Games, Scales and Suslin Cardinals: The Cabal Seminar Volume I (Lecture Notes in Logic, Volume 31)

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Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I The proceedings of the Los Angeles Caltech–UCLA “Cabal Seminar” were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes. Focusing on the subjects of “Games and Scales” (Part I) and “Suslin Cardinals, Partition Properties, Homogeneity” (Part II), each of the two sections is preceded by an introductory survey putting the papers into present context. This volume will be an invaluable reference for anyone interested in higher set theory.

Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the J. S. Guggenheim Memorial Foundation Fellowship and the Carol Karp Prize of the Association for Symbolic Logic, and is a member of the Scientific Research Board of the American Institute of Mathematics. Benedikt L¨owe is Universitair Docent in Logic and Scientific Director of the Graduate Programme in Logic at the Institute for Logic, Language and Computation of the Universiteit van Amsterdam. He is an editor of the Journal of Logic, Language and Information and managing editor of Tbilisi Mathematical Journal. He is a board member of the DVMLG and the EACSL. John R. Steel is Professor of Mathematics at the University of California, Berkeley. Prior to that, he was a professor in the mathematics department at UCLA. He is a recipient of the Carol Karp Prize of the Association for Symbolic Logic and of a Humboldt Prize. Steel is a former Fellow at the Wissenschaftskolleg zu Berlin and the Sloan Foundation.

LECTURE NOTES IN LOGIC

A Publication of The Association for Symbolic Logic This series serves researchers, teachers, and students in the field of symbolic logic, broadly interpreted. The aim of the series is to bring publications to the logic community with the least possible delay and to provide rapid dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board Anand Pillay, Managing Editor Department of Pure Mathematics, School of Mathematics, University of Leeds Jeremy Avigad Department of Philosophy, Carnegie Mellon University Lance Fortnow Department of Computer Science, University of Chicago Vladimir Kanovei Institute for Information Transmission Problems, Moscow Shaughan Lavine Department of Philosophy, The University of Arizona Steffen Lempp Department of Mathematics, University of Wisconsin See end of book for a list of the books in the series. More information can be found at http://www.aslonline.org/books-lnl.html.

LECTURE NOTES IN LOGIC 31

Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I Edited by

ALEXANDER S. KECHRIS California Institute of Technology

¨ BENEDIKT L OWE Universiteit van Amsterdam

JOHN R. STEEL University of California, Berkeley

association for symbolic logic

v

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521899512 © The Association for Symbolic Logic 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008

ISBN-13

978-0-511-45552-0

eBook (EBL)

ISBN-13

978-0-521-89951-2

hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

PART I: GAMES AND S C ALES John R. Steel Games and scales. Introduction to Part I. . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Alexander S. Kechris and Yiannis N. Moschovakis Notes on the theory of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Itay Neeman Propagation of the scale property using games. . . . . . . . . . . . . . . . . . . . . .

75

John R. Steel Scales on Σ11 -sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Yiannis N. Moschovakis Inductive scales on inductive sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Yiannis N. Moschovakis Scales on coinductive sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Donald A. Martin and John R. Steel The extent of scales in L(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Donald A. Martin The largest countable this, that, and the other . . . . . . . . . . . . . . . . . . . . . . 121 John R. Steel Scales in L(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 John R. Steel Scales in K(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Donald A. Martin The real game quantifier propagates scales . . . . . . . . . . . . . . . . . . . . . . . . . 209 John R. Steel Long games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 vii

viii

contents

John R. Steel The length-ù1 open game quantifier propagates scales . . . . . . . . . . . . . . 260 PART II: S US LIN C ARDINALS, PARTITION PROPERTIES, HOMOGENEITY Steve Jackson Suslin cardinals, partition properties, homogeneity. Introduction to Part II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Alexander S. Kechris Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin The axiom of determinacy, strong partition properties, and nonsingular measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Alexander S. Kechris and W. Hugh Woodin The equivalence of partition properties and determinacy . . . . . . . . . . . . 355 Alexander S. Kechris and W. Hugh Woodin Generic codes for uncountable ordinals, partition properties, and elementary embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Alexander S. Kechris A coding theorem for measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Donald A. Martin and John R. Steel The tree of a Moschovakis scale is homogeneous . . . . . . . . . . . . . . . . . . . 404 Donald A. Martin and W. Hugh Woodin Weakly homogeneous trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

PREFACE

This is the first of four volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. We have grouped the papers of the original Cabal Seminar volumes according to their topics. This volume contains the papers on “Games and Scales” (Part I) and “Suslin Cardinals, Partition Properties, Homogeneity” (Part II). Each of the parts contains an introductory survey (written by John Steel and Steve Jackson, respectively) putting the papers into a present-day context. Table 1 gives an overview of the papers in this volume with their original references. This volume must not be understood as a historical edition of old papers. In the 1980s, there were a number of results obtained by the researchers associated with the Cabal Seminar, some of which were intended for a fifth Cabal volume that was never published. We include some of these papers in this volume, together with papers reporting on new developments related to the research of the Cabal Seminar. These papers are Steel’s “Scales in K(R)” and “The Length-ù1 Open Game Quantifier Propagates Scales” in Part Iand “The Equivalence of Partition Properties and Determinacy” and “Generic Codes for Uncountable Ordinals, Partition Properties, and Elementary Embeddings” by Kechris and Woodin, “The Tree of a Moschovakis Scale is Homogeneous” by Martin and Steel, and “Weakly Homogeneous Trees” by Martin and Woodin in Part II. We also added a new expository paper, “Propagation of the Scale Property Using Games” by Neeman, and modernized and made uniform the notation and layout, and we have given the authors the opportunity to make corrections to their original papers. The new LATEX layout has resulted in changes in the numbering of sections and theorems. The typing and design were partially funded by NSF Grant DMS-0100745; Johan van Benthem’s Spinoza project Logic in Action; the Institute for Logic, Language and Computation; the Association for Symbolic Logic; and the DFG-NWO collaborative project “Determinacy and Combinatorics” (DFG: KO 1353/3-1; NWO: DN 61-532). A lot of people were involved in typing, ix

x

PREFACE

Steel

Part I Games and scales Introduction to Part I

new

Kechris, Moschovakis

Notes on the theory of scales

[Cabal i, pp. 1–53]

Neeman

Propagation of the scale property using games

new

Steel

Scales on Σ11 -sets

[Cabal iii, pp. 72–76]

Moschovakis

Inductive scales on inductive sets

[Cabal i, pp. 185–192]

Moschovakis

Scales on coinductive sets

[Cabal iii, pp. 77–85]

Martin, Steel

The extent of scales in L(R)

[Cabal iii, pp. 86–96]

Martin

The largest countable this, that, and the other

[Cabal iii, pp. 97–106]

Steel

Scales in L(R)

[Cabal iii, pp. 107–156]

Steel

Scales in K(R)

new

Martin

The real game quantifier propagates scales

[Cabal iii, pp. 157–171]

Steel

Long games

[Cabal iv, pp. 56–97]

Steel

The length-ù1 open game quantifier propagates scales

new

Jackson

Part II Suslin cardinals, partition properties, homogeneity Introduction to Part II

new

Kechris

Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy

[Cabal ii, pp. 127–146]

Kechris, Kleinberg, Moschovakis, Woodin

The axiom of determinacy, strong partition properties, and nonsingular measures

[Cabal ii, pp. 75–100]

Kechris, Woodin

The equivalence of partition properties and determinacy

new

Kechris, Woodin

Generic codes for uncountable ordinals, partition properties, and elementary embeddings

new

Kechris

A coding theorem for measures

[Cabal iv, pp. 103–109]

Martin, Steel

The tree of a Moschovakis scale is homogeneous

new

Martin, Woodin

Weakly homogeneous trees

new

Table 1.

