The Determinacy of Long Games (De Gruyter Series in Logic and Its Applications)

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The Determinacy of Long Games

Itay Neeman

Walter de Gruyter

de Gruyter Series in Logic and Its Applications 7 Editors: W. A. Hodges (London) R. Jensen (Berlin) M. Magidor (Jerusalem)

Itay Neeman

The Determinacy of Long Games

≥

Walter de Gruyter Berlin · New York

Author Itay Neeman Mathematics Department University of California Box 951555, 6334 MSB LOS ANGELES, CA 90095-1555 USA e-mail: [email protected] Series Editors Wilfrid A. Hodges School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London E1 4NS United Kingdom

Ronald Jensen Institut für Mathematik Humboldt-Universität Unter den Linden 6 10099 Berlin Germany

Menachem Magidor Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem Israel Mathematics Subject Classification 2000: 03E60, 03E55, 03E47 Keywords: determinacy, large cardinals, infinte games

P Printed on acid-free paper which falls within the guidelines E of the ANSI to ensure permanence and durability

Library of Congress Cataloging-in-Publication Data Neeman, Itay, 1972 The determinacy of long games / by Itay Neeman. p. cm. (De Gruyter series in logic and its applications; 7) Includes bibliographical references and index. ISBN 3-11-018341-2 (cloth : alk. paper) 1. Game theory. 2. Determinants. 3. Logic, Symbolic and mathematical. I. Title. II. Series. QA269.N44 2004 519.3dc22 2004021609

ISBN 3-11-018341-2 ISSN 1438-1893 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting using the Authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Cover design: Rainer Engel, Berlin.

To my parents, Avraham and Bruria

Preface This book presents my research into proofs of determinacy for games of countable length. (Further developments on long games, dealing with games of length ω1 , are presented in Neeman [33].) It has been a while since I started studying this topic; some of the methods in Chapters 2 and 3, and certainly my interest in the subject, go all the way back to my Ph.D. dissertation [30]. I am grateful to Tony Martin, John Steel, and Hugh Woodin, first for bringing the study of interactions between determinacy and Woodin cardinals into existence, and second for helping me join it. I am also grateful to Ronald Jensen, for his work on fine structure which is crucial to other parts of my research, and, in the context of this book, for introducing me both to large cardinals and to proofs of determinacy. My research was supported by several organizations: the University of California in Los Angeles; the Society of Fellows at Harvard University; the Alexander von Humboldt Foundation; and the National Science Foundation through grants DMS 98-03292 and DMS 00-94174 (CAREER). I thank them sincerely. There is some background on determinacy, and a synopsis of the current work, in the introduction to this book. Let me here say a few words on the book’s structure. The basic components for proofs of determinacy of long games are presented in Chapter 1. Most important among them are the auxiliary games map associated to a given name, and the concept of a pivot. Neeman [34] gives an informal view of these concepts, and it may be useful as a starting point. (The definitions in Chapter 1 are for the map A discussed in Section 5.1 of [34]. The map discussed in Section 2 of the paper is essentially a special case.) Chapter 2 presents the first application, to games of fixed countable length, and Chapter 3 presents a more elaborate application, to games of continuously coded length. Neeman [34] gives special cases of these applications, and again it may be a useful starting point. Chapters 4, 5, and 6 develop tools for handling much longer games. These tools are applied in Chapter 7 to games ending at ω1 in L of the play. The chapter is written so as to use only the end results in Chapters 5 and 6, and the relevant concepts. These results and most of the concepts appear in Sections 4A, 4B, 4D (4), 4E (7), 5G, 6A, and 6G. In a first reading it may be useful to skim through Chapters 4, 5, and 6, concentrating on these particular sections, and then continue to Chapter 7. As far as prerequisites go, the work here should be accessible to any reader with a knowledge of basic set theory, say from Jech [10] or Kunen [15], some familiarity with Silver indiscernibles, say from Jech [10, §18] or Kanamori [11, §9], and some familiarity with extenders and iteration trees, say from Martin–Steel [18], [19]. For the background knowledge of iteration trees the reader may also consult Appendix A, but the exposition there is very brief. Let me now leave you with the book. I hope you find it both useful and pleasant. Los Angeles, California, October 2004

Itay Neeman

Contents

Preface

vii

Introduction 1 Basic components 1A The auxiliary games map . . . 1A (1) Types . . . . . . . . . 1A (2) The rules of the game . 1B Generic runs . . . . . . . . . . 1C Pivots . . . . . . . . . . . . . 1C (1) The game Apiv [x] . . 1C (2) Constructing σpiv [, x] 1C (3) Properties of σpiv . . . 1D Mirror images . . . . . . . . . 1E Sample application . . . . . . 1F Mixed pivots . . . . . . . . . 1F (1) The game . . . . . . . 1F (2) The strategy σmix [, x] 1F (3) Properties of σmix . . . 2

