıi D Xı0 , f) Y.ıi C1/ D Yı0 [ ¹xj j j  i º. (5) Suppose j W hMk W k < !i ! hMk W k < !i is a countable iteration of .hMk W k < !i; f /. Then j.X(Code) .hMk W k < !i; Sf ; z//  X [ ¹;º: The order is defined as follows: .hNk W k < !i; g; y/ < .hMk W k < !i; f; z/ if .hNk W k < !i; g/ < .hMk W k < !i; f / in

Qmax

and y D z.

t u

710

10 Further results

Remark 10.27. (1) There are natural Q.X/ max variations for each Pmax -variation we .;/ have considered. We shall consider Pmax in Section 10.2.3. We analyze the Q.X/ max -extension first because the analysis is a little more subtle than that of the .;/ Pmax -extension. We also shall use the results of this analysis to simplify presen.;/ tation of Pmax , but this of course is not essential. (2) Suppose .hMk W k < !i; f; z/ 2 Q.X/ max :  Then since .hMk W k < !i; f / 2 Qmax it follows that for all x 2 R \ Mk , x # 2 MkC1 . (3) By (4(d)) and (4(f)), [¹Mk \ Ord j k < !º is the least .[¹P .!/ \ Mk j k < !º/-uniform indiscernible above !1M0 . (4) By (4(f)), Ord \ Mi < .ıi C1/ . Thus S(Code) .M0 ; Sf ; z/ D S(Code) .hMk W k < !i; S; z/jı0 : (5) By (4(g)), X(Code) .hMk W k < !i; S; z/ 2 M0 and further if j W hMk W k < !i ! hMk W k < !i is an iteration, then the two possible interpretations of j.X(Code) .hMk W k < !i; S; z// coincide.

t u

We note the following corollary to Lemma 10.21. Lemma 10.28. Suppose that .hMk W k < !i; f; z/ 2 Q.X/ max : Let h.  ; X ; Y / W  < ıi D S(Code) .hMk W k < !i; Sf ; z/ be the associated sequence and let Y D [¹Y j  < ıº: Let Z D [¹P .!/ \ Mk j k < !º: Let IY be the set of Y -uniform indiscernibles, , such that !1M0   and let IZ be the set of Z-uniform indiscernibles. Then IY D IZ :

t u Q.X/ max .

We now come to the main theorem for the existence of conditions in This theorem is much weaker than the existence theorems we have proved for the other Pmax variations we have analyzed. The reason for this difference lies in the nature of the Q.X/ max conditions. Suppose .hMk W k < !i; f; z/ 2 Q.X/ max :

10.2 Coding into L.P .!1 //

711

Then there must exist x 2 M0 \ R such that x # … M0 . Therefore hH.!1 /M0 ; 2i 6 hH.!1 /; 2i: This rules out the forms of A-iterability which we have used for the analysis of these other Pmax variations. We will of course use A-iterable structures in the analysis of the Q.X/ max -extension, but the actual details of this use will differ slightly when compared to previous instances. One could quite easily develop the analysis of the Pmax -extension along these lines and so these differences are not really fundamental. Theorem 10.29. Suppose that X  P .!/ and that for each t 2 R, t # exists. Suppose x0 2 X and x1 2 R. Then there exists .hMk W k < !i; f; z/ 2 Q.X/ max such that (1) .x0 ; x1 / 2 LŒz, (2) Gf is LŒz-generic where Gf  Coll.!;
1 be the least ¹zk j k < !º-uniform indiscernible above 0 . These are the least two elements of \¹Ck j k < !º. For each k < ! let ık be the least element of CkC1 . Therefore

0 D sup¹ık j k < !º: Construct by induction a sequence hgk ; hk W k < !i of generics such that (1.1) gk  Coll .!; Ck \ ık /, gk is LŒzk -generic, and gk 2 LŒzkC1 , (1.2) hk  Coll .!; ¹ık º/ hk is LŒzk Œgk -generic, and hk 2 LŒzkC1 . Construct by induction a sequence hGk W k < !i of generics such that the following conditions are satisfied. As in the proof of Theorem 6.64 these conditions uniquely specify the generics. For each k < ! let bk D ¹2iC1 j i 2 x0 º [ ¹3iC1 j i 2 zk º:

712

10 Further results

(2.1) Gk  Coll .!; Ck \ 0 / and Gk is LŒzk -generic. (2.2) Gk \ Coll .!; Ck \ ık / D gk . (2.3) Gk \ Coll .!; ¹ık º/ D hk . (2.4) For all ˛ 2 CkC1 \ 0 ,

Gk \ Coll .!; .˛; ˇ// is the LŒzkC1 ŒgŒh-least filter, F , such that a) F is LŒzk ŒgŒh-generic, b) for all  < ˛ and for all i 2 bkC1 , ¹.0; i /º 2 F jColl .!; ¹ º/ $ ¹.0; i /º 2 G0 jColl .!; ¹ º/; c) for all  < ˛ and for all i … bkC1 , ¹.0; !/º 2 F jColl .!; ¹ º/ $ ¹.0; i /º 2 G0 jColl .!; ¹ º/; where g D Gk \ Coll .!; Ck \ ˛/, h D GkC1 \ Coll .!; ¹˛º/, ˇ is the least element of CkC1 above ˛, and for each  < ˛,  is the  th indiscernible of LŒzk  above ˛.

Define

f W !1LŒG0  ! H.!1 /LŒG0 

as follows. Suppose ˛ < !1LŒG0  . Then f .˛/ D ¹p 2 Coll.!; ˛/ j p  2 G0 º where for each p 2 Coll.!; ˛/, p  is the condition in Coll .!; ¹˛º/ such that dom.p  / D dom.p/  ¹˛º and such that p  .k; ˛/ D p.k/ for all k 2 dom.p/. For each k < ! let Mk D L1 ŒzkC1 ŒGkC1  D L1 ŒzkC1 ŒG0 : Thus just as in the proof of Theorem 6.64, .hMk W k < !i; f / 2 Qmax : Let S(Code) .hMk W k < !i; Sf ; z0 / D h.  ; X ; Y / W  < ıi: Thus ı D 0 !. For each ˛ < 0 , X˛ D ;, Y˛ D ¹z0 º and ˛ is the ˛ th indiscernible of LŒz0  above 0 . Further for each i < ! and for each ˛ < 0 , (3.1) Xˇ D ¹x0 º, (3.2) Yˇ D ¹zk j k  i C 1º, (3.3) ˇ is the ˛ th indiscernible of LŒziC1  above 0 , where ˇ D . 0 .i C 1// C ˛. These follow in a straightforward fashion from the definitions of the generics Gk .

10.2 Coding into L.P .!1 //

713

Therefore .hMk W k < !i; f; z0 / 2 Q.X/ max t u

and is as required. Lemma 10.30. Suppose .hNk W k < !i; g; z/ < .hMk W k < !i; f; z/ in

Q.X/ max

and let

j W hMk W k < !i ! hMk W k < !i

be the .unique/ iteration such that j.f / D g. (1) X(Code) .hMk W k < !i; Sf ; z/  X(Code) .hNk W k < !i; Sg ; z/. (2) Suppose that h 2 M0 is a function such that ¹˛ < !1M0 j h.˛/ ¤ f .˛/º 2 .INS /M1 and suppose that x 2 M0 is a subset of ! such that .hMk W k < !i; h; x/ 2 Q.X/ max : Then .hNk W k < !i; j.h/; x/ 2 Q.X/ max and .hNk W k < !i; j.h/; x/ < .hMk W k < !i; h; x/: Proof. By the definition of the order in Q.X/ max , .hNk W k < !i; g/ < .hMk W k < !i; f / in

Qmax .

Therefore, since .j; hMk W k < !i; hMk W k < !i/ 2 N0 ;

for all k < !,

M

N0 N1 \ Mk D INS \ Mk : INS kC1 \ Mk D INS

From this it follows from the definitions that for all i < !, S(Code) .hMk W k < !i; Sg ; z/ is an initial segment of S(Code) .Ni ; Sg ; z/. Therefore S(Code) .hMk W k < !i; Sg ; z/ is an initial segment of S(Code) .hNk W k < !i; Sg ; z/: The first claim of the lemma, (1), follows by the elementarity of j . We prove (2). Note that .hNk W k < !i; j.h/; x/ 2 Q max and .hNk W k < !i; j.h/; x/ < .hMk W k < !i; h; x/ in

Q max .

714

10 Further results

Let h. ˛ ; X˛ ; Y˛ / W ˛ < ıi D S(Code) .Ni ; Sg ; z/: and let ˛0 be such that for all i < !, S(Code) .hMk W k < !i; j.Sf /; z/ D S(Code) .Ni ; j.Sf /; z/j˛0 : Since .hMk W k < !i; h; x/ 2 Q.X/ max it follows that

.hMk W k < !i; j.h/; x/ 2 Q.X/ max :

By an argument similar to that just given, for all i < !, S(Code) .hMk W k < !i; j.Sh /; x/ is an initial segment of S(Code) .Ni ; j.Sh /; x/. Let h. ˛0 ; X˛0 ; Y˛0 / W ˛ < ı 0 i D S(Code) .hNk W k < !i; j.Sh /; x/ and let ˛00 be such that for all i < !, S(Code) .hMk W k < !i; j.Sh /; x/ D S(Code) .Ni ; j.Sh /; x/j˛00 : Let ı0 be such that

Y.ı0 0 C1/ D Yı00 [ ¹t º

for some t 2 M0 such that t # … M0 . This uniquely specifies ı0 as the witness for .hMk W k < !i; j.h/; x/ 2 Q.X/ max to clause (4) in the definition of Q.X/ max . Thus ˛00 D ı0 C !1N0 ! and so ˛00 has cofinality !. Necessarily ı 0 > ˛00 . Let Z D [¹P .!/ \ Mk j k < !º and let

Z  D [¹P .!/ \ Mk j k < !º:

Since .hMk W k < !i; j.f // is an iterate of .hMk W k < !i; f / and since .hMk W k < !i; f / 2 Q max ; it follows that the Z  -uniform indiscernibles above !1N0 coincide with the Z-uniform indiscernibles above !1N0 . Since .hMk W k < !i; j.h/; x/ 2 Q.X/ max it follows that the Y˛0 0 -uniform indiscernibles above !1N0 coincide with the Z  -uniform indiscernibles above !1N0 . Further these coincide with the Y˛0 -uniform indiscernibles above !1N0 .

10.2 Coding into L.P .!1 //

715

Finally j.h/.˛/ D g.˛/ for all ˛ < !1N0 such that ˛ is a Z-uniform indiscernible and such that ˛ > !1M0 . Therefore by induction on  it follows that if ˛0 C  < ı then 0 Y.˛0 C / n Y˛0 D Y.˛ n Y˛0 0 0 C / 0

and

0

0 X.˛0 C / n X˛0 D X.˛ n X˛0 0 : 0 C / 0

0

t u

(2) follows.  Remark 10.31. There is an important difference between Q.X/ max and Qmax . Suppose

.hMk W k < !i; f; z/ 2 Q.X/ max ; and h 2 M0 is a function such that ¹˛ < !1M0 j h.˛/ ¤ f .˛/º 2 .INS /M1 : Then in general .hMk W k < !i; h; x/ … Q.X/ max for any choice of x. This will cause problems in the analysis that follows. This difficulty does not arise in the case of Qmax . t u Lemma 10.32. Suppose that X  P .!/. Suppose that .hNk W k < !i; g; x/ 2 Q.X/ max ; .hMk W k < !i; f; z/ 2 Q.X/ max ; and t  ! codes hMk W k < !i. Suppose t 2 LŒx and G is LŒx-generic where G  Coll.!;
716

10 Further results

Proof. (3) is an immediate consequence of (2) and the definition of the order on Q.X/ max . (2) follows from (4) since .hMk W k < !i; f; z/ 2 N0 : To see this suppose k W hNk W k < !i ! hk.Nk / W k < !i is a countable iteration. Then by elementarity it follows that k.X / D k.X(Code) .hNk W k < !i; Sg ; x// [ k.j.X(Code) .hMk W k < !i; Sf ; z/// where X D X(Code) .hNk W k < !i; j.Sf /; z/: Therefore k.X /  X. We construct the iteration j to satisfy (1) and (4). Fix c0 2 X(Code) .hMk W k < !i; Sf ; z/ and let b0 D ¹2iC1 j i 2 c0 º [ ¹3iC1 j i 2 xº: Define a function g0 W !1N0 ! N0 by perturbing g as follows. Let C be the set of uniform [¹P .!/ \ Mk j k < !ºindiscernibles below !1N0 and above !1M0 . Let D  C be the set of  2 C such that C \  has ordertype  . For each ˛ < !1N0 , g0 .˛/ D g.˛/ unless ˛ D  C ˇ where  2 D, ˇ <  and ˇ is the ˇ th element of C past  . In this case g0 .˛/ D g.˛/ if ¹.0; i /º 2 f .ˇ/ and i 2 b0 , otherwise g0 .˛/ D ¹¹.0; !/º _ p j p 2 g.˛/º: Let j W hMk W k < !i ! hMk W k < !i be the iteration of length .!1 /N0 determined by g0 . Clearly j 2 LŒxŒG. We come to the key claims. Let S(Code) .hMk W k < !i; Sf ; z/ D h  ; X ; Y W  < ıi; let j.S(Code) .hMk W k < !i; Sf ; z// D h  ; X ; Y W  < ı  i; and let S(Code) .hNk W k < !i; j.Sf /; z/ D h  ; X ; Y W  < ı  i:

10.2 Coding into L.P .!1 //

717

The claims are the following: (1.1) For all   ı  ,

.  ; X ; Y / D .  ; X ; Y /:

(1.2) S(Code) .hNk W k < !i; Sg ; x/ is the sequence,       h ı i:  C ; Xı  C n Xı  ; .Yı  C n Yı  / [ ¹xº W ı C  < ı

The first claim is immediate. The point is that [¹j..INS /Mk / j k < !º D .INS /N1 \ .[¹Mk j k < !º/ and that

[¹Mk \ Ord j k < !º

is the least [¹P .!/ \ Mk j k < !º-uniform indiscernible above !1N0 . The latter implies that sup¹  j  < ı  º D [¹Mk \ Ord j k < !º by clause (4(h)) in the definition of Q.X/ max . We prove the second claim. From the first claim   Yı  D [¹Y j  < ı º

and

 ı  D [¹Mk \ Ord j k < !º:

Let Y D [¹Y j  < ıº: Q.X/ max ,

it follows that the Y -uniform indiscernibles are exactly From the definition of the [¹R \ Mk j k < !º-uniform indiscernibles. For each ˇ < !1N0 , let ˇ be the ˇ th Y -uniform indiscernible above !1N0 . The key point is that the Y -uniform indiscernibles above !1N0 coincide with the Yı  -uniform N0 N0 indiscernibles above !1 . Therefore for each ˇ < !1 , if ˇ 0 then ˇ D ı  Cˇ : The iteration giving j was constructed using the function g0 , therefore j.f / and g0 agree on the critical sequence of the iteration. However the critical sequence of the iteration is exactly the set of Y -uniform indiscernibles between !1M0 and !1N0 and this is the set C specified above in the definition of g0 . Thus for each ˛ 2 C , g.˛/ D g0 .˛/ D j.f /.˛/: This proves (1). For each i  ! let

Si D ¹˛ j ¹.0; i /º 2 g.˛/º

and let Ti D ¹˛ j ¹.0; i /º 2 j.f /.˛/º:

718

10 Further results

For each i  !, let SQi be the set computed from Si and let TQi be the set computed from Ti , each computed relative to [¹Nk j k < !º. Thus for each ˇ < !1N0 and for each i < !, if i 2 b0 and if ˇ 2 Ti then ˇ 2 TQi otherwise ˇ 2 TQ! . For each i < !, Ti is stationary in [¹Nk j k < !º and so putting everything together, if  D ı  C !1N0 then  (2.1) YC1 D Yı  [ ¹xº,  D Xı (2.2) XC1  [ ¹c0 º,  (2.3) C1 is the least indiscernible of LŒx above !1N0 ,

(2.4) for each i  !,

SQi \ Z D TQi \ Z where Z is the set of indiscernibles of  such that  is an indiscernible of LŒx and such that  2 [¹Nk j k < !º:

Thus j has the desired properties and this proves the lemma.

t u

The proof of Lemma 10.32 adapts to prove that Q.X/ max is !-closed. Lemma 10.33. Suppose that X  P .!/ and that for all t 2 R, t # exists. Then Q.X/ max is !-closed. Proof. Suppose hpi W i < !i is a strictly decreasing sequence of conditions in Q.X/ max and that for each i < !, pi D .hMki W k < !i; fi ; z/: Let f D [¹fi j i < !º. For each i < ! let ji W hMki W k < !i ! hMO ki W k < !i be the iteration such that ji .fi / D f . This iteration exists since hpi W i < !i is a strictly decreasing sequence in Q.X/ max . We note the following properties of .hMO kk W k < !i; f; z/. (1.1) .hMO kk W k < !i; f / 2 Q max . (1.2) Let Then

Y D Y(Code) .hMO kk W k < !i; f; z/: [¹MO kk \ Ord j k < !º 0

is the least Y -uniform indiscernible above .!1 /M0 . (1.3) Suppose that

j W hMO kk W k < !i ! hj.MO kk / W k < !i is a countable iteration, then X(Code) .hj.MO kk / W k < !i; j.f /; z/  X [ ¹;º:

10.2 Coding into L.P .!1 //

719

By Theorem 10.29 there exists .hNk W k < !i; g; x/ 2 Q.X/ max such that x codes hMO kk W k < !i and such that the filter H  Coll.!;
The difficulty is that,

k

.hMO kk W k < !i; f; z/ … Q.X/ max :

Nevertheless the properties (1.1)–(1.3) suffice to implement the proof of Lemma 10.32. This yields an iteration j W hMO k W k < !i ! hj.MO k / W k < !i k

k

such that (2.1) j 2 N0 , (2.2) for all k < !,

Ok

.INS /j.Mk / D j.MO kk / \ .INS /N1 ; (2.3) .hNk W k < !i; j.f /; z/ 2 Q.X/ max . Thus .hNk W k < !i; j.f /; z/ < pi t u

for all i < !. One corollary of Lemma 10.30 and Lemma 10.32 is that Q.X/ max is homogeneous.

Lemma 10.34. Suppose that X  P .!/ and that for all t 2 R, t # exists. Then Q.X/ max is homogeneous. Proof. This follows by Lemma 10.30(2). Suppose that .hMk W k < !i; f; z/ 2 Q.X/ max and that

.hMk0 W k < !i; f 0 ; z 0 / 2 Q.X/ max :

Let t 2 R code the pair .hMk W k < !i; hMk0 W k < !i/. By Theorem 10.29, there exists .hNk W k < !i; g; x/ 2 Q.X/ max such that t 2 LŒx and such that the filter G  Coll.!;
720

10 Further results

By the iteration lemma, Lemma 10.32, there exist iterations j W hMk W k < !i ! hj.Mk / W k < !i and

j 0 W hMk0 W k < !i ! hj 0 .Mk0 / W k < !i

such that .hNk W k < !i; j.f /; z/ 2 Q.X/ max ; such that

.hNk W k < !i; j 0 .f 0 /; z 0 / 2 Q.X/ max ;

and such that for each  < !1N0 , if  is an indiscernible of LŒt  then j.f /./ D g./ D j 0 .f 0 /./: For each q 2 Q.X/ max let .X/ Q.X/ max jq D ¹p 2 Qmax j p < qº:

By Lemma 10.30(2) the partial orders Q.X/ max j.hNk W k < !i; j.f /; z/; 0 0 0 Q.X/ max j.hNk W k < !i; j .f /; z /;

and Q.X/ max j.hNk W k < !i; g; x/ are isomorphic. Finally .hNk W k < !i; j.f /; z/ < .hMk W k < !i; f; z/ and

.hNk W k < !i; j 0 .f 0 /; z 0 / < .hMk0 W k < !i; f 0 ; z 0 /:

t u

We fix some additional notation. Suppose p 2 Q.X/ max and p D .hMk W k < !i; f; z/: Then P .!/.p/ D [¹Mk \ P .!/ j k < !º: Another corollary of Lemma 10.32 is the following lemma. Lemma 10.35. Suppose that X  P .!/ and that for all t 2 R, t # exists. Suppose that .hMk W k < !i; f; z/ 2 Q.X/ max and that x0 2 X. Then there exists .hNk W k < !i; g; z/ 2 Q.X/ max such that .hNk W k < !i; g; z/ < .hMk W k < !i; f; z/ and such that x0 2 X(Code) .hNk W k < !i; Sg ; z/.

10.2 Coding into L.P .!1 //

721

Proof. Let t 2 R code the pair .hMk W k < !i; x0 /. By Theorem 10.29 there exists a condition .hNk W k < !i; g; x/ 2 Q.X/ max such that (1.1) t 2 LŒz, (1.2) G is LŒz-generic where G  Coll.!;
t u

A similar, though easier, argument establishes: Lemma 10.36. Suppose that X  P .!/ and that for all t 2 R, t # exists. Suppose that .hMk W k < !i; f; z/ 2 Q.X/ max and that y0  !. Then there exists .hNk W k < !i; g; z/ 2 Q.X/ max such that .hNk W k < !i; g; z/ < .hMk W k < !i; f; z/ and such that y0 2 Y(Code) .hNk W k < !i; g; z/.

t u

Lemma 10.37. Suppose that X  P .!/. Suppose that hD˛ W ˛ < !1 i is a sequence of dense subsets of Q.X/ max . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ . Suppose that .M; T; ı/ 2 H.!1 / is such that: (i) M is transitive and M  ZFC. (ii) ı 2 M \ Ord, and ı is strongly inaccessible in M .

722

10 Further results

(iii) T 2 M and T is a tree on !  ı. (iv) Suppose P 2 Mı is a partial order and that g  P is an M -generic filter with g 2 H.!1 /. Then hM Œg \ V!C1 ; pŒT  \ M Œg; 2i  hV!C1 ; Y; 2i: Suppose that hp˛ W ˛ < !1M i is a sequence of conditions in Q.X/ max such that (v) hp˛ W ˛ < !1M i 2 M , (vi) for all ˛ < !1M , p˛ 2 D˛ , (vii) for all ˛ < ˇ < !1M , pˇ < p˛ : Suppose g 

Coll.!; !1M /

is M -generic and let Z D [¹Z˛ j ˛ < !1M º where for each ˛ < !1M , Z˛ D P .!/.p˛ / . Suppose  is a Z-uniform indiscernible, !1M <  < !2M and that hAi W i < !i 2 M Œg is a sequence of subsets of !1M . Then for each q 2 Coll.!; / there exists a condition .hMk W k < !i; h; z/ 2 D! M 1

such that the following hold. (1) .hMk W k < !i; h; z/ 2 M Œg; (2)  < !1M0 ; (3) q 2 h./; (4) For each i < !, for each ˛ < !1M , and for each k < !, ˛ 2 Ai if and only if ˛ 2 .SQi /Mk where

Si D ¹ˇ < !1M0 j ¹.0; i /º 2 h.ˇ/º

and where for each ˛ < !1M , ˛ is the ˛ th Z-uniform indiscernible above !1M0 ; (5) h.!1M / D g; (6) For all ˛ < !1M , (7) hp˛ W ˛ < !1M i 2 M0 .

.hMk W k < !i; h; z/ < p˛ I

10.2 Coding into L.P .!1 //

723

Proof. We work in M Œg. For each ˛ < !1M let .hMk˛ W k < !i; f˛ ; z/ D p˛ and let

j˛ W hMk˛ W k < !i ! hMQ k˛ W k < !i

the iteration of hMk˛ W k < !i determined by f D [¹f˛ j ˛ < !1M º: Let h˛k W k < !i be a strictly increasing sequence which is cofinal in !1M such that h˛k W k < !i 2 M Œg: For each k < ! let Thus Let

Nk D MQ 0 k : ˛

.hNk W k < !i; f / 2 Q max : j W hNk W k < !i ! hNOk W k < !i

be a countable iteration, of limit length, such that j.!1N0 / >  and such that q 2 j.f /./: The iteration exists since the critical sequence of any iteration of hNk W k < !i is an initial segment of the Z-uniform indiscernibles above !1N0 . Let hi W i < !i be O

an increasing sequence of elements of Z n , cofinal in !1N0 . For each i < ! let ˛ ˛ ki W hM i W k < !i ! hMO i W k < !i k

k

be the (unique) iteration such that ki .f˛i / D j.f /ji , and let ˛ pO˛i D .hMO k i W k < !i; fO˛i ; z/

be the corresponding condition in Q.X/ max . Note that for all ˛ < ˛i < !1M , pO˛i < p˛ : Now choose a condition p 2

Q.X/ max

\ M Œg such that for all i < !, p < pO˛i

and such that hAk W k < !i 2

M0.p/

where we let

.hMk.p/ W k < !i; f.p/ ; z/ D p: Let

˛ Z  D [¹P .!/ \ hMO 0 i j i < !º:

724

10 Further results

Let IZ be the class of Z-uniform indiscernibles and let IZ  be the class of Z  -uniform indiscernibles. Thus N IZ  D IZ n !1 0 : Further Z  2 M0.p/ and Z  is countable in M0.p/ . The key point is the following. Let M