PREFACE

xi

laying out, and proofreading the papers. We should like to thank (in alphabetic order) Edgar Andrade, Stefan Bold, Samson de Jager, Leona Kershaw, Tomasz Polacik, Doroth´ee Reuther, and Philipp Rohde for their important contribution. Very special thanks are due to Samson de Jager, who coordinated the typesetting effort in the final two years of the project. REFERENCES

Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.

The Editors Alexander S. Kechris, Pasadena, CA ¨ Benedikt Lowe, Amsterdam John R. Steel, Berkeley, CA

PART I: GAMES AND SCALES

GAMES AND SCALES INTRODUCTION TO PART I

JOHN R. STEEL

The construction and use of Suslin representations for sets of reals lies at the heart of descriptive set theory. Indeed, virtually every paper in descriptive set theory in the Cabal Seminar volumes deals with such representations in one way or another. Most of the papers in the section to follow focus on the construction of optimally definable Suslin representations via gametheoretic methods. In this introduction, we shall attempt to put those papers in a broader historical and mathematical context. We shall also give a short synopsis of the papers themselves, and describe some of the work done later to which they are related. §1. Some definitions and history. A tree on a set X is a subset of X 0, we must impose a definability restriction on our ä 12n+1 -Borel representation, since again, it could be that every set of reals is ùe1 + 1-Borel. One way to do that is to assume full AD, and Martin showed that indeed, assuming AD, every ∆12n+1 set is e ä 12n+1 -Borel. So we have e Theorem 3.3 (Martin, Moschovakis). Assume AD; then the ∆12n+1 sets of e reals are precisely the ä 12n+1 -Borel sets. e See [Mos80, 7D.9]. This fully generalizes Suslin’s 1917 theorem to the higher levels of the projective hierarchy. §5 and §9 introduce inner models, obtained from Suslin representations, which have certain degrees of correctness. In §5, it is shown that for n ≥ 2, there is a unique, minimal Σ1n -correct inner model Mn∗ containing all the ordinals; the model is obtained by closing under constructibility and an optimally definable Skolem function for Σ1n . (Kechris and Moschovakis call this model Mn —not to be confused with Mn ; see below.) §9 considers the model L[T ], where T is the tree of a Π12n+1 scale on a complete Π12n+1 set. These models have proved more important in later work than the Mn∗ . It is shown that if n = 0, then L[T ] = L; in particular, L[T ] is independent of the Π12n+1 scale and complete set chosen. Moschovakis conjectured that L[T ] is independent of these choices if n > 0 as well, and more vaguely, that it is a “correct higher level analog of L”. Becker’s paper [Bec78] contains an excellent summary of what was known in 1977 about the models of §5 and §9. The independence conjecture, which

12

JOHN R. STEEL

inspired a great deal of work, became the third Victoria Delfino problem. Harrington and Kechris [HK81] made a significant advance by showing that the reals of L[T ], where T is the tree of any Π12n+1 scale on a complete Π12n+1 set, are the largest countable Σ12n+2 set of reals, and hence independent of the choice of T . Building on this work, Moschovakis made a step forward in the late 70’s with the introduction of the model HΓ, for Γ a pointclass which resembles Π11 in a certain technical sense, and has the scale property. (See [Mos80, 8G.17 ff.].) Assuming ∆12n -determinacy, the pointclass Π12n+1 is an example of such a Γ, e many more examples. The model H is of the form L[U ], where but there are Γ U is a universal ∃R Γ (in the codes) subset of the prewellordering ordinal of Γ, and one can think of it as a fragment of HOD corresponding to Γdefinability. Using the Harrington-Kechris work, Moschovakis showed that HΓ is independent of the universal set and Γ-norm used to define U , that it includes L[T ], for the tree T of a Γ scale on a complete Γ set, and that R ∩ HΓ is the largest countable ∃R Γ set of reals. (See [Mos80, 8G.17 ff.].) Moschovakis’ results require a bit more than Γ-determinacy. The independence of L[T ] was finally provedeby Becker and Kechris [BK84], who showed Theorem 3.4 (Becker, Kechris 1984). Let Γ be a pointclass which resembles Π11 and has the scale property, and suppose AD holds in L(Γ, R). Let T be the tree of any Γ-scale on a complete Γ set; then L[T ] = HΓ . e The Becker-Kechris proof makes heavy use of a class of games introduced by Martin in order to obtain an approximation to Theorem 3.4. Not long after the last of the Cabal Seminar volumes appeared, our understanding of the large cardinal side of the “equivalence” between large cardinals and determinacy caught up with our understanding of the determinacy side. This equivalence is mediated by the canonical inner models for large cardinal hypotheses, which are sometimes called extender models. We can now identify each of the models of §5 and §9 as an extender model, and thereby understand it much more deeply than we could using only pure descriptive set theory. For example, most nontrivial facts in the first order theory of L[T ] (e.g., that the GCH, and Jensen’s diamond and square principles, hold in L[T ]) seem to require its identification as an extender model for proof. The identifications are as follows: Here and in the rest of the paper, for 0 ≤ n ≤ ù, we let Mn be the minimal iterable proper class extender model with n Woodin cardinals. If L[M |ã] n ≥ 2 is even, then Mn∗ is L[Mn−2 |ã], where ã is least such that ã = ù1 n−2 M and L[Mn−2 |ã] is Σ1n -correct. (For n > 2, we have that ã < ù1 n−2 .) If n is ∗ odd, then Mn is the minimal proper class extender model Q such that if S is an initial segment of Q projecting to ù, then Mn−2 (S)# is an initial segment of Q. These identifications are implicit in [Ste95B]. Finally, if n ≥ 3 is odd, and T is the tree of a Π1n scale on a complete Π1n set, then there is an iterate Q