3

1

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15 16 16 20 23 27 28 30 35 37 39 43 43 48 49

Games of fixed countable length 2A General games and iteration games 2B Limits . . . . . . . . . . . . . . . 2B (1) The basic definitions . . . 2B (2) I wins . . . . . . . . . . . 2B (3) II wins . . . . . . . . . . 2B (4) The third case . . . . . . . 2B (5) Summary . . . . . . . . . 2C Successors . . . . . . . . . . . . . 2D Limits again . . . . . . . . . . . . 2D (1) II wins . . . . . . . . . . 2D (2) I wins . . . . . . . . . . . 2D (3) Determinacy . . . . . . . 2E Universally Baire sets . . . . . . .

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51 51 56 57 59 65 66 68 68 73 75 79 80 82

Games of continuously coded length 3A Codes . . . . . . . . . . . . . . 3B First determinacy result, part I . 3B (1) Names . . . . . . . . . 3B (2) The basic step . . . . . .

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x

Contents

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96 104 106 109 109 112 119 120 124 124 129

Pullbacks 4A Codes . . . . . . . . . . . . . . . . . . . 4B Woodin’s extender algebra . . . . . . . . 4C Names, part I . . . . . . . . . . . . . . . 4C (1) Removing obstructions . . . . . . 4C (2) Relative successors . . . . . . . . 4C (3) Relative limits and records . . . . 4D Names, part II . . . . . . . . . . . . . . . 4D (1) Woodin limits of Woodin cardinals 4D (2) Relative successors . . . . . . . . 4D (3) Compositions . . . . . . . . . . . 4D (4) Summary . . . . . . . . . . . . . 4E Mirror images . . . . . . . . . . . . . . . 4E (1) Removing obstructions . . . . . . 4E (2) Mirrored successor game . . . . . 4E (3) Relative limits and records . . . . 4E (4) Woodin limits of Woodin cardinals 4E (5) Relative successors . . . . . . . . 4E (6) Compositions . . . . . . . . . . . 4E (7) Summary . . . . . . . . . . . . .

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175 175 181 183 183 187 194 194 197 199 201

3C

3D

3E 4

3B (3) I wins . . . . . . . . . . . . . . 3B (4) Discussion . . . . . . . . . . . 3B (5) Internal ultrapowers vs. copying First determinacy result, part II . . . . . 3C (1) II wins . . . . . . . . . . . . . 3C (2) Determinacy . . . . . . . . . . A slight improvement . . . . . . . . . . 3D (1) 02 functions . . . . . . . . . . 3D (2) 01+α functions . . . . . . . . . Variation . . . . . . . . . . . . . . . . . 3E (1) A sketch of the proof . . . . . .

5 When both players lose 5A Saturation . . . . . . . . . . . . . 5B Successors, basic step . . . . . . . 5C Relative limits . . . . . . . . . . . 5C (1) Basic step . . . . . . . . . 5C (2) Construction . . . . . . . 5D Woodin limits of Woodin cardinals 5D (1) Basic step . . . . . . . . . 5D (2) Impossibility . . . . . . . 5D (3) Another impossibility . . . 5D (4) Construction . . . . . . .

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xi

Contents

5E 5F 5G

Relative successors . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6 Along a single branch 6A The game . . . . . . . . . . . . . . . 6A (1) How to leap . . . . . . . . . . 6A (2) The skipping game . . . . . . 6B Successors, basic step . . . . . . . . . 6C Relative limits . . . . . . . . . . . . . 6C (1) Finite . . . . . . . . . . . . . 6C (2) Infinite . . . . . . . . . . . . 6C (3) Construction . . . . . . . . . 6D Woodin limits of Woodin cardinals . . 6D (1) Hopeful . . . . . . . . . . . . 6D (2) Not hopeful . . . . . . . . . . 6D (3) Combined . . . . . . . . . . . 6E Relative successors and compositions 6F Skips . . . . . . . . . . . . . . . . . 6F (1) Routes . . . . . . . . . . . . 6F (2) Extensions . . . . . . . . . . 6F (3) Closing skips . . . . . . . . . 6F (4) Discussion . . . . . . . . . . 6F (5) Safe positions . . . . . . . . . 6F (6) The strategy . . . . . . . . . . 6G Conclusion . . . . . . . . . . . . . . 7

Games which reach local cardinals 7A Shifted payoff . . . . . . . . . 7B Layout . . . . . . . . . . . . . 7B (1) Extensions . . . . . . 7C Basic step . . . . . . . . . . . 7C (1) Obstruction free . . . 7C (2) I-acceptably obstructed 7C (3) Summary . . . . . . . 7D Construction . . . . . . . . . . 7E The main theorem . . . . . . . 7F Determinacy . . . . . . . . . .