.p/

h˛ W ˛ < !1 0 i be the increasing enumeration of IZ  and let I D ¹˛Cˇ j ˛ D ˛ and 0 < ˇ < !1M º: Suppose that fO 2 M.p/ is a function such that 0

.p/

M (1.1) dom.fO/ D !1 0 , .p/

M0

(1.2) for all ˛ < !1

,

fO.˛/  Coll.!; ˛/

and fO.˛/ is a filter, .p/

M0

(1.3) for all ˛ 2 !1 Then

n I,

f.p/ .˛/ D fO.˛/:

.hMk.p/ W k < !i; fO; z/ 2 Q.X/ max

and for each i < !,

.hMk.p/ W k < !i; fO; z/ < pO˛i :

Since hAk W k < !i 2 M0.p/ , we can choose fO so that requirement (4) of the lemma is satisfied by the condition .hMk.p/ W k < !i; fO; z/ by modifying fOjI if necessary. But this implies that requirement (4) is satisfied by any condition q 2 Q.X/ max such that .p/ O q < .hM W k < !i; f ; z/: k

Let pO D .hMk.p/ W k < !i; fO; z/. Finally by (vi), hM Œg \ V!C1 ; pŒT  \ M Œg; 2i  hV!C1 ; Y; 2i: and so M Œg \ D! M 1

is dense in

Q.X/ max

\ M Œg. Let .hMk W k < !i; h; z/ 2 D! M \ M Œg 1

be a condition such that O .hMk W k < !i; h; z/ < p: The condition .hMk W k < !i; h; z/ is as required.

t u

10.2 Coding into L.P .!1 //

725

Lemma 10.38. Suppose that X  P .!/. Suppose that hD˛ W ˛ < !1 i is a sequence of dense subsets of Q.X/ max . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ and suppose that .M; I/ 2 H.!1 / is such that .M; I/ is strongly Y -iterable. Let ı 2 M be the Woodin cardinal associated to I. Suppose t  !, t codes M and .hNk W k < !i; f; z/ 2 Q.X/ max ; is a condition such that t 2 LŒz. Let 2 Mı be a normal .uniform/ measure and let .M  ;  / be the !1N0 -th iterate of .M; /. Suppose that Gf is LŒz-generic where Gf  Coll.!;
j W hNk W k < !i ! hNk W k < !i

is an iteration and let 

hp˛ W ˛ < .!1 /N0 i D j.hp˛ W ˛ < !1N0 i/: 

Then for all ˛ < .!1 /N0 , p˛ 2 D˛ . Proof. We fix some notation. Suppose that F  Coll.!;
F W !1N0 ! .!1N0 ; /

be the LŒt -least function such that F is onto. Let D  C be the set of  2 C such that  is the ordertype of C \  . Let E be the set of  < !1N0 such that for some ˇ,

728

10 Further results

(2.1) Lˇ Œt   ZFC, (2.2)  is an inaccessible cardinal in .!1 /Lˇ Œt . Since t codes .M; /, E  D. Further E 2 LŒt  and in LŒt , E contains a subset which is a club in !1N0 . For each  2 E let ˇ be the least ordinal, ˇ, satisfying (2.1)–(2.2), and let F be the function F as computed in Lˇ Œt . The point of all of this is reflection. Let  > be an ordinal such that L Œt   ZFC and suppose X  L Œt  is a countable elementary substructure containing t and F . Let  D X \ !1N0 . Then  2 E and F is the image of F under the collapsing map. For each ˇ < ! N0 let Tˇ D ¹ 2 E j ¹.0; ˇ/º 2 f . /º: Thus

hTˇ j ˇ < !1N0 i 2 LŒzŒGf 

and for each ˇ < !1N0 , Tˇ … .INS /N1 . We modify G0 to obtain G as follows. For each ˛ < !1N0 , GjColl.!; ˛/ D G0 jColl.!; ˛/; unless ˛ D . C /

M

for some  2 E. In this case GjColl.!; ˛/ D ¹p _ q j q 2 G0 jColl.!; ˛/º:

where p D ¹.0; F .ˇ//º and  2 Tˇ . For each  2 C such that C \  is bounded in  , GjColl.!; ˛/ D G0 jColl.!; ˛/ for all but at most one ˛ in the interval Œ  ;   where   is the largest element of C \ . Further for this one possible exception, GjColl.!; ˛/ D ¹p _ q j q 2 G0 jColl.!; ˛/º for some condition p 2 Coll.!; ˛/. Finally G0 j0 D Gj0 where 0 is the least element of C . Thus by induction on  2 C it follows that for all  2 C , Gj is M  -generic for Coll.!; < /. Therefore G is M  -generic for Coll.!;
10.2 Coding into L.P .!1 //

Let

729

hp˛ W ˛ < !1N0 i

be the interpretation of   by G. For each ˛ < !1N0 let .hMk˛ W k < !i; f˛ ; z˛ / D p˛ : For all ˛ < ˇ, pˇ < p˛ . Therefore for all ˛ < ˇ, z˛ D zˇ and f˛  fˇ : Let x D z0 and let h D [¹f˛ j ˛ < !1N0 º. We finish by proving (3.1) .hNk W k < !i; h; x/ 2 Q.X/ max , (3.2) for all ˛ < !1N0 , p˛ 2 D˛ and .hNk W k < !i; h; x/ < p˛ : (3.2) is an immediate consequence of (3.1) and the definitions. We prove (3.1). Let Z D [¹P .!/.p˛ / j ˛ < !1N0 º: Thus by (1.3)(b), Z D P .!/ \ M ŒG: For each k < ! let Ak D ¹˛ <  j ¹.0; k/º 2 GjColl.!; ˛ C 1/º and let Sk D ¹˛ <  j ¹.0; k/º 2 GjColl.!; ˛/º: Fix k < ! and as above let be the successor cardinal of !1N0 as computed in M  . From the definition of G0 , it follows that for each ˛ 2 C , ˛ 2 .SQk /N0 if and only if ˛ 2 Bk where ˛ is the ˛ th Z-uniform indiscernible above . By the modification of G0 to produce G, .SQk /N0 \ .!1N0 ; / D ;: For each ˛ < !1N0 let j˛ W hMk˛ W k < !i ! hMO k˛ W k < !i be the iteration such that j.f˛ / D h. For each ˛ < ˇ < !1N0 , pˇ < p˛ and so S(Code) .hMO k˛ W k < !i; Sh ; x/  S(Code) .hMO kˇ W k < !i; Sh ; x/:

730

10 Further results

For each ˛ < !1N0 , let h ˇ ; Xˇ ; Yˇ W ˇ < ı˛ i D S(Code) .hMO k˛ W k < !i; Sh ; x/ and let

ı D sup¹ı˛ j ˛ < !1N0 º D Ord \ .[¹MO 0˛ j ˛ < !1N0 º/:

Since h D [¹f˛ j ˛ < !1N0 º and since for all  2 C , f . / D h. /; it follows that for each ˛ 10.2 Coding into L.P .!1 //

735

where Zi is the closure of ¹ai º under F and where ji W hMki W k < !i ! hji .Mki / W k < !i is the iteration such that ji .fi / D fG . For each i < ! let Xi D ji Œai  D ¹ji .b/ j b 2 ai º: Thus for each i < !, Xi  X and further X D [¹Xi j i < !º: We note that for each i < !, since ji .ai / D Ni , ji .A \ ai / D A \ Ni : For each i < ! and let Di be the set of .hMk W k < !i; f; z/ < .hMki W k < !i; fi ; z/ such that

j  .A \ ai / D A \ j  .ai /

and such that for all countable iterations j W hMk W k < !i ! hj.Mk / W k < !i; it is the case that j.A \ j  .ai // D A \ j.j  .ai //, where j  W hMki W k < !i ! hj  .Mki / W k < !i is the iteration such that j  .fi / D f . We claim that for some q 2 G, ¹p < q j p 2 Q.X/ max º  Di : Assume toward a contradiction that this fails. Then for all q 2 G there exists p 2 G such that p < q and p … Di . However G is L.; R/-generic and so there must exist .hMk W k < !i; f; z/ 2 G and an iteration j W hMk W k < !i ! hj.Mk / W k < !i such that (3.1) .hMk W k < !i; f; z/ < .hMki W k < !i; fi ; z/, (3.2) j.A \ j  .ai // ¤ A \ j.j  .ai // where j  W hMki W k < !i ! hj  .Mki / W k < !i is the iteration such that j  .fi / D f , (3.3) .hj.Mk / W k < !i; j.f /; z/ 2 G.

736

10 Further results

But this contradicts the fact that ji .A \ ai / D A \ Ni . Therefore for some q 2 G, ¹p < q j p 2 Q.X/ max º  Di : Note that Di is definable in the structure hH.!2 /; A; G; 2i from ai . Therefore Di \ Zi ¤ ; and so .hMkiC1 W k < !i; fiC1 ; z/ 2 Di : For each i < n < !, let ji;n W hMki W k < !i ! hji;n .Mki / W k < !i be the iteration such that ji;n .fi / D fn and let ji;! W hMki W k < !i ! hji;! .Mki / W k < !i be the iteration such that ji;! .fi / D [¹fn j n < !º: Thus for all i < !, hji;! .Mki / W k < !i 2 jiC1;! .M0iC1 /: The key points are that MX D [¹ji;! .Mki / j i; k < !º D [¹ji;! .ai / j i < !º: and that for each i < !, ji;! .ai / D NiX where NiX is the image of Ni under the collapsing map. These identities are easily verified from the definitions. Finally suppose jO W MX ! MO X is a countable iteration. For each i < !,

jO.hjiC1;! .MkiC1 / W k < !i/

is an iterate of hMkiC1 W k < !i. Further for each i < !, hMkiC1 W k < !i 2 Di : Therefore for each i < !, jO.A \ NiX / D jO.jiC1;! .A \ ji;iC1 .ai /// D A \ jO.jiC1;! .ji;iC1 .ai /// D A \ jO.NiX /:

10.2 Coding into L.P .!1 //

737

X However for each i < !, NiX is transitive and NiX 2 NiC1 . Further

MX D [¹NiX j i < !º: Therefore and so

MO X D [¹jO.NiX / j i < !º jO.A \ MX / D A \ MO X : t u

Therefore MX is A-iterable.

The motivation for considering Q.X/ max was to investigate whether the assumption that the nonstationary ideal on !1 is !1 -dense implies that the inner model L.P .!1 // is close to the inner model L.R/ as the covering theorems might suggest. The next theorem shows that this is not the case. Note that (5) in the statement of the theorem is marginally stronger than the conclusions of the covering theorems. Theorem 10.42. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC C “ ‚ is regular”: Suppose that X  P .!/ is a set such that X 2 L.; R/: Q.X/ max

is !-closed and homogeneous. Further, suppose G  Q.X/ Then max is L.; R/generic. Then for each A 2 , L.A; X; R/ŒG  ZFC and in L.A; X; R/ŒG the following hold. (1) The nonstationary ideal on !1 is !1 -dense. (2) L.P .!1 // D L.X; R/ŒG. (3) X is a definable (as a predicate) in the structure hH.!2 /; 2i from fG . (4) ı12 D !2 . (5) Suppose S  !1 is stationary and f W S ! Ord: Then there exists g 2 L.A; X; R/ such that ¹˛ 2 S j f .˛/ D g.˛/º is stationary.