INTRODUCTION TO PART I

13

of Mn−1 such that L[T ] = L[Q|ä 1n ]. This identification is implicit in [Ste95A], e where the parallel fact with the pointclass Πn replaced by ΣL(R) , and Mn−1 1 replaced by Mù , is proved. So we have Theorem 3.5 (Steel 1994). Assume there are ù Woodin cardinals with a measurable above them all, and let Γ = Π12n+1 or Γ = ΣL(R) ; then HΓ is an 1 iterable extender model. In a similar vein, the prewellordering and scale theorems of §2 - §4 can now be proved using extender models. In the prewellordering case, the proof is due to Woodin, and in the scale case, to Neeman; in neither case is the proof published, but see [Ste95B]. These proofs require significantly more theory than the comparsion game approach, but in some ways they give deeper insight into the meaning of the norms being constructed. Finally, Suslin and ∞-Borel representations are related to Lebesgue measurability, the Baire property, and the perfect set property in §10 and §11. Solovay’s breakthrough results from 1966 on the regularity of Σ12 sets under large cardinal hypotheses [Sol66] are thereby extended to other pointclasses. A basic result on the existence of largest countable sets is proved (in effect): Theorem 3.6 (Kechris, Moschovakis). Suppose Γ is adequate, ù-parametrized, has the scale property, and is closed under ∃R , and suppose all Γ games e are determined; then there is a largest countable Γ set of reals. When it exists, the largest countable Γ set is called CΓ . The theorem is implicit in the proof of Theorem 11B-2, which proves the existence of CΓ for Γ = Σ12n . Kechris’ paper [Kec75] contains further basic information in this area. The sets CΓ are quite important, partly because many of them show up naturally as the set of reals in some canoncal inner model. For example, Solovay showed that CΣ12 = R ∩ L [KM78B, 11B-1], and we now know that for any n, CΣ12n+2 = R ∩ M2n . (See [Ste95B]. Note that M0 = L.) In general, under the hypotheses of Theorem 3.4, we have C∃R Γ = R ∩ HΓ . (See [Mos80, 8G.29].) Kechris [Kec75] shows that assuming Π12n+1 - determinacy, there is a largest e countable Π12n+1 set of reals CΠ12n+1 . This result is due to Guaspari, Kechris, and Sacks for n = 0, in which case CΠ12n+1 has an inner-model-theoretic meaning as the set of reals ∆12n+1 -equivalent to the first order theory of some level of M2n projecting to ù. It is open whether this characterization of CΠ12n+1 holds also for n > 0. Propagation of the scale property using games [Nee07]. Scales on Σ11 sets [Ste83B]. e Moschovakis unified his results on scale propagation under the real quantifiers into a single theorem on the propagation of scales under the game

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JOHN R. STEEL

quantifier on ù. Letting A ⊆ R × R, we put ayA(x, y) ⇔ ∃n0 ∀n1 ∃n2 ∀n3 ...A(x, hni : i < ùi), where we interpret the right hand side as meaning its quantifier string has a Skolem function, that is, that player I wins the game on ù with payoff Ax = {y : A(x, y)}. We write aA for {x : ayA(x, y)}, and if Γ is a pointclass, we set aΓ = {aA : A ∈ Γ}. The following is often called the third periodicity theorem. It dates from approximately 1973; see [Mos73] or [Mos80, 6E]. Theorem 3.7 (Moschovakis). Let Γ be an adequate, ù-parameterized pointclass closed under quantification over ù, and suppose Γ(x)-determinacy holds for all reals x. Suppose Γ has the scale property; then (a) aΓ has the scale property, and (b) if G is a game on ù with payoff set in Γ, and the player whose payoff is Γ has a winning strategy in G, then that player has a aΓ winning strategy. The proof involves a more sophisticated comparison game: given a norm ϕ on A, one gets a norm on aA using comparison games in which the two players play out the games with payoff Ax1 and Ax2 simultaneously, in different roles on the two boards, each trying to win in his role as player I with lower ϕ-norm than the other. The first paper in the present pair gives a thorough exposition of the proof of this theorem. (See also [Mos80, 6E].) It is easy to see that aΠ1n = Σ1n+1 , and assuming Σ1n -determinacy, that e Thus Theorem 3.7 aΣ1n = Π1n+1 . Setting Σ10 = Σ01 , this is true for n = 0 as well. subsumes Theorem 3.1. Part (b) of Theorem 3.7, on the existence of canonical winning strategies, is very useful. In the special case of projective sets, we get Corollary 3.8 (Moschovakis). Assume ∆12n -determinacy, and let G be a e with Σ1 payoff has a winning game with Σ12n payoff, and suppose the player 2n strategy; then he has a ∆12n+1 winning strategy. Moschovakis’ proof used Σ12n -determinacy, but Martin later showed this e so we have stated the theorem in its sharper follows from ∆12n -determinacy, e form. Of course, we also get ∆12n+2 strategies for games won by a player with Π12n+1 payoff from Corollary 3.8, but this already follows easily from the basis theorem for Π12n+1 . It is easy to see that these definability bounds on winning strategies are optimal. It is natural to ask what are the optimally definable scales and winning strategies for the projective pointclasses which zig when they should have zagged, that is, for Σ12n+1 and Π12n+2 . The second paper in this pair gives part of the answer. Let α-Π1 be the α th level of the difference hierarchy over Π1 1

1

INTRODUCTION TO PART I

(see [Ste83B]). and let Λ0 =

[

15

ùk-Π11 .

k n, αi (n) = knx , thus αi → α x and hx, αi i → hx, α x i. Also for i > n, ϕn (x, αi ) = ëxn , hence by the limit property of scales, hx, α x i ∈ P and ϕn (x, α x ) ≤ ëxn . Then certainly α x ∈ P0x . But also α x (0) = k0x and ϕ0 (x, α x ) ≤ ëx0 , i.e., ϕ0 (x, α x ) = ëx0 . Thus α x ∈ P1x . A similar argument shows inductively that for all n, α x ∈ Pnx . ⊣ (3)

Put now P ∗ (x, α) ⇐⇒ ∃αP(x, α) ∧ α = α x . Clearly P ∗ ⊆ P and ∃αP(x, α) =⇒ ∃!αP ∗ (x, α). To complete the proof it will be enough to show that P ∗ ∈ Γ. It is eas˘ And this follows from the ier to show that the complement of P ∗ is in Γ. computation

NOTES ON THE THEORY OF SCALES

39

¬P ∗ (x, α) ⇐⇒ n n ¬P(x, α) ∨ P(x, α) ∧ ∃n∃â P(x, â)  ∧ (∀i < n) α(i) = â(i)
∧ ϕi (x, α) = ϕi (x, â) ∧ ϕn (x, â) < ϕn (x, α) ∨ (ϕn (x, â) = ϕn (x, oo α)
∧ â(n) < α(n)) ⇐⇒ n  ¬P(x, α) ∨ ∃n∃â ((∀i < n)(SΓ˘ (i, x, â, x, α) ∧ SΓ˘ (i, x, α, x, â) ∧ α(i) = â(i)) ∧ (SΓ˘ (n, x, â, x, α) ∧ ¬SΓ (n, x, α, x, â) ∨ (â(n) < α(n) ∧ SΓ˘ (n, x, â, x, α)

o ∧ SΓ˘ (n, x, α, x, â))) .