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268 268 270 276 277 280 281 284 285 289 292

A Extenders, generic extensions, and iterability

301

Bibliography

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Index

313

Introduction

The determinacy of infinite games has been central to the development of modern set theory. From its humble, anecdotal beginnings in the 1930s and ’50s, the subject of determinacy has grown to provide the dividing line between the realm of definable sets of reals and the realm of the axiom of choice. It has had conceptual influences on the study of forcing, and substantial concrete influences on the study of large cardinals. The basic definitions are quite simple. Let C ⊂ ωω , that is let C be a set of infinite sequences of natural numbers. Define Gω (C), the length ω game with payoff set C, to be played as follows: Players I and II collaborate to produce an infinite sequence x = $x(i) | i < ω% of natural numbers. They take turns as in Diagram 1, I picking x(i) for even i and II picking x(i) for odd i. If at the end the sequence x they produce belongs to C then player I wins; and otherwise player II wins. The game Gω (C), or any other game for that matter, is determined if one of the two players has a winning strategy, namely a strategy for the game that wins against all possible plays by the opponent. The set C is said to be determined if the corresponding game Gω (C) is determined. I II

x(0)

x(2) x(1)

...... x(3)

......

Diagram 1. The game Gω (C).

For illustrative purposes it is helpful to express determinacy by means of logical quantifiers. The statement that player I has a winning strategy in Gω (C) is the natural interpretation of the quantifier string: ∃ x(0) ∀ x(1) ∃ x(2) ∀ x(3) . . . . . . . . .

$x(i) | i < ω% ∈ C

(1)

where the quantifiers range over natural numbers. The statement that II has a winning strategy on the other hand is the natural interpretation of the string: ∀ x(0) ∃ x(1) ∀ x(2) ∃ x(3) . . . . . . . . .

$x(i) | i < ω% ∈ C

(2)

where again the quantifiers range over natural numbers. Determinacy then is the statement that one of the equations holds, in other words that equation (2) is the negation of equation (1). This would simply be a matter of logic if the quantifier strings were finite. But the strings here are of length ω, and determinacy is far from trivial. Indeed, one cannot expect determinacy to hold for all sets: using the axiom of choice, more specifically using a wellordering of the real line, a straightforward construction produces a non-determined set C. Surprisingly, it has turned out that one can expect determinacy to hold for “concrete” sets of reals, meaning sets which are definable, over the real line and in fact more.

2

Introduction

Following standard usage let ω δ.) We call u the κ-type of x0 , . . . , xn−1 in Vη (relative to δ) if u is the unique (κ, n)-type which is realized by x0 , . . . , xn−1 in Vη . We work “relative to δ” throughout this subsection, whether this is mentioned explicitly or not. We say that a (κ, n)-type u is realizable if it is realized by some x0 , . . . , xn−1 in some Vη . Note that if u is realized by x0 , . . . , xn−1 in Vη , then projm τ (u) is realized by x0 , . . . , xm−1 in Vη . Remark 1A.4. Our definition of types here differs from that of [32] in two ways. First, we allow constants from Vκ ∪ {κ} in κ-types, rather than just constants from Vκ . This will save us from passing to κ + ω-types later on. Secondly, we build the parameter δ into the definition through the constant δ. Definition 1A.5. If the formula “there exists a largest ordinal,” and the formula “ κ, δ, v0 , . . . , vn−1 ∈ Vν , where ν is the largest ordinal” are both elements of the (κ, n)-type u we define u− = { φ( δ, c0 , . . . , ck , v0 , . . . , vn−1 ) | k ∈ N, c0 , . . . , ck ∈ Vκ ∪ {κ}, and δ, c0 , . . . , ck , v0 , . . . , vn−1 ] where ν the formula “Vν |= φ[ is the largest ordinal” is an element of u}. If κ, δ, x0 , . . . , xn−1 ∈ Vη and u is realized by x0 , . . . , xn−1 in Vη+1 then u− is defined and is realized by the same x0 , . . . , xn−1 in Vη . Definition 1A.6. Let u be a (κ, n)-type, and let w be a (τ, m)-type. We say that w is a subtype of u (and write w < u) if • τ < κ; • m ≥ n; and