738

10 Further results

Proof. (1)–(4) follow from Theorem 10.40. (5) follows from (1), Lemma 10.41 and from Theorem 3.42 using the chain condition of Q.X/ max to reduce to the case that f WS !ı where ı < ‚L.A;X;R/ , cf. the proof of Lemma 6.79.

t u

Perhaps our covering theorems do not capture all the covering consequences of the assumption that the nonstationary ideal is !1 -dense, particularly if in addition large cardinals are assumed to exist. Theorem 10.44 is the version of Theorem 10.42 which addresses this question. Theorem 10.43. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADR : Then for each set A 2  there exists an inner model L.S; R/ such that (1) S  Ord and A 2 L.S; R/, (2) HODSL.S;R/  “There is a proper class of Woodin cardinals”.

t u

Theorem 10.44. Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  DC C ADR : Suppose that X  P .!/ is a set such that X 2 L.; R/: Suppose that G  Q.X/ max is L.; R/-generic. Then there is an inner model N  L.; R/ŒG containing the ordinals, R and G such that: (1) N  ZFC C “There is a proper class of Woodin cardinals”; (2) N  “INS is !1 -dense”; (3) X 2 N and X is 1 -weakly homogeneously Suslin in N ; (4) N  X 2 L.P .!1 //.

t u

10.2 Coding into L.P .!1 //

739

.;/ 10.2.3 Pmax .;/ We define and briefly analyze Pmax which is the version of Q.X/ max which corresponds to .;/ Pmax but with X D ;. Our interest in Pmax lies in Theorem 10.70. This theorem shows that Martin’s Maximum CC .c/ C Strong Chang’s Conjecture

together with all the …2 consequences of ./ for the structure hH.!2 /; Y; 2 W Y  R; Y 2 L.R/i does not imply ./. One corollary is that for the characterization of ./ using the “converse” of the absoluteness theorem (Theorem 4.76), it is essential that the predicate INS be added to the structure. For this application we need only consider the case when X D ;; i. e. we are in effect just defining the version of Q.;/ max which corresponds to Pmax . However all the results, including the absoluteness theorem (Theorem 10.55), .X/ in the obvious fashion. We have chosen to concentrate on the special generalize to Pmax .;/ case of Pmax because this case suffices for our primary applications (and the notation is slightly simpler). Strong Chang’s Conjecture is discussed in Section 9.6. .;/ extension of L.R/ are The iteration lemmas necessary for the analysis of the Pmax .;/ actually simpler to prove than those for Qmax . Further the iteration lemmas necessary for the analysis of the Q.;/ max extension of L.R/ are in turn (slightly) simpler than those required for the analysis of the Q.X/ max extensions for general X. We shall use the notation from Section 10.2.1. Suppose that A  !1 . Let SA denote the sequence hSi W i < !i where for each i < !, Si is the set of ˛ < !1 such that (1) ˛ is a limit ordinal, (2) ˛ C i C 1 2 A, (3) i D min¹j < ! j ˛ C j C 1 2 Aº. Thus SA is a sequence of pairwise disjoint subsets of !1 . .;/ Definition 10.45. Pmax is the set of triples

hM; a; zi such that the following hold. (1) M is a countable transitive set such that M  ZFC C ZC: (2) M is iterable. (3) M 

AC .

740

10 Further results

(4) Let hSi W i < !i D .Sa /M . For each i < !, Si … .INS /M . (5) Suppose that C  !1M is closed and unbounded with C 2 M. Then there exists a closed cofinal set D  C such that D 2 LŒx for some x 2 R \ M. (6) X(Code) .M; Sa ; z/ D ¹;º. (7) Y(Code) .M; Sa ; z/ D P .!/ \ M. The order is defined as follows: hM1 ; a1 ; z1 i < hM0 ; a0 ; z0 i if z1 D z0 and there exists an iteration j W M0 ! M0 such that (1) j.a0 / D a1 , 

(2) .INS /M0 D .INS /M1 \ M0 .

t u .;/

.;/ The nontriviality of Pmax is an immediate corollary of the analysis of L.; R/Qmax where   P .R/

is a pointclass closed under continuous preimages such that L.; R/  ADC C “‚ is regular”: Remark 10.46. The use of the analysis of the Q.;/ max -extension (Theorem 10.42) to .;/ .;/ is for expediency. If one defines Pmax using sequences of obtain conditions in Pmax .;/ models (as in the definition of Qmax ) then it is much easier to produce conditions. The conditions can be constructed directly without reference to Q.;/ t u max . Theorem 10.47. Suppose that A  R and that L.A; R/  ADC : Then there exists .;/ hM; a; zi 2 Pmax

such that (1) A \ M 2 M and hM \ V!C1 ; A \ M; 2i  hV!C1 ; A; 2i, (2) M is A-iterable, (3) X(Code) .M; a; z/ D ¹;º.

10.2 Coding into L.P .!1 //

741

Proof. Let G  Q.X/ max be L.A; R/-generic where X D ;. Fix 0 to be least such that L 0 .A; R/ŒG  ZFC C ZC; and let B 2 P .R/ \ L.A; R/ be such that  ı 11 .B/ > 0 . By Theorem 10.40 and Lemma 10.41, L.A; R/ŒG  ZFC and further the following hold in L.A; R/ŒG. (1.1) X(Code) .SfG ; zG / D ¹;º. (1.2) Y(Code) .SfG ; zG / D P .!/. (1.3)

AC

holds.

(1.4) The set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . (1.5) The set ¹X  hH.!2 /; B; 2i j MX is B-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Let SfG D hSi W i < !i and let AG D ¹˛ C i C 1 j ˛ is a limit ordinal and ˛ 2 Si º: Thus SAG D hSi \ C W i < !i where C is the set of countable limit ordinals and so by (1.1), in L.A; R/ŒG, X(Code) .SAG ; zG / D ¹;º: By (1.4) and Lemma 4.24, the set of ¹Y  L 0 .A; R/ŒG j Y is countable and MY is strongly iterableº contains a club in P!1 .M /. Here MY is the transitive collapse of Y . Thus, by (1.5), there exists a countable elementary substructure, X  L 0 .A; R/ŒG; such that .;/ hM; a; zi 2 Pmax

and satisfies the requirements of the lemma, where (2.1) z D zG , (2.2) M is the transitive collapse of X , (2.3) a D AG \ X \ !1 D AG \ .!1 /M .

t u

742

10 Further results

.;/ It is convenient to organize the analysis of Pmax following closely that of Q.X/ max . The reason is simply that most of the proofs adapt easily to the new context. The next four lemmas summarize the basic iteration facts that one needs. These lemmas are direct analogs of the lemmas we proved as part of the analysis of Q.X/ max . We leave the details to the reader. .;/ , such as the !-closure The first two easily yield elementary consequences for Pmax .;/ and homogeneity of Pmax , the latter two allow one to complete the basic analysis.

Lemma 10.48. Suppose hM1 ; a1 ; z1 i < hM0 ; a0 ; z0 i in

.;/ Pmax

and let

j W M0 ! M0

be the .unique/ iteration such that j.a0 / D a1 . (1) X(Code) .M0 ; Sa0 ; z0 /  X(Code) .M1 ; Sa1 ; z1 /. (2) Suppose that b0 2 M0 is such that for each i < !, Sia0 M Sib0 2 .INS /M0 ; where hSia0 W i < !i D .Sa0 /M0 and hSib0 W i < !i D .Sb0 /M0 . Suppose that x0 2 M0 is a subset of ! such that .;/ : hM0 ; b0 ; x0 i 2 Pmax .;/ Then hM1 ; j.b0 /; x0 i 2 Pmax and hM1 ; j.b0 /; x0 i < hM0 ; b0 ; x0 i.

t u

.;/ As we have indicated, the iteration lemmas required for the analysis of Pmax are .;/ .;/ routine generalizations of those for Qmax . The situation for Pmax is actually quite a bit .;/ conditions are simpler and there is more freedom in less complicated since the Pmax constructing iterations.

Lemma 10.49. Suppose that .;/ ; hM1 ; a1 ; z1 i 2 Pmax .;/ hM0 ; a0 ; z0 i 2 Pmax ;

t  ! codes M0 , and that t 2 LŒz1 . Let hSia0 W i < !i D .Sa0 /M0 ; hSia1 W i < !i D .Sa1 /M1 ; and let C be the set of  < !1M1 such that  is an indiscernible of LŒt . Then there exists an iteration j W M0 ! M0 such that j 2 M1 and such that: (1) for each i < !, C \ j.Sia0 / D C \ Sia1 ; .;/ (2) hM1 ; j.a0 /; z0 i 2 Pmax ;

(3) hM1 ; j.a0 /; z0 i < hM0 ; a0 ; z0 i.

t u

10.2 Coding into L.P .!1 //

743

Lemma 10.50. Suppose that hD˛ W ˛ < !1 i .;/ is a sequence of dense subsets of Pmax . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ . Suppose that .M; T; ı/ 2 H.!1 / is such that:

(i) M is transitive and M  ZFC. (ii) ı 2 M \ Ord, and ı is strongly inaccessible in M . (iii) T 2 M and T is a tree on !  ı. (iv) Suppose P 2 Mı is a partial order and that g  P is an M -generic filter with g 2 H.!1 /. Then hM Œg \ V!C1 ; pŒT  \ M Œg; 2i  hV!C1 ; Y; 2i: .;/ Suppose that hp˛ W ˛ < !1M i is a sequence of conditions in Pmax such that

(v) hp˛ W ˛ < !1M i 2 M , (vi) for all ˛ < !1M , p˛ 2 D˛ , (vii) for all ˛ < ˇ < !1M , pˇ < p˛ : Suppose g  Coll.!; !1M / is M -generic and let Z D [¹Z˛ j ˛ < !1M º where for each ˛ < !1M , Z˛ D P .!/ \ M˛ and hM˛ ; a˛ ; z0 i D p˛ . Suppose  is a Z-uniform indiscernible, !1M <  < !2M and that hAi W i < !i 2 M Œg is a sequence of subsets of !1M . Then for each m < !, there exists a condition hN ; a; zi 2 D! M 1

such that the following hold where hSi W i < !i D .Sa /N ; and where for each ˛ < !1M , ˛ is the ˛ th Z-uniform indiscernible above !1N . (1) hN ; a; zi 2 M Œg. (2)  < !1N and  2 Sm . (3) For each i < ! and for each ˛ < !1M , ˛ 2 Ai if and only if ˛ 2 .SQi /N . (4) For all ˛ < !1M , hN ; a; zi < p˛ . (5) hp˛ W ˛ < !1M i 2 N .

t u

744

10 Further results

Lemma 10.51. Suppose that hD˛ W ˛ < !1 i .;/ is a sequence of dense subsets of Pmax . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ and suppose that

.M; I/ 2 H.!1 / is such that .M; I/ is strongly Y -iterable. Let ı 2 M be the Woodin cardinal associated to I. Suppose t  !, t codes M and .;/ hN ; a; zi 2 Pmax ;

is a condition such that t 2 LŒz. Let 2 Mı be a normal .uniform/ measure and let .M  ;  / be the !1N -th iterate of .M; /. Then there exists a sequence hp˛ W ˛ < !1N i 2 N and there exists .b; x/ 2 N such that .;/ (1) hN ; b; xi 2 Pmax ,

(2) for all ˛ < !1N , p˛ 2 D˛ and hN ; b; xi < p˛ ; (3) there exists an M  -generic filter g  Coll.!;
t u

.;/ is homogeneous. Lemma 10.53. Suppose that L.R/  ADC . Then Pmax

t u

We adopt the usual notational conventions. A filter .;/ G  Pmax

is semi-generic if for all ˛ < !1 there exists a condition hM; a; zi 2 G such that ˛
!:

Again by the chain condition of P , it suffices to prove that .P!1 .˛//V is cofinal in

P

.P!1 .˛//V ; where ˛ D !2V . But this is immediate.

t u

Theorem 9.134, Theorem 9.135, and Theorem 9.136 (these are the theorems of Section 9.7 concerning ideals on !2 ) are immediate corollaries of the boundedness theorems, Theorem 10.62 and Theorem 10.63, together with the next lemma. Lemma 10.65. Suppose that I  P .!2 / is a normal uniform ideal such that ¹˛ j cof.˛/ D !º 2 I: Let P D hP .!2 / n I; i. Suppose that either (1) I is !3 -saturated, or (2) I is !-presaturated and that P is @! -cc, or (3) 2@2 D @3 and

P

.!1 /V D .!1 /V : Then P is weakly proper. Proof. (1) is an immediate corollary of Theorem 10.64. The proof of (2) is straightforward. The relevant observation is that since the ideal I is !-presaturated and since ¹˛ < !2 j cof.˛/ D !º 2 I; it follows that for each k < !, V //V .cof.!kC1

P

> !:

Since P is @! -cc, every countable set of ordinals in V P is covered by a set in V of cardinality (in V ) less than @! . This combined with the observation above, yields (2). The proof of (3) is a straightforward adaptation of the proof of Theorem 10.64, again one shows that for each k < !, V .cof.!kC1 //V

P

> !;

and of course one is only concerned with those values of k < ! such that P is not !kC1 -cc; i. e. with cardinals below the chain condition satisfied by P .