⊣ Again we should mention the trivial observation that if Γ is adequate, then Scale(Γ) implies Scale(Γ). e 3.2. Establishing the Scale property. Theorem 3.2. Scale(Π11 ). ˆ Corollary 3.3 (The classical Novikoff-Kondo-Addison Theorem). Unif(Π11 ). Proof of Theorem 3.2. Let A ⊆ X, A ∈ Π11 . Then for some recursive function f : X → R x ∈ A ⇐⇒ f(x) ∈ WO. For α ∈ WO put α↾n = {m : m ϕ1 (x, α ′ ) we are done, otherwise ϕ1 (xn , α) = ϕ1 (xni , αi ) =

46

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

ϕ1 (x, α ′ ) and then we look (if n ≥ 2) at ϕ2 , etc. In any case this shows that pϕ0 (xn , α),ϕ1 (xn , α), . . . , ϕn (xn , α)q ≥ pϕ0 (x, α ′ ), . . . , ϕn (x, α ′ )q, i.e., player II wins. Thus we have described a winning strategy for player II in Gn (x, xn ), so x ≤ n xn . ⊣ (Claim 3.18) ⊣ (Theorem 3.14) 3.4. The zig-zag picture. It follows from the results of this section that the pictures given in Section 2.4 for the Prewellordering property hold also for the Scale and Uniformization properties, i.e., under the stated hypotheses these properties hold for the circled pontclasses. That, assuming PD, they hold only for the circled pointclasses we will prove in the next section. 3.5. The Martin-Solovay Uniformization Theorem. From the results of this section it is obvious that Det(∆12 ) =⇒ Unif(Π12 , Π13 ). e However Martin and Solovay had obtained a similar theorem from weaker hypotheses before these results were proved, namely ∀α(α # exists) =⇒ Unif(Π12 , ∆14 ), see [MS69]; this in turn was strengthened by [Man71] to ∀α(α # exists) =⇒ Unif(Π12 , Π13 ). These proofs (and in fact the statements of the theorems) involve the theory of indiscernibles with which we are not concerned here. We will state one by-product of this work which will be useful later. Theorem 3.19. Assume that there exists a measurable cardinal or (the weaker hypothesis) that for each α, α # exists, let u1 = ℵ1 , u2 , u3 , . . . , uù be the first ù + 1 uniform indiscernibles. Then: 1. un ≤ ℵn , uù ≤ ℵù and cf(un+1 ) = cf(u2 ). 2. If AC holds, then uù < ℵ3 . 3. Every Π12 set A admits a Π13 scale on uù , i.e., a scale hϕn : n ∈ ùi with each ϕn : A → uù . ((1) and (2) are due to Solovay, (3) is implicit in [MS69, Man71]) §4. Bases. One of the most interesting corollaries of uniformization results is the computation of bases for pointclasses. If Γ is a pointclass and C a set of reals, put Basis(Γ, C ) ⇐⇒ for each A ∈ Γ, A ⊆ R, A 6= ∅ =⇒ A ∩ C 6= ∅. If Λ is a pointclass, we often abbreviate Basis(Γ, Λ) ⇐⇒ Basis(Γ, {α : the set {hn, mi : α(n) = m} ∈ Λ}).

NOTES ON THE THEORY OF SCALES

47

4.1. Computation of bases. It is immediately obvious that Unif(Γ, Γ′ ) implies Basis(Γ, {α : {α} ∈ Γ′ })

(4.1.1)

and it is easy to see that if Γ is adequate and Basis(Γ, C ) holds, then Basis(∃R Γ, {α : (∃â)(â ∈ C ∧ α is recursive in â)}). (4.1.2) From this and the results in §3 it is clear that

and

Det(∆12n ) implies Basis(Σ12n+2 , ∆12n+2 ) e

(4.1.3)

PD implies Basis(Σ12n , ∆12n ), n ≥ 2. On the other hand we have Theorem 4.4. Det(∆12n ) implies ¬ Basis(Σ12n+1 , ∆12n+1 ). e Proof. Since Det(∆12n ), we have PWO(Π12n+1 ). But this has a consequence that {α : α ∈ ∆12n+1 e } ∈ Π12n+1 . (This is announced in [AM68].) From this the result follows immediately. (For another proof see [MS69].) ⊣ The periodicity phenomenon is again clear in (4.1.3) and Theorem 4.4. Theorem 4.5 (Martin, Solovay, Mansfield, [MS69, Man71]). If α # exists for all reals α, then there exists a fixed Π13 singleton α0 (i.e., {α0 } ∈ Π13 ) such that Basis(Σ13 , {â : â is recursive in α0 }). Theorem 4.6 (Moschovakis, [Mos71A]). If Det(∆12n ) holds, then there exe ists a fixed Π12n+1 singleton α0 such that Basis(Σ12n+1 , {â : â is recursive in α0 }).

Proof. By (4.1.2) it will be enough to find a Π12n+1 singleton α0 such that every Π12n set contains a real recursive in α0 . Let B ⊆ ù × R be a universal Π12n set. Uniformize B by some B ∗ ∈ Π12n+1 . Then B ∗ ⊆ B and ∃αB(n, α) ⇐⇒ ∃!αB ∗ (n, α). Define B ∗∗ (n, α) ⇐⇒ B ∗ (n, α)∨(∀â(¬B(n, â))∧α = ët0}. Then B ∗∗ ∈ Π12n+1 and ∀n∃!αB ∗∗ (n, α). Put α ∈ C ⇐⇒ ∀nB ∗∗ (n, (α)n ). For this proof choose (α)n so that α is completely determined by {(α)n : n ∈ ù}. Thus C is a singleton and C ∈ Π12n+1 . If C = {α0 } we show that every Π12n set A contains a real recursive in α0 . In fact if A ∈ Π12n we have a ∈ A ⇐⇒ hn0 , αi ∈ B, for some n0 . Then A 6= ∅ =⇒ ∃α(hn0 , αi ∈ B), so hn0 , (α)n0 i ∈ B i.e., (α)n0 ∈ A. ⊣