758

10 Further results

Since 2@2 D @3 , P is !4 -cc in V and so all cardinals above !3V are preserved. Therefore we need consider only the cases k  2. For k D 0 this is immediate and the case k D 1 follows by appealing to the generic ultrapower associated to the V -generic filter G  P . This leaves the case k D 2; i. e. !3V . But this case now follows by Lemma 9.120. t u Lemma 10.68, below, isolates the application of Lemma 10.59 within the proof of Theorem 10.69. This lemma in turn requires the following two lemmas. Lemma 10.66. Suppose that hS˛ W ˛ < !1 i is a sequence of stationary subsets of !1 and that h ˛ W ˛  !1 i is a closed increasing sequence of cardinals such that for each ˛ < !1 , ˛C1 is measurable and such that !1 < . Suppose that S  !1 is stationary and let Z be the set of X 2 P!1 . / such that (1) X \ !1 2 S , (2) For each ˛  X \ !1 , ordertype.X \ ˛ / 2 S˛ : Then Z is stationary in P!1 . /. Proof. Suppose T  !1 is stationary. Let GT be the game played on !1 for T :  The players alternate choosing countable ordinals, i , for i < ! with Player I choosing i for i even. Player I wins if sup¹i j i < !º 2 T: Since T is stationary, Player II cannot have a winning strategy. For each ˛ < !1 let G˛ be the game of length ! .1 C ˛/ defined as follows. A play of the game is an increasing sequence h W  < ! .1 C ˛/i of countable ordinals. Player I chooses  for  even and Player II chooses  for  odd. Player II wins if for some ˇ  ˛, sup¹ j  < ! .1 C ˇ/º … Sˇ : We claim that for each ˛, Player II cannot have a winning strategy in G˛ . This is easily proved by induction on ˛. Let ˛0 be least such that Player II has a winning strategy in G˛0 and let  W !1
10.2 Coding into L.P .!1 //

759

If ˛0 is a limit ordinal then again one can construct a winning strategy for Player II in the game GS˛0 by using an increasing ! sequence cofinal in ˛0 . One constructs a strategy   W !1
sup¹ j  < ! .1 C ˇ/º 2 Sˇ ;

(1.2) sup¹i j i < !º D sup¹ j  < ! .1 C ˛0 /º: The first condition, (1.1), is arranged by appealing to the induction hypothesis; i. e. that Player II does not have a winning strategy in G˛ for any ˛ < ˛0 and for any choice of hS W   ˛i. This proves the claim that for each ˛, Player II cannot have a winning strategy in G˛ . Fix a countable elementary substructure X  H. C / such that X \ !1 2 S and such that h ˛ W ˛ < !1 i 2 X: We claim there exists

Y  H. C /

such that (2.1) X  Y , (2.2) X \ !1 D Y \ !1 , (2.3) for each ˛  X \ !1 , ordertype.Y \ ˛ / 2 S˛ : If not then Player II has a winning strategy in G˛ where ˛ D X \ !1 . This follows from the following observation. Suppose Z  H. C / is a countable elementary substructure and 2 Z is a measurable cardinal. Let 2 Z be a normal measure on and let  2 \¹A 2 Z j A 2 º: Let ZŒ D ¹f ./ j f 2 Zº:

760

10 Further results

Then (3.1) ZŒ  H. C /, (3.2) Z \ V D ZŒ \ V . Using this it is straightforward to prove the claim above; if Y  H.!2 / does not exist then Player II has a winning strategy in G˛ where ˛ D X \ !1 . Thus Y  H. C / exists as required and the lemma follows. t u Lemma 10.67. Suppose that hSi W i < !i is a sequence of pairwise disjoint stationary subsets of !1 and that there exist !1 many measurable cardinals. Then there is a partial order P such that P is .!; 1/-distributive and such that if G  P is V -generic then .INS /V D .INS /V ŒG \ V and in V ŒG there exists a sequence hTi W i < !i of pairwise disjoint subsets of !1 and an ordinal  such that: (1) For each i < !, Ti  !1 and for each S 2 P .!1 / \ V n INS , both S \ Ti … INS and S n Ti … INS . (2) !1 <  < !2 and cof. / D !1 . (3) There exists a closed cofinal set C   such that for each i < !, hC; SQi \ C; 2i Š h!1 ; Ti ; 2i: (4) Suppose that  W !1 !  is a surjection and that  <  . a) Suppose that i < !, S D ¹˛ < !1 j ordertype.Œ˛/ 2 Si º; and that S is stationary. Then for each k < !, both S \ Tk and S n Tk are stationary in !1 . b) Suppose that cof./ D !1 , C   is closed and cofinal, S  !1 is stationary and that for some i < !, hC; SQi \ C; 2i Š h!1 ; S; 2i: Then for each k < !, both S \ Tk and S n Tk are stationary in !1 .

10.2 Coding into L.P .!1 //

761

Proof. Suppose that is a cardinal and that S  !1 is stationary. Let P . ; S / denote the partial order where: (1.1) P . ; S / is the set of pairs .f; c/ such that a) c  !1 , c is closed, and c is countable, b) f W max.c/ ! and for all ˛ 2 c, ordertype.f Œ˛/ 2 S: (1.2) .c0 ; f0 /  .c1 ; f1 / if a) c0 D c1 \ .max.c0 / C 1/, b) f0  f1 . Suppose that is measurable or a countable limit of measurable cardinals. Suppose that A  !1 is stationary and that g  P . ; S / is V -generic then in V Œg: (2.1) V !  V . (2.2) 2 SQ . (2.3) A is stationary in !1 . This follows from Lemma 10.66. The key point is that by Lemma 10.66, ¹X 2 P!1 . / j X \  2 A and ordertype.X / 2 S º is stationary in P!1 . /. More generally suppose h ˛ W ˛ < !1 i is a closed increasing sequence of cardinals such that for each ˛ < !1 , ˛C1 is measurable. Suppose that hA˛ W ˛ < !1 i is a sequence of stationary subsets of !1 and that Y g P . ˛ ; A˛ / ˛
is V -generic where the product partial order is computed with countable support. Then in V Œg: (3.1) V !  V . (3.2) For each ˇ < !1 , ˇ 2 AQˇ . (3.3) For each ˇ < !1 let gˇ D g \

Y

P . ˛ ; A˛ /:

˛ 777

From Lemma 10.79 and Lemma 10.80 one easily obtains the homogeneity and the .;;B/ !-closure of Pmax . .;;B/ is Lemma 10.83. Suppose that B  R and that L.B; R/  ADC . Then Pmax !-closed. t u .;;B/ Lemma 10.84. Suppose that B  R and that L.B; R/  ADC . Then Pmax is homogeneous. t u

We adopt the usual notational conventions. Suppose that B  R. A filter .;;B/ G  Pmax

is semi-generic if for all ˛ < !1 there exists a condition hM; a; zi 2 G such that ˛ AC

10.3 Bounded forms of Martin’s Maximum

787

Corollary 10.96 (Bounded Martin’s Maximum). Suppose that either: (i) There is a measurable cardinal, or (ii) INS is precipitous. Then 2@0 D 2@1 D @2 .

t u

In the presence of large cardinals, Bounded Martin’s Maximum implies that ı12 D !2 . This is an immediate corollary of the results of Chapter 3. Lemma 10.97 (Bounded Martin’s Maximum). Assume there is a Woodin cardinal with a measurable above. Then  ı 12 D !2 . Proof. Let ı be the least Woodin cardinal. By Shelah’s theorem, Theorem 2.64, there exists a semiproper partial order P of cardinality ı such that V P  “INS is !2 -saturated”: Clearly V P  “There is a measurable cardinal ”; and so by Theorem 3.17, V P  “ı12 D !2 ”: The lemma follows by applying Bounded Martin’s Maximum to P .

t u

The axiom ./ implies a very strong form of Bounded Martin’s MaximumCC . This is the content of the next theorem, Theorem 10.99, which in essence is simply a reformulation of the fundamental absoluteness theorem, Theorem 4.64, for Pmax . Remark 10.98. The requirement on N , in Theorem 10.99, that for each partial order P 2 N, 1 N P  “ 2 -Determinacy”; can be reformulated in terms of large cardinals. In fact, since R  N , it is equivalent to the assertion that for each set a 2 N , with a  Ord, .M1 .a//# 2 N where M1 .a/ is computed in V . M1 .a/ denotes the minimum (iterable) fine structure model of ZFC C “There is a Woodin cardinal” containing the ordinals and constructed relative to the set a. The formal definition involves the fine structure theory of Mitchell and Steel .1994/. t u Theorem 10.99 (Axiom ./). Suppose that for each partial order P , 1 V P  “ 2 -Determinacy”:

788

10 Further results

Suppose that N is a transitive inner model such that (i) P .!1 /  N , (ii) N  ZFC, (iii) for each partial order P 2 N , 1 N P  “ 2 -Determinacy”:

Then N  “Bounded Martin’s MaximumCC ”. Proof. By Lemma 10.94, Bounded Martin’s MaximumCC is equivalent to the following: (1.1) Suppose that P is a partial order such that .INS /V D .INS /V

P

\VI

i. e. such that P is stationary set preserving. Then P

hH.!2 /; INS ; 2iV †1 hH.!2 /; INS ; 2iV : Thus it suffices to show that (1.1) holds in N . In fact this follows by an argument which is essentially identical to that used to prove Theorem 4.69. Fix a …1 formula .x0 / in the language for the structure hH.!2 /; INS ; 2i and fix a set A  !1 such that hH.!2 /; INS ; 2i  ŒA: Clearly we may suppose that A … L.R/ and so by Theorem 4.60, A is L.R/generic for Pmax . Let GA  Pmax be the L.R/-generic filter given by A. Thus by Theorem 4.67 there is a condition h.M; I /; ai 2 GA such that the following holds. (2.1) Suppose

j W .M; I / ! .M ; I  /

is a countable iteration and let a D j.a/. Let N be any countable, transitive, model of ZFC such that: 

a) .P .!1 //M  N ; 

b) !1N D !1M ; c) Q3 .S /  N , for each S 2 N such that S  !1N ; d) If S  !1N , S 2 M and if S … I  then S is a stationary set in N . Then

hH.!2 /; 2; INS iN  Œa :

10.3 Bounded forms of Martin’s Maximum

789

It follows that (2.1) is expressible as a …13 statement about t where t 2 R codes h.M; I /; ai. The theorem now follows by a simple absoluteness argument, noting that from the hypothesis that for every partial order P , 1 NP   2 -Determinacy; it follows that for every partial order P ,

N †1 N P I 4 1 i. e. that †4 statements with parameters from V are absolute between N and N P . Fix a set E  Ord such that E 2 N and such that H. /N 2 LŒE where  D .jP [ H.!2 /jCC /N . Suppose that G  P is N -generic and assume toward a contradiction hH.!2 /; INS ; 2i  .:/ŒA: Let g  Coll.!; sup.E// be N ŒG-generic. Then E # together with the iteration j W .M; I / ! .M ; I  / which sends a to A, witness that (2.1) fails in N ŒGŒg which contradicts N †1 N ŒGŒg: 4

t u

Remark 10.100. One corollary of Theorem 10.99 is that the consistency of Bounded Martin’s MaximumCC is relatively weak even in conjunction with large cardinal axioms. For example if ZFC C “There is a proper class of Woodin cardinals” is consistent then so is ZFC C “There is a proper class of Woodin cardinals” C Bounded Martin’s MaximumCC : In contrast, ZFC C “There is a measurable cardinal” C Martin’s Maximum implies the consistency of ZFC C “There is a proper class of Woodin cardinals” and much more. The latter is by results of Steel combined with results of Schimmerling. t u Suppose that   P .R/ is a pointclass closed under continuous preimages such that

L.; R/  ADC C “ ‚ is regular”:

790

10 Further results 2P max

The basic analysis of L.R/

2P max

generalizes to L.; R/

. In particular,

Theorem 10.101. Suppose A  R and that L.A; R/  ADC : Suppose G  2 Pmax is L.A; R/-generic. Then L.A; R/ŒG  ZFC and in L.A; R/ŒG, (1) ./ holds, (2) INS is not !2 -saturated.

t u

The next lemma is an immediate corollary of the analysis of Pmax as summarized in Theorem 10.101. 2

Lemma 10.102. Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R : Suppose that G  2 Pmax is L.; R/-generic. N  L.; R/ŒG containing R [ ¹Gº such that

Then there is an inner model

(1) N  ZFC C ./, (2) for all partial orders P 2 N , 1 N P  “ 2 -Determinacy”; (3) N  “INS is not !2 -saturated”. Proof. By Theorem 9.14 there exists a pointclass 0   such that L.0 ; R/  ADC C AD R : Therefore, and this is all we require, there exists A 2  such that (1.1) L.A; R/  ADC , (1.2) .‚0 /L.A;R/ < .‚/L.A;R/ . Let ı D .‚/L.A;R/ and let M D .HOD/L.A;R/ \ Vı : By Theorem 9.113, ı is a Woodin cardinal in .HOD/L.A;R/ and so M  ZFC: It follows that for each partial order P 2 M , 1 M P  “ 2 -Determinacy”; and further that M.R/ is a set (symmetric) extension of M . The latter is easily proved by a variation of Vopenka’s argument that every set of ordinals is set generic over HOD. Let N D M.R/ŒG: Thus N is a set generic extension of M . It follows that N is as required.

t u

10.3 Bounded forms of Martin’s Maximum

791

By altering the choice of the inner model, N , in the proof of Lemma 10.102, one can also prove the following variation. Lemma 10.103. Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R : Suppose that G  2 Pmax is L.; R/-generic. N  L.; R/ŒG containing R [ ¹Gº such that

Then there is an inner model

(1) N  ZFC C ./, (2) N  “There exists a proper class of Woodin cardinals ”, (3) N  “INS is not !2 -saturated”.

t u

As an immediate corollary of Theorem 10.99 and Lemma 10.103 it follows that Bounded Martin’s MaximumCC does not imply that INS is !2 -saturated. The basic method can be used to show that a number of consequences of Martin’s Maximum are not implied by Bounded Martin’s Maximum. Two interesting questions are:  Assume Bounded Martin’s MaximumCC and that there exists a proper class of Woodin cardinals. – Must INS be semi-saturated? – Must INS be precipitous? Corollary 10.104. Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R . Suppose that G  2 Pmax is L.; R/-generic. Then there is an inner model N  L.; R/ŒG containing R [ ¹Gº such that (1) N  Bounded Martin’s MaximumCC , (2) N  “There exists a proper class of Woodin cardinals ”, (3) N  “INS is not !2 -saturated”. Proof. By the Martin–Steel Theorem, for all partial orders P 2 N , 1 N P  “ 2 -Determinacy”: The corollary follows from this, Theorem 10.99 and Lemma 10.103.

t u

The closure on N , in Theorem 10.99(iii), cannot be significantly weakened, though it can be weakened slightly. We state two closely related theorems which illustrate this. We require a definition.