48

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

A well known basis theorem says that for some fixed Σ11 subset of ù, say A0 , we have Basis(Σ11 , {α : α is recursive in A0 }). The following question is open: Does this generalize (under any reasonable hypothesis) to Σ12n+1 , n ≥ 1?1 4.2. Independence results. It is clear that the weakest basis result one can expect for a (“lightface”) pointclass Γ is Basis(Γ, {α : α is ordinal definable}). But even such a weak result is not provable in ZFC for Γ beyond Σ12 as the next theorem shows. Theorem 4.7 (L´evy, [L´ev66]). In ZFC alone we cannot prove Basis(Π12 , {α : α is ordinal definable}). Proof. It is enough to show that if M is a countable model of ZF+V=L and α is a real Cohen generic over M , then in N = M [α] there is a Π12 set containing no ordinal definable real. In fact, in N , consider the set A = {â : â 6∈ L}. Then A = Π12 and A 6= ∅. But A cannot contain an ordinal definable real, since all such reals belong already to M = LN (because the notion of forcing is homogeneous). ⊣ 1 Of course we have a basis theorem for Σ3 assuming, for example, that there exists a measurable cardinal (Theorem 4.6). Unfortunately we cannot go further even with this stronger hypothesis. Theorem 4.8. (L´evy’s method for Theorem 4.7 using a key result of Silver [Sil71].) In ZFC + “there exists a measurable cardinal”, we cannot prove Basis(Π13 , {α : α is ordinal definable}). Proof. Repeat the proof of 4.7, but now start with an M which is a countable model of ZF+V=L[ì], where ì is a normal measure on a cardinal κ. ⊣ §5. Partially playful universes. We outline here a construction which (granting PD) yields for each n ≥ 3 a model M n of ZF+AC such that M n |= Det(∆1n−1 ), e M n |= R admits a Σ1n+1 -good wellordering. In particular the zig-zag picture of §2.4 for the scale property in M n has only finitely many teeth, i.e., the scale property settles on the Σ side for k ≥ n + 1. The results are due to Moschovakis. 1 Martin and Solovay have shown in 1972 that this generalization is false for n ≥ 1, granting Det(∆12n ). They also show that in Theorem 4.6, α0 can be any Π12n+1 singleton which is not ∆12n+1 . e This turns out to be the correct generalization of the Kleene Basis Theorem for Σ11 . See [MS].

NOTES ON THE THEORY OF SCALES

49

Fix n ≥ 3, let k be the largest even integer less than n such that k = n − 1 or k = n − 2 and assume Det(∆1k ). By the Second Periodicity Theorem we have e Unif(Π1n−1 , Π1n ) whether n is odd or even, so let Pn−1 (m, α, â) be the standard Π1n−1 universal ∗ relation and let Pn−1 (m, α, â) be the Π1n relation that comes out of the proof of the Second Periodicity Theorem such that ∗ Pn−1 (m, α, â) =⇒ Pn−1 (m, α, â), ∗ (∃â)Pn−1 (m, α, â) =⇒ (∃!â)Pn−1 (m, α, â).

(∗) (∗∗)

Finally define Fn∗ (m, α)

=

  the unique â  

ët0

∗ such that Pn−1 (m, α, â), if (∃â)Pn−1 (m, α, â), if ∀â¬Pn−1 (m, α, â).

Clearly Fn∗ is a function whose graph is Π1n . Let M be a model of ZF, transitive and containing all ordinals (for brevity, standard model). We call M Σ1n -correct if for every Σ1n formula ϑ(α1 , . . . , αℓ ), α1 , . . . , αℓ ∈ M =⇒ (ϑ(α1 , . . . αℓ ) ⇐⇒ M |= ϑ(α1 , . . . , αℓ )). Lemma 5.1. Assume Det(∆1k ). A standard model M of ZF+DC is Σ1n -correct e F ∗ , i.e., if and only if M is closed under n α ∈ M =⇒ Fn∗ (m, α) ∈ M.

Proof. Assume first that M is Σ1n -correct. Notice that if for some α ∈ M and some m1 , m2 , ∀â(Pn−1 (m1 , α, â) ⇐⇒ ¬Pn−1 (m2 , α, â)), then the same equivalence holds in M (it is expressible by a Π1n formula); thus hm1 , m2 , αi codes a ∆1n−1 set in M if and only if it does in the world. This e k ≤ n − 1, and it is now easy to verify that applies to ∆1k sets, since e M |= Det(∆1k ). e Hence the Second Periodicity Theorem holds in M , so that (∗) and (∗∗) hold. Now if for some m, α ∈ M, (∃â)Pn−1 (m, α, â), then M |= (∃â)Pn−1 (m, α, â), ∗ ∗ hence for some â ∈ M, M |= Pn−1 (m, α, â), hence Pn−1 (m, α, â) in the world ∗ and â = Fn (m, α) ∈ M . To prove the converse, assume that M is closed under Fn∗ and then show by induction on i ≤ n that M is Σ1i -correct. This part of the proof does not need the assumption that M |= DC. We omit the details. ⊣

50

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

(Actually neither direction of the equivalence needs the assumption M |= DC, but the proof is a bit more complicated.) Define by induction on the ordinal î, M0n = ∅, n Mî+1 = Mîn ∪ {x ⊆ Mîn : x is definable in hMîn , ∈↾Mîn i}

∪ {Fn∗ (m, α) : α ∈ Mîn }, [ [ Mçn = Mîn if ç = ç > 0, î=< 89:; ?>=< Σ13 Σ12 Σ14 ??   ??   ??     89:; ?>=< 89:; ?>=< Π1 Π1 Π1 Π1 Σ11

1

2

3

4

?>=< 89:; Σ15

...

Π15

Another interesting model, M ù , can be obtained by closing under all = 3, 4, 5, . . . . This satisfies PD and has the same zig-zag picture as V (assuming PD of course), but in M ù , R admits a very simple (hyperanalytic) good wellordering. Kechris has shown by indiscernibility considerations that Fn∗ , n

M3 $ M4 $ M5 $ · · · and in fact for each n ≥ 3, there is an α ∈ M n+1 \M n . §6. Trees. We show here that the existence of a scale on a set A yields a representation for A in terms of a tree on ordinals which is very similar to the classical representation for Σ11 sets. This is the key to the applications of scales described in the remainder e of this paper. 6.1. Notation for trees. A tree on some set C is a set T of finite sequences from C such that if hc0 , c1 , . . . , ck i ∈ T ∧ i ≤ k, then hc0 , . . . , ci i ∈ T ; in particular every non-empty tree contains the empty sequence h i. A branch through (or of) a tree T on C is any function f ∈ ù C such that for all n, (hf(0), . . . , f(n)i ∈ T ). Put [T ] = the set of all branches through T and call T wellfounded if [T ] = ∅, i.e., if T has no infinite branches. The idea here is a bit clearer if we consider the relation ≻ of proper extension on finite sequences. hc0 , . . . , ck i ≻ hd0 , . . . , dℓ i ⇐⇒ k < ℓ ∧ c0 = d0 ∧ . . . ∧ ck = dk ; T is wellfounded if and only if ≺ ↾T has no infinite descending chains, i.e., if and only if ≺ ↾T is wellfounded. We can now assign an ordinal rank to every sequence of a wellfounded tree in the canonical way we do this for any wellfounded relation, |u|T = sup{|v|T + 1 : v ∈ T, u ≻ v} (where sup(∅) = 0) and define the rank of T , |T | = sup{|u|T : u ∈ T } = |h i|T .

52

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

By convention let also |u|T = −1, if u 6∈ T . We shall often look at the subtree of T starting from some sequence, Tu = {v : u av ∈ T }, where u av is concatenation of sequences. Most useful for us will be trees of pairs, i.e., trees on sets C = A × B— usually C = ù × κ for some ordinal κ. A typical member of a tree T on A × B is a sequence hha0 , b0 i, ha1 , b1 i, . . . , han , bn ii and a branch through T is a function f ∈ ù (A × B). It will be convenient to represent each branch f by the pair hg, hi, g ∈ ù A, h ∈ ù B which determines it, f(n) = hg(n), h(n)i. For each fixed g ∈

ù

A now, we can define a new tree T (g) on B by

T (g) = {hb0 , . . . , bn i : hhg(0), b0 i, . . . , hg(n), bn ii ∈ T }. In the typical case when T is a tree on ù × κ, for each α ∈ R we will have a tree on κ T (α) = {hî0 , . . . , în i : hhα(0), î0 i, . . . , hα(n), în ii ∈ T }; notice that the function α 7→ T (α) is continuous in a strong sense, i.e., ¯ + 1) =⇒ hî0 , . . . , în i ∈ T (â). hî0 , . . . , în i ∈ T (α) ∧ α(n ¯ + 1) = â(n 6.2. κ-scales and their trees. Let hϕn : n ∈ ùi be a scale on A; we call hϕn : n ∈ ùi a κ-scale, if every ϕn is a function on A into κ, i.e., if the length of each prewellordering ≤ϕn is ≤ κ. With each κ-scale hϕn : n ∈ ùi on A we define the associated tree T on ù × κ by T = {hhα(0), ϕ0 (α)i, hα(1), ϕ1 (α)i, . . . , hα(n), ϕn (α)ii : α ∈ A}. Theorem 6.1. Let A be a pointset, A ⊆ R, hϕn : n ∈ ùi a κ-scale on A, T the associated tree. Then α ∈ A ⇐⇒ T (α) is not wellfounded ⇐⇒ (∃f)(α, f) ∈ [T ]. (this is an idea implicit in many of the classical proofs.)

NOTES ON THE THEORY OF SCALES

53

Proof. If α ∈ A, then hϕ0 (α), ϕ1 (α), ϕ2 (α), . . .i is a branch through T (α). Conversely, suppose hî0 , î1 , î2 . . .i is a branch through T (α), i.e., for each n, hhα(0), î0 i, . . . , hα(n), în ii ∈ T ; by the definition of T , there must exist reals α0 , α1 , . . . in A, so that for each n, hhαn (0), ϕ0 (αn )i,hαn (1), ϕ1 (αn )i, . . . , hαn (n), ϕn (αn )ii = hhα(0), î0 i, hα(1), î1 i, . . . , hα(n), în ii. This implies immediately that limn αn = α and for m ≤ n, ϕm (αn ) = îm , so that by the basic property of scales α ∈ A. ⊣ Kechris has shown that a converse to Theorem 6.1 is true, namely: if α ∈ A ⇐⇒ T (α) is not wellfounded, where T is a tree on ù × κ, then A admits a κ ù -scale. This shows a connection between the notion of scale and some ideas of Mansfield in [Man70]. 6.3. Computing lengths of scales. Theorem 6.2. If A ⊆ X × R admits a κ-scale, κ ≥ ù, then ∃R A admits a κ ù -scale. Proof. See the proof of Theorem 3.6. Σ11

⊣

set admits an ù -scale. e Proof. Every closed set admits an ù-scale. Now define Theorem 6.3. Every

ù

⊣

ä 1n = sup{î : î is the length of a ∆1n prewellordering of R}. e e Classically it is known that ä 11 = ℵ1 . Clearly every Π12n+1 -norme on a set has length ≤ ä 12n+1 . Thus: e e Theorem 6.4. Assume Det(∆12n ). Then every Π12n+1 set admits a ä 12n+1 -scale e e e (by the Periodicity Theorem 3.2).

Corollary 6.5. (a) Every Π11 set admits a ℵ1 -scale. e (b) Every Σ12 set admits a ℵù 1 -scale. e Corollary 6.6. Assume Det(∆12n ). Then every Σ12n+2 set admits a (ä 12n+1 )ù e e e scale. From §3.5 we also have

Theorem 6.7 (Martin, Solovay, [MS69]). If α # exists for all reals α, then every Π12 set admits a uù -scale. e Corollary 6.8 (Martin, [Mar70B]). If α # exists for all reals α, then every Σ13 set admits a (uù )ù -scale. If we also assume AC, then every Σ13 set admits a e e κ-scale, with κ < ℵ3 .

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The reader should have noticed that in this section a considerable change in our attitude towards scales has happened. We started worrying not only about definability of a scale but also about its length. Later sections will show why. §7. Computing lengths of wellfounded relations. 7.1. The Kunen-Martin theorem. Recall that for a wellfounded relation < we put for x ∈ Field( â øn (α, â) = pα(0), â(0), ϕ0 (α, â), . . . , α(n), â(n), ϕn (α, â)q. Then notice the following limit property of høn : n ∈ ùi: If αi > âi for all i, and for each n, øn (αi , âi ) is eventually constant, then limi (αi , âi ) = (α, â) exists and α > â.

55

NOTES ON THE THEORY OF SCALES

Define now f by induction on the length of sequences: f(h i) = h i f(hα0 i) = h i f(hα0 , α1 i) = hø0 (α0 , α1 )i f(hα0 , α1 , α2 i) = hø0 (α0 , α1 ), ø1 (α0 , α1 ), ø1 (α1 , α2 ), ø0 (α1 , α2 )i, and in general f(hα0 , . . . , αn−1 , αn i) = f(hα0 , . . . , αn−1 i) a

høn−1 (α0 , α1 ), øn−1 (α1 , α2 ), . . . , øn−1 (αn−1 , αn ), øn−2 (αn−1 , αn ), . . . , ø0 (αn−1 , αn )i.

The idea is to include in f(hα0 , . . . , αn i) all øj (αi , αi+1 ) for i ≤ n − 1, j ≤ n − 1. The diagram below explains the way we have done it: ø0

ø1

ø2

ø3

...

hα0 , α1 i ◦

◦

◦

◦

...

hα1 , α2 i ◦

◦

◦

◦

...

hα2 , α3 i ◦

◦

◦

◦

...

hα3 , α4 i ◦

◦

◦

◦

...

.. .

.. .

.. .

.. .

.. .

Clearly f is an ≺-preserving (on T − {h i}) map from T onto a set of finite sequences S on κ ù = ë. Thus it will be enough to show that ≺ ↾S is wellfounded. Assume not, towards a contradiction. Then we have f(hα00 i) ≻ f(hα01 , α11 i) ≻ f(hα02 , α12 , α22 i) ≻ · · · for some α00 , α01 , α11 , . . . , such that α01 > α11 , α02 > α12 > α22 etc. Then in the diagram ø0 (α01 , α11 ) ø0 (α02 , α12 ) ø1 (α02 , α12 )

ø1 (α12 , α22 ) ø0 (α12 , α22 )

ø0 (α03 , α13 ) ø1 (α03 , α13 )

ø1 (α13 , α23 ) ø0 (α13 , α23 )

.. .

.. .

.. .

.. .

ø2 (α03 , α13 ) . . . .. .