792

10 Further results

1 Definition 10.105. ZF 2 -Determinacy:

(1) For all x 2 R, x # exists. (2) Suppose 1 .x; y/ and 2 .x; y/ are †12 formulas and a 2 R are such that for all transitive models, M , of ZF, if a 2 M then ¹b 2 R \ M j M  1 Œa; bº D .R \ M / n ¹b 2 R \ M j M  2 Œa; bº: Then the set ¹b 2 R j 1 Œa; bº is determined.

t u

Remark 10.106. Of course, (2) of Definition 10.105 implies (1), and so an equivalent notion is obtained by eliminating (1) from the definition. t u Theorem 10.107. Suppose that for each partial order P , 1 V P  “ 2 -Determinacy”: Then there exists a transitive inner model N containing the ordinals such that

(1) P .!1 /  N , (2) N  ZFC, 1 (3) for each partial order P 2 N ,N P  “ ZF 2 -Determinacy”,

(4) N  “Bounded Martin’s Maximum fails”. Proof. For each set a  Ord let L# .a/ denote the minimum inner model M such that (1.1) M  ZFC, (1.2) Ord  M , (1.3) a 2 M , (1.4) for all b 2 M , b # 2 M . We prove that N  “Bounded Martin’s Maximum fails” where, abusing notation slightly, N D L# .P .!1 //: The proof of the theorem is similar.

10.3 Bounded forms of Martin’s Maximum

793

Let G0  Coll.!1 ; H.!2 // be N -generic. Thus there exists A  !1 such that A 2 N ŒG0  and such that N ŒG0  D L# .A/: Let

A D ¹˛ < !1 j j˛jL

# .A\˛/

D !º:

The key point is that for each S 2 .P .!1 / n INS /N , A \ S … .INS /N ŒG0  : Let C0  A be N ŒG0 -generic for PA where PA is the partial order of countable subsets of A , which are closed in !1 , ordered by extension (Harrington forcing). Thus in N ŒG0 ŒC0 : (2.1) C0  !1 , and C0 is closed and unbounded. (2.2) For each ˛ 2 C0 , j˛jL

# .A\˛/

D !:

However, for each S 2 .P .!1 / n INS /N , A \ S … .INS /N ŒG0  ; and so .INS /N D N \ .INS /N ŒG0 ŒC0  : Assume toward a contradiction that N  “Bounded Martin’s Maximum”: Therefore by Lemma 10.93, hH.!2 /; 2iN †1 hH.!2 /; 2iN ŒG0 ŒC0  : But this implies, by (2.1) and (2.2), that in N there exist AO  !1 and CO  !1 such that: (3.1) CO  !1 , and CO is closed and unbounded. (3.2) For each ˛ 2 CO , j˛jL

# .A\˛/ O

D !:

O  exists, which is a contradiction. But by the hypothesis of the lemma, .CO ; A/

t u

The second theorem, Theorem 10.108, is closely related Theorem 9.75 and Theorem 9.81, which show that closure properties of P .!1 / transfer upwards to closure properties of P .!2 /, assuming WRP.2/ .!2 /. Theorem 10.108. Suppose that 1 V Coll.!;!1 /  “ 2 -Determinacy”:

794

10 Further results

Suppose that N is a transitive inner model such that (i) P .!1 /  N , (ii) N  ZFC, (iii) N  “Bounded Martin’s MaximumCC ”. Then for each partial order P 2 N , 1 N P  “ ZF 2 -Determinacy”:

Proof. We prove that for each set a  Ord, if a 2 N then a# 2 N ; i. e. that for each partial order P 2 N , 1 NP  “ …  1 -Determinacy”: The proof of the theorem is similar. Fix a  Ord with a 2 N . Assume toward a contradiction that a# … N: Let  D sup.a/ and suppose that G0  Coll.!1 ;  / be N -generic. Thus there exists A  !1 such that A 2 N ŒG0  and such that A# … N ŒG0 . Therefore there exists b0 2 H.!1 / such that in N ŒG0 , ¹˛ < !1 j b0 2 .A \ ˛/# º is stationary and co-stationary in !1 . However .INS /N D .INS /N ŒG0  \ N and so by Lemma 10.94, hH.!2 /; INS ; 2iN †1 hH.!2 /; INS ; 2iN ŒG0  since N  “Bounded Martin’s MaximumCC ”. Therefore there exists AO  !1 such that AO 2 N and such that ¹˛ < !1 j b0 2 .A \ ˛/# º is stationary and co-stationary in !1 . But by the hypothesis of the lemma, AO# exists and so the set ¹˛ < !1 j b0 2 .A \ ˛/# º cannot be both stationary and co-stationary. This is a contradiction.

t u

Theorem 10.99 can be improved to provide a characterization of ./. First we .;;B/ note the following corollary of the analysis of Pmax which rules out one possible characterization.

10.3 Bounded forms of Martin’s Maximum

795

Theorem 10.109. Suppose   P .R/ is a pointclass closed under continuous preimages such that L.; R/  AD R C “ ‚ is regular”: .;;B/ Let B  R be a set in L.R/. Suppose that G0  Pmax is L.; R/-generic and let L.;R/ M D ¹a 2 .HOD/ ŒRŒG0 j j rank.a/ < ‚º where ‚ D .‚/L.;R/ . Then:

(1) For each partial order P 2 M ,

M P  ADL.R/ :

(2) M  :./. (3) Suppose that N  M is a transitive inner model such that a) P .!1 /M  N , b) N  ZFC, 1 c) For each partial order P 2 N , N P  “ 2 -Determinacy”. Then N  “Bounded Martin’s MaximumCC ”.

t u

Definition 10.110 (Feng–Magidor–Woodin). Suppose that A  R. Then A is universally Baire if for any compact Hausdorff space X and any continuous function,  W X ! R; the set ¹a 2 X j .a/ 2 Aº has the property of Baire in X . t u The next theorem gives a useful characterization of the sets A  R which are universally Baire. Theorem 10.111 (Feng–Magidor–Woodin). Suppose A  R. equivalent.

The following are

(1) A is universally Baire. (2) Suppose that P is an infinite partial order and let ı D 2jP j : Then there exist trees S; T on !  ı such that A D pŒT  and such that if G  P is V -generic then in V ŒG, pŒT V ŒG D RV ŒG n pŒS V ŒG : u t Thus if A  R is universally Baire and P is a partial order, then A has an unambiguous interpretation in V P . If G  P is V -generic then we let AG denote the interpretation of A in V ŒG. It is easily verified that AG D [¹pŒT V ŒG j T 2 V and A D pŒT V º: In the presence of suitable large cardinals, the universally Baire sets are exactly the sets which are 1 -homogeneously Suslin. This is a corollary of Theorem 2.32, Theorem 10.111, and the principal theorem of .Martin and Steel 1989/.

796

10 Further results

Theorem 10.112. Suppose there is a proper class of Woodin cardinals and that A  R. Then following are equivalent. (1) A is universally Baire. (2) A is 1 -weakly homogeneously Suslin. (3) A is 1 -homogeneously Suslin.

t u

The following lemma is an immediate corollary of the definition of a universally Baire set. Lemma 10.113. Let   P .R/ be the set of universally Baire sets. Then  is a -algebra closed under continuous preimages. u t 1 †12 -sets is answered by Clearly every † 1 -set is universally Baire. The situation for  the following theorem of .Feng, Magidor, and Woodin 1992/.

Theorem 10.114 (Feng–Magidor–Woodin). The following are equivalent. (1) For every set X , X # exists. 1 (2) Every † 2 -set is universally Baire.

t u

Corollary 10.115. Suppose A  R. Then the following are equivalent. (1) .A; R/# exists and .A; R/# is universally Baire. (2) Each set B 2 L.A; R/ \ P .R/ is universally Baire. Proof. Every set B 2 L.A; R/\P .R/ is a continuous preimage of .A; R/# . Therefore (1) implies (2). Suppose that (2) holds. By Theorem 10.114, .A; R/# exists. Note that .A; R/# is naturally a countable union of sets in L.A; R/ and so by Lemma 10.113, .A; R/# is universally Baire. t u It is open whether the assumption that every projective set in universally Baire implies generic absoluteness for projective statements. The following theorem gives a sufficient condition which is implied in many cases by an appropriate determinacy hypothesis. Theorem 10.116. Suppose that A  R and that for each set B  R  R, if B is definable in the structure hH.!1 /; A; 2i then there exists a choice function f WR!R such that

10.3 Bounded forms of Martin’s Maximum

797

(i) for all x 2 R if .x; y/ 2 B for some y 2 R then .x; f .x// 2 B, (ii) f 1 ŒO is universally Baire for each open set O. Suppose that P is a partial order and that G  P is V -generic. Then hH.!1 /; A; 2iV  hH.!1 /; AG ; 2iV ŒG :

t u

Lemma 10.117 (Feng–Magidor–Woodin). Suppose that for each partial order P , V P  ADL.R/ : Then R# is universally Baire.

t u

There are two approximate converses to Lemma 10.117. Theorem 10.118. Assume L.R/  AD and that R# is universally Baire. Then for each partial order P , V P  ADL.R/ :

t u

Theorem 10.119. The following are equivalent. (1) For each partial order P , V P  ADL.R/ : (2) R# is universally Baire and for each partial order P , if G  P is V -generic then .R# /G D .R# /V ŒG :

t u

Remark 10.120. (1) Theorem 10.119 is proved using basic method for proving Theorem 5.104; i. e. the proof uses core model methods. We note that the theorem is false at the projective level. (2) It is open whether the actual converse to Theorem 10.117 holds. We shall need the generalization of Lemma 10.117 to L.R# /. Lemma 10.121. Suppose that for each partial order P , V P  ADL.R / : #

Then .R# /# is universally Baire.

t u

798

10 Further results

Definition 10.122. A-Bounded Martin’s Maximum: (1) A  R is universally Baire. (2) Suppose that P is a partial order such that .INS /V D .INS /V

P

\ V:

Suppose that D  P .P / is a collection of predense subsets of P , each with cardinality !1 , such that jDj  !1 . Suppose that h W  < !1 i is a sequence of terms for elements of AG . Then there exists a filter F  P such that: a) For all D 2 D, F \ D ¤ ;; b) For each  < !1 , ¹.i; j / 2 !  ! j for some p 2 F ; p  .i; j / 2 º 2 A:

t u

Definition 10.123. A-Bounded Martin’s MaximumCC : (1) A  R is universally Baire. (2) Suppose that P is a partial order such that .INS /V D .INS /V

P

\ V:

Suppose that D  P .P / is a collection of predense subsets of P , each with cardinality !1 , such that jDj  !1 . Suppose that h W  < !1 i is a sequence of terms for stationary subsets of !1 and that h W  < !1 i is a sequence of terms for elements of AG . Then there exists a filter F  P such that: a) For all D 2 D, F \ D ¤ ;; b) For each  < !1 , ¹.i; j / 2 !  ! j for some p 2 F ; p  .i; j / 2 º 2 AI c) For each  < !1 , ¹˛ < !1 j for some p 2 F ; p  ˛ 2  º is stationary in !1 .

t u

As an immediate corollary of Theorem 10.111 we obtain the following. Theorem 10.124. (1) (Martin’s Maximum) Suppose that A  R is universally Baire. Then A-Bounded Martin’s Maximum holds. (2) (Martin’s MaximumCC ) Suppose that A  R is universally Baire. Then t u A-Bounded Martin’s MaximumCC holds.

10.3 Bounded forms of Martin’s Maximum

799

Theorem 10.127 provides an equivalence to ./ in terms of a strong form of Bounded Martin’s Maximum. We shall prove several versions. Suppose that A  R. We denote by  †1! .A/ the pointclass of all sets B  R such that B is definable in the structure hV!C1 ; A; 2i from real parameters; these are the sets B  R which are projective in A. The final theorem we shall need for the proof of Theorem 10.127 is the following. Theorem 10.125. Assume R# exists and that every set which is  †1! .R# / is determined. Suppose that A0  R, A0 2 L.R/ and that A0 is definable in L.R/ from x0 and indiscernibles of L.R/. Suppose that M0 is a countable transitive model of ZFC such that: (i) Suppose that g  Coll.!; H.!2 /M0 / is an M0 -generic filter with g 2 H.!1 /. Then R# \ M0 Œg D .R# /M0 Œg and hM0 Œg \ V!C1 ; R# \ M0 Œg \ V!C1 ; 2i  hV!C1 ; R# ; 2i. (ii) x0 2 M0 . Then there exists .M1 ; I1 ; ı/ 2 H.!1 / such that (1) M1 is transitive and M1  ZFC C “ı is a Woodin cardinal”, (2) I1 D .I
and such that x D [¹.gjk/ j k < !º: Let A  Ord be a set such that (3.1) .!2 /LŒA D !2 , (3.2) ¹; .s; t /º 2 LŒA, (3.3) !3 is a measurable cardinal in LŒA. Let PA D P \ LŒA. By (3.1), PA is simply the partial order for Namba forcing as defined in LŒA.  defines a term A 2 LŒAPA for a real. Since there is a measurable cardinal in LŒA, LŒAPA  “For all y 2 R, y # exists”: Let ı be the least strongly inaccessible cardinal in LŒA. Thus ı < !3V : Fix a tree T 2 LŒA on !  ı such that if g   PA is LŒA-generic then in LŒAŒg  , 

pŒT  D ¹.y; y # / j y 2 RLŒAŒg  º: As usual we regard the infinite branches of T as triples .y; z; f / where y 2 ! ! , z 2 ! ! and f 2 ı ! . Thus working in LŒA there exists a condition .s  ; t  /  .s; t / in PA and a function

 W t ! T

such that if b is an infinite branch of t  then [¹  .bjk/ j k < !º is an infinite branch .y; z; f / of T such that y D [¹.bjk/ j k < !º. The condition .s  ; t  / 2 PA and the function   are constructed by a fusion argument, analogous to the construction of .s; t / and . Here though one cannot require that   is length preserving. We return to V Œg. By genericity we may suppose that .s  ; t  / 2 g. We now prove that in V Œg, x # exists and further that !3V is an indiscernible of LŒx. Since .s  ; t  / 2 g, there exists z 2 RV Œg such that .x; z/ 2 pŒT . By absoluteness it follows that z D x# for otherwise in LŒA there must exist .x  ; z  / 2 pŒT  such that z  ¤ .x  /# contradicting the choice of T . This proves x # exists in V Œg. We finish by proving that !3V is an indiscernible of LŒx. In fact a simple boundedness argument shows that for each ordinal , the . C 1/th indiscernible of LŒx is below the least ordinal above  which is admissible relative to T , cf. the proof of Theorem 10.62. However !3V is a cardinal in LŒT  so it follows that !3V is an indiscernible of LŒx.

830

10 Further results

In summary we have proved that in V Œg, for all x 2 RV Œg , x # exists and that !3V is an indiscernible of LŒx. However !1V Œg D !1V and so it follows that .ı12 /V Œg  !3V .

t u

Theorem 10.193 (Axiom ./CC ). Suppose that for each A  !2 , A exists. Suppose that g is V -generic for Namba forcing. Then .ı12 /V Œg D .!3 /V : Proof. Clearly !3 D .‚/L.;R/ : Recall that AG denotes the set [¹a j h.M; I /; ai 2 Gº. By modifying G if necessary we can suppose that !1 D .!1 /LŒAG  and so, since AC holds, there exists a surjection  W !2 ! R which is †1 definable in the structure hH.!2 /; INS ; 2i from AG . Fix an ordinal ˛ 2 !3 n !2 and fix a set B 2  such that B codes a a surjection  W R ! ˛: Let F  G be the set of

h.M; I /; ai 2 G

such that (1.1) M  ZFC C ./, (1.2) B \ M 2 M, (1.3) .M; I / is B-iterable, (1.4) hV!C1 \ M; B \ M; 2i  hV!C1 ; B; 2i. By Lemma 4.52 and Lemma 4.56, F is dense in G. Suppose that h.M; I /; ai 2 F and let jG W .M; I / ! .M ; I  / be the (unique) iteration such that jG .a/ D AG : It follows from (1.1)–(1.4) that hV!C1 \ M ; 2i  hV!C1 ; 2i

10.6 The Axiom ./C

831

and so by (1.1) it follows that  hH.!2 /M ; 2i  hH.!2 /; 2i:  Therefore j.!2 /M 2 M . Now suppose that hh.Mk ; Ik /; ak i W k < !i is a decreasing sequence in F and for each k < ! let jGk W .Mk ; Ik / ! .Mk ; Ik / be the (unique) iteration such that jGk .ak / D AG : Let x 2 R code hh.Mk ; Ik /; ak i W k < !i. Then (2.1) Œsup¹Mk \ Ord j k < !º D [¹R \ Mk j k < !º, (2.2) ordertype.¹ ı .ˇ/ j ˇ 2 [¹Mk \ Ord j k < !ºº/  x where x is the least ordinal, , above !1 such that L Œx is admissible. The theorem easily follows. Define f W !2
Then s 2 M where

jG W .M; I / ! .M ; I  /

is the iteration such that j.a/ D AG . (3.2) Suppose s 2 !2
t u

832

10 Further results

There are three analogs of ./C in the context of CH: Version I: Suppose that X  !1 . Then there exists a set A  R such that  L.A; R/  ADC ,  there is a filter g  Coll.!1 ; R/ such that (1) g is L.A; R/-generic, (2) X 2 L.A; R/Œg.

t u

Version II: Suppose that X  !1 . Then there exists a pointclass   P .R/ such that  L.; R/  ZF C DC C ADR ,  there is a filter g  Coll.!1 ; R/ such that (1) g is L.; R/-generic, (2) X 2 L.; R/Œg.

t u

Version III: Suppose that X  !1 . Then there exists a pointclass   P .R/ such that  L.; R/  ZF C DC C ADR ,  there is a filter g  Coll.!1 ; R/ such that (1) g is L.; R/-generic, (2) X 2 L.; R/Œg, (3) .INS /L.;R/Œg D INS \ L.; R/Œg.

t u

Remark 10.194. Clearly, assuming CH, (Version I) and (Version II) are each expressible by a …2 sentence in the structure hH.!2 /; 2i: Further (Version III) is expressible by a …2 sentence in the structure hH.!2 /; INS ; 2i: It is a corollary of the proof of Theorem 10.180 that (Version III) cannot be implied by CH in -logic. In fact one can show that the sentence “There exists a partial order P such that RV D .R/V V cannot be a validity of -logic.

P

P

and such that

 (Version III)” t u

10.6 The Axiom ./C

833

We conjecture that (Version I) is implied by CH if there exists a measurable Woodin cardinal. This conjecture is implied by the following stronger conjecture for which we make the following definition. Suppose that G  H.!1 /: 

Let G be the set of b 2 H.!2 / such that for all countable X  hH.!2 /; G ; 2i; if b 2 X then bX 2 G where bX is the image of b under the transitive collapse of X . The stronger conjecture is:  (Long Game Conjecture) Assume that ı is a measurable Woodin cardinal and that AR is ı-homogeneously Suslin. Suppose that G  H.!1 / and that G 2 L.A; R/. Then there exists a set B  R such that (1) B is ı-homogeneously Suslin, (2) A 2 L.B; R/, (3) in L.B; R/Pmax , the integer game of length !1 given by G  is determined t u where G  is as defined in L.B; R/Pmax . Remark 10.195. (1) It is not difficult to show that assuming ./, there is definable integer game of length !1 which is not determined. (2) It is a corollary of Long Game Conjecture that if ı is a measurable Woodin cardinal then for every set A  R, if A is ı-homogeneously Suslin and if G 2 P .H.!1 // \ L.A; R/; then the integer game of length !1 given by G  is determined (no assumption concerning CH is made). The Long Game Conjecture is true if this corollary is provable from the existence of a measurable Woodin cardinal, provided certain iterability assumptions hold in V . (3) Neeman has shown that if there is a measurable Woodin cardinal then the Long Game Conjecture follows directly from iterability assumptions for V , Neeman .2004/. t u Theorem 10.196 (Long Game Conjecture). Assume there exists a proper class of measurable Woodin cardinals. Then there exists a universally Baire set A  R such that the following holds. Suppose that X  !1 , Y  !1 and that !1 D .!1 /LŒX D .!1 /LŒY  : Suppose that A \ LŒX  2 LŒX  and that A \ LŒY  2 LŒY . Then LŒX  LŒY :

t u

834

10 Further results

Theorem 10.197 (Long Game Conjecture, CH). Assume there exists a proper class of measurable Woodin cardinals. Suppose that X  !1 . Then there exists a set A  R such that (1) L.A; R/  ADC , (2) there is a filter g  Coll.!1 ; R/ such that a) g is L.A; R/-generic, b) X 2 L.A; R/Œg.

t u

We now consider the problem of obtaining (Version II) from CH. This is closely related to the question concerning CH, listed at the end of the previous section:  Does there exist a sentence ‰ such that ZFC C CH C ‰ is -consistent and such that for all †2 sentences, , either – ZFC C CH C ‰ ` “H.!2 /  ”, or – ZFC C CH C ‰ ` “H.!2 /  :”? It is convenient to define a slight strengthening of (Version II). (Version II)C : Suppose that X  !1 and that A  R is universally Baire. Then there exists a pointclass   P .R/ such that  A 2 ,  L.; R/  ZF C DC C ADR ,  there is a filter g  Coll.!1 ; R/ such that (1) g is L.; R/-generic, (2) X 2 L.; R/Œg.

t u

We remark that the assumptions (i)–(iii) of Theorem 10.198 should hold in any fine structural inner model in which there exists a proper class of measurable Woodin cardinals. Further it seems quite plausible that (i), (iii) and a sufficient fragment of (ii) are provable consequences of the existence of a proper class of measurable Woodin cardinals; i. e. that the stronger theorem, obtained by eliminating the assumptions (i)– (iii), is actually true. Theorem 10.198. Assume there exists a proper class of measurable Woodin cardinals and that: (i) Long Game Conjecture holds; (ii) WHIHC holds;

10.7 The Effective Singular Cardinals Hypothesis

835

(iii) For each universally Baire set, A  R, there exists Q ı/ 2 H.!1 / .M; E; such that a) M  ZFC, b) EQ 2 M is a weakly coherent Doddage (in M) witnessing ı is a Woodin cardinal in M, c) in M there is a measurable Woodin cardinal above ı, d) M is A-closed, Q has a universally Baire iteration scheme which is weakly A-good. e) .M; E/ Then the following are equivalent. (1) ZFC C CH  (Version II)C . (2) For each †2 sentence, , either a) ZFC C CH C “H.!2 / †2 H.!2 /V b) ZFC C CH C “H.!2 / †2 H.!2 /

Coll.!1 ;R/

V Coll.!1 ;R/

” ` “H.!2 /  ”, or ” ` “H.!2 /  :”.

t u

Suppose that LŒE is a Mitchell–Steel inner model with a superstrong cardinal, and a proper class of Woodin cardinals, in which the countable initial segments of LŒE are -iterable for every . Then one can show that in LŒE, the †2 theory of H.!2 / is not finitely axiomatized over ZFC in -logic. With additional assumptions one can also show that in LŒE, (Version II) must fail. Thus any attempt to prove (Version II) from CH would seem to require large cardinals beyond superstrong.

10.7 The Effective Singular Cardinals Hypothesis Assume there is a proper class of Woodin cardinals. Suppose is an uncountable cardinal and that g  Coll.!; / is V -generic. Suppose in V Œg there exists a prewellordering .RV Œg ; g / such that in V Œg: (1) g is 1 -homogeneously Suslin; (2) g has length !2 . Must there exist in V an 1 -homogeneously Suslin prewellordering of length CC ? By Theorem 9.132, if there is a proper class of Woodin cardinals and if the nonstationary ideal on !2 is semi-saturated then the answer is no, with D !1 . The case when is a singular strong limit cardinal seems particularly interesting. A positive answer is an effective form of the Singular Cardinals Hypothesis.