56

ALEXANDER S. KECHRIS AND YIANNIS N. MOSCHOVAKIS

each column consists of identical ordinals, thus for each n and for each i j, øn (αji , αj+1 ) becomes constant for large enough i. Thus for each j, i i hαj , αj+1 i → hαj , αj+1 i and αj > αj+1 , i.e., α0 > α1 > α2 > · · · , a contradiction. ⊣ Corollary 7.2. Every Σ11 wellfounded relation has length < ℵ1 (classical e result).

Corollary 7.3. Every Σ12 wellfounded relation has length < ℵ2 . Thus if e of R, the Continuum Hypothesis holds. (Martin; there exists a Σ12 wellordering e by an unpublished forcing argument before scales were introduced.)

Corollary 7.4. Assume ∀α(α # exists). Then every Σ13 wellfounded relae tion has length < (uù )+ . If we also assume AC, then every Σ13 wellfounded e by Martin, 1 + relation has length < ℵ3 . (That ä 3 ≤ (uù ) was already shown e [Mar70B].)

7.2. Projective ordinals. We introduced in §6 the projective ordinals ä 1n and e first we mentioned that ä 11 = ℵ1 (this follows also independently from our e corollary in 7.1). By the results in § 7.1, it is then clear that

ä 12 ≤ ℵ2 (Martin) e ∀α(α # exists) + AC =⇒ ä 13 ≤ ℵ3 (Martin) e Det(∆12n ) =⇒ ä 12n+2 ≤ (ä 12n+1 )+ (Kunen, Martin) e e e (To prove 7.2.3 recall Theorem 6.4.)

(7.2.1)

(7.2.2) (7.2.3)

Det(∆12 ) + AC =⇒ ä 14 ≤ ℵ4 (Kunen, Martin) (7.2.4) e e Open Problem 7.5. Is it true that assuming AC (and any other reasonable hypotheses), ä 1n ≤ ℵn , n ≥ 5? e We shall mention some other known results about the projective ordinals in the last section. §8. Construction principles. A construction principle for a pointclass Γ asserts, roughly speaking, that every set in Γ can be expressed, in some canonical way, in terms of sets in a simpler pointclass Γ′ . A classical example is the result that every analytic (Σ11 ) set can be expressed both as a union and an e intersection of ℵ1 Borel sets. 8.1. Inductive analysis of projection of trees. Let T be a tree on ù × κ. We write A = p[T ] iff α ∈ A ⇐⇒ ∃f(hα, fi ∈ [T ]) ⇐⇒ T (α) is not wellfounded.

NOTES ON THE THEORY OF SCALES

57

Theorem 8.1. Let T be a tree on ù × κ and A = p[T ]. Put for 1 ≤ î < κ + and u a finite sequence from κ, Aîu = {α : |T (α)u | < î}, where for any tree J we abbreviate |J | < î ⇐⇒ J is wellfounded and |J | < î. Then, if lh(u) = n we have A0u = {α : hhα(0), u0 i, . . . , hα(n), un−1 ii 6∈ T } \ î Aî+1 = Aîu ∪ Auaç u ç0

î 0, cf(ℵn ) > ù, thus if T (α) is not wellfounded there is a î < ℵn such that T î (α) is not wellfounded, where T î = {hhk0 , î0 i, . . . , hkm , îm ii ∈ T : î0 , . . . , îm ≤ î}. Thus α ∈ A ⇐⇒ ∃în < ℵn [T în (α) is not wellfounded]. If în < ℵn we can replace T în by an isomorphic tree on ù × ℵn−1 , say T1în . Then α ∈ A ⇐⇒ (∃în < ℵn )(T1în (α) is not wellfounded). If n − 1 > 0 we have again T1în (α) is not wellfounded ⇐⇒ (∃în−1 < ℵn−1 )((T1în )în−1 is not wellfounded) and we proceed similarly. After at most n steps we get α ∈ A ⇐⇒∃în < ℵn ∃în−1 < ℵn−1 . . . ∃î1 < ℵ1 (T în ,...,î1 (α) is not wellfounded) where T în ,...,î1 is a tree on ù×ù. But then {α : T în ,...,î1 (α) is not wellfounded} is a Σ11 set, thus it is the union of ℵ1 Borel sets and the proof is complete. ⊣ e Open Problem 8.8. Prove assuming AC (and any other reasonable hypotheses) that every Σ1n+1 set is the union of ℵn Borel sets, n ≥ 4. Notice that a e solution to the problem at the end of §7 solves this problem too. §9. Constructibility in the tree associated with a scale. In §6 we associated with each κ-scale hϕn : n ∈ ùi on a set A ⊆ R a tree T on ù × κ. We introduce and study here the models L[T ], where T comes from a complete Π12n+1 set, granting Det(∆12n )—these are the basic tools for the results in the e theorem in the present section is that the tree that next two sections. The key comes from a complete Π11 set is in fact constructible. 9.1. The models L[T 2n+1 ]. Theorem 9.1 (Folk-type result). Let T be the tree associated with a κ-scale on some set A, let Q ⊆ R × R and assume that for some recursive f : R × R → R, Q(α, â) ⇐⇒ f(α, â) ∈ A.

NOTES ON THE THEORY OF SCALES

61

Then for some tree S ∈ L[T ] on ù × κ ∃âQ(α, â) ⇐⇒ S(α) is not wellfounded. ⇐⇒ (∃â ∈ L[T, α])Q(α, â)

(1) (2)

Proof. We have ∃âQ(α, â) ⇐⇒ ∃â(f(α, â) ∈ A) ⇐⇒ ∃â∃ã(f(α, â) = ã ∧ ã ∈ A) ¯ ⇐⇒ ∃â∃ã(∀nR(α(n), ¯ â(n), ã(n)) ¯ ∧ T (ã) is not wellfounded) where R is recursive. Put S ′ = {hha0 , hb0 , c0 , î0 ii, . . . , hak , hbk , ck , îk iii : hhc0 , î0 i, . . . , hck , îk ii ∈ T ∧ R(pa0 , . . . , akq, pb0 , . . . , bkq, pc0 , . . . , ckq)}. Then S ′ is a tree on ù × (ù × ù × κ), S ′ ∈ L[T ] and ∃âQ(α, â) ⇐⇒ S ′ (α) is not wellfounded ⇐⇒ ∃â ∈ L[T, α]Q(α, â) where the last equivalence follows from the usual absoluteness of wellfoundedness. To get instead of S ′ a tree on ù × κ fix a 1-1 mapping from ù × ù × κ onto κ in L[T ] and replace S ′ by an isomorphic tree on ù × κ, call it S. ⊣ Fix now a complete Π12n+1 set P2n+1 ⊆ R. (P2n+1 is such that for every A ⊆ X, A ∈ Π12n+1 we can find f : X → R recursive such that x ∈ A ⇐⇒ f(x) ∈ P2n+1 .) Assuming Det(∆12n ), let hϕn : n ∈ ùi be a fixed for the e let T 2n+1 be the associated tree. T 2n+1 is discussion Π12n+1 scale on P2n+1 and 1 a tree on ù × ä 2n+1 . e Theorem 9.2. Assume Det(∆12n ). Then for every Σ12n+2 set A we can find a tree S ∈ L[T 2n+1 ] such that e α ∈ A ⇐⇒ S(α) is not wellfounded.