836

10 Further results

Definition 10.199. Effective Singular Cardinals Hypothesis: Assume there is a proper class of Woodin cardinals. Suppose that is a singular strong limit cardinal and that g  Coll.!; / is V -generic. Suppose that M  V Œg is a transitive inner model such that in V Œg: (1) R  M ; (2) M  ZF C AD; (3) Every set A 2 P .R/ \ M is 1 -homogeneously Suslin. Then ‚M < . CC /V .

t u

Remark 10.200. There are two natural variations of the Effective Singular Cardinals Hypothesis: (1) One could require that GCH holds below , or (2) that the Effective Generalized Continuum Hypothesis holds below . The Effective Generalized Continuum Hypothesis is the obvious variation of Effective Singular Cardinals Hypothesis:  Suppose that is an infinite cardinal and that g  Coll.!; / is V -generic. Suppose that M  V Œg is a transitive inner model such that in V Œg: – R  M; – M  ZF C AD; – Every set A 2 P .R/ \ M is 1 -homogeneously Suslin. Then ‚M < . CC /V .

t u

We give a brief summary of a few relevant results and which are proved in Chapter 7 of .Woodin 2010b/. These results are primarily concerned with the following related problem. Suppose is a singular strong limit cardinal and that  2 V Coll.!;/ 1

is a term for an -homogeneously Suslin set of reals. Must  be equivalent to a term which is definable from parameters in H. C /? It is convenient to introduce the notion of a term relation. Definition 10.201. Suppose that is a cardinal and that  2 V Coll.!;/ is a term for a subset P .!/. The term relation of  is the set of pairs .p; / such that

10.7 The Effective Singular Cardinals Hypothesis

837

(1)  !  Coll.!; /, (2) p 2 Coll.!; /, (3) p  x 2 , where x 2 V Coll.!;/ is the term for a subset of ! given by ; ŒŒn 2 x  D _¹q 2 Coll.!; / j .n; q/ 2 º:

t u

The case when has uncountable cofinality is easily dealt with. Theorem 10.202. Suppose that is a strong limit cardinal of uncountable cofinality. Suppose that  2 V Coll.!;/ is a term for an 1 -homogeneously Suslin subset of P .!/. Then the term relation for  is definable, from parameters, in the structure hH. C /; 2i: Proof. Without loss of generality we may suppose that   H. C /. Let T be a term for a weakly homogeneous tree in V Coll.!;/ with projection  and let F be term for a function which witnesses that T is weakly homogeneous. Since every every countably complete measure in V Coll.!;/ which concentrates on finite sequences of ordinals extends uniquely a measure in V , there exists (uniquely) a partial function  W ¹.s; t; q/ j s 2 !
838

10 Further results

Let G be the set of countably generated filters g  Coll.!; /. Thus G 2 H. C / as is the function  W G ! P .R/ defined by .g/ D A g . Let R be the term relation for  . Then is is easily verified that .p; / 2 R if and only if S is stationary in P!1 .H. // where S is the set of countable elementary substructures Y  hH. /; 2i such that for some set D  Y \ Coll.!; /, (1.1) p 2 Y , (1.2) D is dense in Y \ Coll.!; /, (1.3) if g 2 G , p 2 g, and D \ g ¤ ;, then Ig . / 2 A g : Thus R is definable in H. C / from .G ; /.

t u

As one might expect, the case when has cofinality ! is more subtle. Theorem 10.203. Let be a singular strong limit cardinal of cofinality !. Suppose that there exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below . Then there exists a closed cofinal set C  such that: (1) j.C / D C ; (2) Suppose  2 C and cof. / D !. Then jV j D  and there exists a term  2 V Coll.!;/ for a subset of P .!/ such that, a)  is a term for a set which is < -weakly homogeneously Suslin, b) every set in L .VC1 / \ P .VC1 / is †1 definable from parameters in the structure, hH. C /; R ; 2i; where R is the term relation of . Remark 10.204. The large cardinal hypothesis:  There exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below ,

t u

10.7 The Effective Singular Cardinals Hypothesis

839

yields a structure theory for L.VC1 / which in many aspects is analogous to the structure theory for L.R/ in the context of ADL.R/ . Note that by Kunen’s theorem on the nonexistence of an elementary embedding of V to V , must be the ! th element of the critical sequence of j . The next theorem shows that from this hypothesis one obtains a weak failure of the Effective Singular Cardinals Hypothesis. The proof of this theorem and of related theorems can be found in Chapter 7 of .Woodin 2010b/. t u Theorem 10.205. Suppose that there exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below and that g  Coll.!; / is V -generic. Then in V Œg there exists a transitive inner model M  L.VC1 /Œg such that (1) RV Œg  M , (2) M  ZF C ADC , (3) . CC /L.VC1 / < ‚M .

t u

Remark 10.206. It is a natural conjecture that the inner model M of Theorem 10.205 can be chosen such that .‚/L.VC1 /Œg D ‚M : It is immediate that .‚/L.VC1 /Œg is simply the least ordinal  such that in L.VC1 /,  is not the surjective image of VC1 . We denote this ordinal by ‚L.VC1 / , this is the natural generalization of ‚L.R/ to L.VC1 /. This in turn suggests the following problem. Suppose there exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below . Must ‚L.VC1 / < CC ‹

t u

Chapter 11

Questions

The following is a list of questions, including many which have appeared in earlier chapters. The order simply reflects roughly the place within the book where the question is discussed, either explicitly or implicitly, and there is significant overlap among various of these questions. Comments have been asserted in italics for those questions which either have been solved or otherwise affected by developments of which I am aware since the first edition. (1) Assume L.R/  AD. Must ‚L.R/  !3 ? (2) Can there exist countable transitive models M and M  such that M  ZFC C “The nonstationary ideal on !1 is saturated”; 

M is an iterate of M , and such that M 2 M  ? (3) Suppose that the nonstationary ideal on !1 is !2 -saturated and that L.R/  AD: Must ı12 D !2 ? (4) Suppose that N is a transitive inner model containing the ordinals such that N  ZFC and such that for each countable set  N there exists a set  2 N with jjN D ! and such that   . a) Suppose that for each set X , X # exists. Must ı 1 D .ı12 /N ‹

2

b) Suppose that for each partial order P , V P  ADL.R/ : Must .HOD/L.R/ D .HOD/L.R

N/

‹

(5) Suppose that INS is !2 -saturated and that P .!1 /# exists. Suppose that A  !1 and let A D sup¹.!2 /LŒZ j Z  !1 ; A 2 LŒZ; and RLŒA D RLŒZ º: Must A < !2 ?

11 Questions

841

(6) Assume there exists a proper class of Woodin cardinals. Do either of the following imply :CH? a) Every function f W !1 ! !1 is bounded on a closed cofinal subset of !1 by a canonical function. b) Suppose that A  R is universally Baire and that f W !1 ! A: Then there exists a tree T on !  !1 such that that A D pŒT  and such that ¹˛ < !1 j f .˛/ 2 pŒT j˛º contains a closed cofinal subset of !1 . Solved by Larson and Shelah: The answer is no. (7) Suppose the nonstationary ideal on !1 is !1 -dense. a) Must c D !2 ? b) Must  ı 12 D !2 ? c) Must ‚L.R/  !3 ? (8) Assume Martin’s Maximum.c/. Suppose that   P .R/ is a pointclass, closed under continuous preimages, such that a) L.; R/  ADC , b) !3 D .‚/L.;R/ . Suppose that G  Pmax is an L.; R/-generic filter such that G 2 V and such that P .!1 / D P .!1 /G : Must L.; R/ŒG  ADR ‹ (9) Assume ./. Suppose  W Œ!1 2 ! ¹0;1º is a partition with no homogeneous rectangle for 0 of (proper) cardinality @1 . Must there exist a set X  !1 such that E .3/ ŒX  is nonstationary in !1 ? Justin Moore has proved that the associated partition relation is false .Moore 2006/, but the status of this question remains unclear.   (10) Assume  . a) Must INS be semi-saturated? b) Must HODR  AD?

842 (11)

11 Questions

a) Suppose that the Axiom of Condensation holds. Does strong condensation hold for H.!2 /? b) Suppose that N is a transitive inner model of ZFC in which the Axiom of Strong Condensation holds and that covering fails for N . Must N  LŒx for some x 2 R?

(12) (Conjecture) The following are equiconsistent. a) ZFC C Martin’s Maximum.c/ C “JNS is weakly presaturated”. b) ZFC C CH C “INS jS is !1 -dense for a dense set of S 2 P .!1 / n INS ”. c) ZF C ADR C “‚ is regular”. (13) Assume there is a measurable cardinal. a) Is is possible for every function f W !2 ! !2 to be bounded by a canonical function pointwise on a closed unbounded set? b) Can the nonstationary ideal on !2 be semi-saturated? c) Let I be the nonstationary ideal on !2 restricted to the ordinals of cofinality !1 . Can the ideal I be semi-saturated? d) Suppose that there exists a normal uniform ideal I  P .!2 / such that I is semi-saturated and contains JNS . Suppose that J  P .!2 / is a normal uniform semi-saturated ideal. Must JNS  J ‹ The motivation for this question is rendered irrelevant by Shelah’s theorem on ˘! .!2 /. The natural conjecture now is that the answer to (a)–(c) is negative. (14) Is Martin’s Maximum + MIH consistent? (15) Assume Martin’s Maximum. a) Can ./C hold?

  ?    c) Can L.P .!! //   ?

b) Can L.P .!3 // 

11 Questions

843

(16) (Conjectures) a) There exists a regular (uncountable) cardinal and a definable partition of ¹˛ < j cof.˛/ D !º into infinitely many stationary sets. b) Suppose that there is a proper class of supercompact cardinals. Then (a) holds. c) Assume Martin’s Maximum. There is a definable wellordering of the reals. These conjectures are all implied by the HOD-Conjecture of .Woodin 2010b/ where a number of relevant results are proved. (17) Suppose that   P .R/ is a pointclass closed under continuous preimages such that L.; R/  ADC and let M D .HOD/L.;R/ . Suppose that a  !1 is a countable set such that M Œa  ./: Must .!1 /

M

< .!1 /

M Œa

?

(18) Assuming the existence of some large cardinal: a) Must there exist a semiproper partial order P such that V P  ./‹ b) Must there exist a semiproper partial order P such that V P  “ INS is !1 -dense ”‹ (19) Suppose that 1 and 2 are †2 sentences such that both ZFC C 1 and ZFC C 2 are each -consistent. Is ZFC C 1 C “V P  2 for some semiproper P ” -consistent? (20) Suppose that 0 is 1 huge. Suppose that G0  Coll.!; < 0 / is V -generic and that G1  .Coll.!1 ; < 1 //V ŒG0  is V ŒG0 -generic. Can ODR -Determinacy hold in L.P .!1 //V ŒG1  ?

844

11 Questions

(21) Suppose that 1 and 2 are …2 sentences (in the language for the structure hH.!2 /; INS ; 2i) such that both ZFC C CH C “hH.!2 /; INS ; 2i  1 ” and ZFC C CH C “hH.!2 /; INS ; 2i  2 ” are -consistent. Let  D .1 ^ 2 /. Is ZFC C CH C “hH.!2 /; INS ; 2i  ” -consistent? Solved by Aspero, Larson, and Moore in fall, 2009: The answer is no. (22) Can there exist a sentence ‰ such that for all †2 sentences, , either  ZFC C CH C ‰ ` “H.!2 /  ”, or  ZFC C CH C ‰ ` “H.!2 /  :”; and such that ZFC C CH C ‰ is -consistent? The reformulation with CH C ‰ replaced by either generic-˘ or ˘, is also open and discussed in Woodin .2003/. (23) (Conjecture) Assume there exists a proper class of Woodin cardinals. Let  be a †2 sentence. Then the following are equivalent. a) ZFC C  is -consistent. b) There exists a partial order P such that V P  . This is the  Conjecture. (24) Are the following mutually consistent? a) .ZF C DC/ There exists a cardinal such that for every cardinal , there exists an elementary embedding j WV !V with cp.j / D and j. / > . b) .ZF C DC/ For all x 2 R, ¹xº is OD if and only if for some A 2  1 , x is OD in L.A; R/. (25) Assume there exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below . Define ‚L.VC1 / to be; sup¹˛ 2 Ord j there exists a surjection, f W P . / ! ˛, with f 2 L.VC1 /º: Must

‚L.VC1 / < CC ‹

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Index

A-Bounded Martin’s Maximum, 798 A-Bounded Martin’s MaximumCC , 798 A(Code) .S; z; B/, 769 A-closed structure, 808 ADC Conjecture, 821 ADC , 611 A-iterable model, M , 73 A-iterable structure, hMk W k < !i, 224 Axiom of Strong Condensation, 499 Axiom ./, 184 Axiom ./C , 827 Axiom ./ CC , 827 Axiom  , 241 Œˇ˛ , 493 Œˇ 850

Index

I