Theorem 9.3. Assume Det(∆12n ). Then Σ12n+2 formulas are absolute for e L[T 2n+1 ].

Proof. We show successively that for 2n + 2 ≥ k ≥ 2, Σ1k formulas are absolute for L[T 2n+1 ]. For k = 2 this is Shoenfield’s theorem, while for k ≥ 3 we proceed using (3) and (4) of Theorem 9.1. ⊣ Unfortunately, except for the case n = 0, which we shall study in the rest of this section, there is practically nothing known about the internal structure of L[T 2n+1 ].2

2 It has been recently shown by Harrington and Kechris that for all n ≥ 0, R ∩ L[T 2n+1 ] is the largest countable Σ12n+2 set of reals (as conjectured by Moschovakis) and that additionally,

L[T 2n+1 ] |= “R has a ∆12n+2 -good wellordering”.

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9.2. Absoluteness of closed games. Let S be a set of even finite sequences from a set A. We define the game GS as follows: I a0 a1 .. .

II b0 b1 .. .

I plays a0 , a1 , . . . and II plays b0 , b1 . . . , ai , bi ∈ A. Then I wins iff for some n, ha0 , b0 , . . . , an , bn i ∈ S. Clearly the game is open in I.

The following is a folk-type result. Theorem 9.4. Let M |= ZF+DC and M ⊇ Ord. Let A, S ∈ M , and assume A is wellorderable in M . Then player I has a winning strategy in GS iff M |= player I has a winning strategy in GS and similarly for player II. Moreover the player who has a winning strategy has a winning strategy (for the game in the world) which lies in M . Proof. For each ha0 , b0 , . . . , an , bn i consider the subgame GS (a0 , b0 , . . . , an , bn ) defined by; I α

II â

I plays α, II plays â and I wins iff for some m ha0 , b0 , . . . , an , bn iahα(0), â(0), . . . , α(m), â(m)i ∈ S.

Then define S0 = S S î = {ha0 , b0 , . . . , an , bn i : ∃an+1 ∈ A∀bn+1 ∈ A ∃ç < î(ha0 , b0 , . . . , an+1 , bn+1 i ∈ S ç ). Then for each î, ha0 , b0 , . . . , an , bn i ∈ S î =⇒ player I has a winning strategy in GS (a0 , b0 , . . . , an , bn ). Using this we show: Claim 9.5. Player II has a winning strategy in G if and only if ∀î(h i 6∈ S î ). Proof of Claim 9.5. If player II has a winning strategy in GS = GS (h i), then player I has no winning strategy in GS (h i), thus for all î, h i 6∈ S î . Conversely assume that for each î, h i 6∈ S î . We describe a winning strategy for player II in GS as follows: If player I plays a0 , player II plays the least b0 (in a fixed wellordering of A) such that ∀î(ha0 , b0 i 6∈ S î ). Such a b0 exists, because otherwise for all b, there exists a î such that ha0 , bi ∈ S î . Let g(b) = least such î and find î0 > all g(b), b ∈ A. Then ∀b∃î < î0 (a0 , b) ∈ S î , thus h i ∈ S 0 , a contradiction. Similarly if player I plays a1 , player II picks the least b1 such that ∀î(ha0 , b0 , a1 , b1 i 6∈ S î ), etc. ⊣ (Claim 9.5) These results suggest that L[T 2n+1 ] is a correct higher level analogue of L. Their proof uses determinacy of all hyperprojective sets. See [HK77].

NOTES ON THE THEORY OF SCALES

63

Since the above equivalence was proved under the assumption “ZF+DC+A is wellorderable” and since î 7→ S î is clearly an absolute map and M ⊇ Ord, it is immediate the “player II has a winning strategy” is absolute for M , thus the same is true for “player I has a winning strategy.” Moreover the argument above clearly provides a winning strategy for player II which lies in M and wins in the world, thus it will be enough in order to complete the proof to show that when player I has a winning strategy we can find one (who wins in the world also) in M . Notice that Player I has a winning strategy ⇐⇒ ∃î(h i ∈ g S ) and check that the following is a winning strategy for player I which lies in M . Put î0 = least î such that h i ∈ S î . If î0 = 0, player I has already won. If î0 > 0, let player I play the least a0 such that for every b, ∃î < î0 (ha0 , bi ∈ S î ). If now player II plays b0 , let î1 = least î < î0 such that ha0 , b0 i ∈ S î . If î1 = 0, player I has already won, otherwise let player I play the least a1 such that for all b, ∃î < î1 (ha0 , b0 , a1 , bi ∈ S î ) etc. (Notice that î0 > î1 > · · · , so this cannot go on.) ⊣ (Theorem 9.4) 9.3. Proof that T 1 ∈ L. Suppose A ⊆ ℵ1 . We let Code(A) = {α ∈ WO : |α| ∈ A}. Similarly, if T is a tree on ù × ℵ1 , we let Code(T ) = {hk0 , α0 , . . . , kn , αn i : hhk0 , |α0 |i, . . . , hkn , |αn |ii ∈ T }. We say that Code(T ) is in Γ iff {pk0 , α0 , . . . , kn , αnq : hk0 , α0 , . . . , αn i ∈ Code(T )} ∈ Γ where pk0 , α0 , . . . , kn , αnq = hn, k0 , . . . , kn , α0 (0), . . . , αn (0), α0 (1), . . . , αn (1), . . .i ∈ R. Lemma 9.6 (Kechris). Let A ⊆ R, A ∈ Π11 and assume hϕn : n ∈ ùi is a on A. Then the tree T associated with hϕn : n ∈ ùi is Σ12 in the codes.

Π11 -scale

Proof. We have hk0 , α0 , . . . , kn , αn i ∈ Code(T ) ⇐⇒ α0 , . . . , αn ∈ WO ∧(∃α)(α ∈ A ∧ ϕ0 (α) = |α0 | ∧ α(0) = k0 ∧ . . . ∧ ϕn (α) = |αn | ∧ α(n) = kn ). The result follows immediately if we can show that for each n, α ∈ A ∧ â ∈ WO ∧ ϕn (α) = |â|

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is a Σ12 relation in α, â, n. But each ϕn is a Π11 -norm, thus every initial segment of ≤ϕn =≤n will have countable length (since ä 11 = ℵ1 ). From this we have e α ∈ A ∧ â ∈ WO ∧ ϕn (α) = |â| ⇐⇒ α ∈ A ∧ â ∈ WO ∧ ∃ä((∀m)(m ≤â m =⇒ (ä)m ≤n α) ∧ (∀ã)(ã ≤n α =⇒ (∃m)(m ≤â m ∧ (ä)m ≤n α ∧ α ≤n (ä)m ) ∧ ∀m, ℓ(m