18

1 Basic components

• the formula “there is an ordinal ν and vn , . . . , vm−1 ∈ Vν such that w is realized by v0 , . . . , vm−1 in Vν ” is an element of the type u. Note that w ∈ Vκ , since τ < κ. Thus the formula listed in the last item above can literally be an element of the type u. The definition of a subtype makes no mention of realizability but only stipulates that one particular formula belongs to u. It is immediate then that the property w < u is absolute for any two models of Set Theory which have w and u as elements. Definition 1A.7. We say that a (τ, m)-type w exceeds the (κ, n)-type u, if • τ > κ; • m ≥ n; and • there exist ordinals ν, η, and x0 , . . . , xm−1 ∈ Vν such that – w is realized by x0 , . . . , xm−1 in Vν , – u is realized by x0 , . . . , xn−1 in Vη , and – ν + 1 < η. ν, η, x0 , . . . , xm−1 are said to witness that w exceeds u. The definition of an exceeding type does refer to realizability. Note the essential difference between “w < u” and “w exceeds u” for a type u which is realized by x0 , . . . , xn−1 in Vη . In both cases w must be a type with at least n variables which is realized at a rank below η. If w is a subtype of u then dom(w) < dom(u), while w can exceed u only if dom(w) > dom(u). Definition 1A.8. Let κ < λ be ordinals, E a λ-strong extender with crit(E) = κ and u a type with dom(u) = κ. Let iE : V → Ult(V, E) be the ultrapower embedding. We define StretchE λ (u) to be equal to projλ (iE (u)). iE (u) in Definition 1A.8 is a type in Ult(V, E) with domain iE (κ). iE (κ) is at least as large as λ by Fact 2 in Appendix A, since E is λ-strong. So projλ (iE (u)) in the definition makes sense. Lemma 1A.9. Let M and N be models of ZFC∗ . Suppose that M&κ + 1 = N &κ + 1. Let u be a type in M with dom(u) = κ. Let E ∈ M and assume that M |= “E is a λ-strong extender with crit(E) = κ.” N Then E can be applied to N and StretchE λ (u) (as computed in M) is equal to projλ(iE (u)).

Proof sketch. The ultrapower embeddings iEN and iEM agree on subsets of M&κ, and u is a subset of M&κ. # Definition 1A.10. A (κ, n)-type u is called elastic just in case that u− is defined and u contains the following two formulae:

1A The auxiliary games map

19

• “ δ is an inaccessible cardinal”. • “Let ν be the largest ordinal. Then for all λ < δ there exists a λ-strong extender − ) is realized (relative to (u δ) by E ∈ V& δ such that crit(E) = κ and StretchE λ v0 , . . . , vn−1 in Vν .” The definition of elasticity, like the definition of the notion of a subtype, simply requires certain formulae to belong to u. Elasticity is therefore absolute between any two models of Set Theory which contain u. The main formula in Definition 1A.10 could not, formally, be an element of any type, since it makes a reference to u− which is not an allowed constant. However u− is clearly definable (uniformly over all realizable types). Strictly speaking we should replace “u− ” by its definition, namely “the set of κ } ∪ { δ} which are satisfied by v0 , . . . , vn−1 in Vν , formulae with constants in V κ ∪ { where ν is the largest ordinal.” − Ordinarily if u is realized by x0 , . . . , xn−1 in Vν+1 then StretchE λ (u ) is realized Ult(V,E) by iE (x0 ), . . . , iE (xn−1 ) in ViE (ν) , and relative to iE (δ). The requirement that it must also be realized by x0 , . . . , xn−1 in Vν , and relative to δ, gives strong additional information about E. In the terminology of [18], the existence of an elastic type which is realized by x0 , . . . , xn−1 in Vη+1 and has domain κ implies that κ is η − δ reflecting in the parameters x0 , . . . , xn−1 relative to δ. Thus the existence of realizable elastic types is stronger than the mere existence of extenders. Indeed, to obtain many elastic realizable types we need a Woodin cardinal. Definition 1A.11. Let δ be a cardinal. • Let H ⊂ Vδ . Let λ < δ. An extender E is λ-strong wrt H if it is λ-strong and if in addition iE (H ) ∩ Vλ = H ∩ Vλ . • A cardinal κ < δ is λ-strong wrt H if it is the critical point of an extender which is λ-strong wrt H . • κ is 0 we require κn > κn−1 . If n = 0 we require κn > rank(X). (b) proj3 (un ) is elastic. (c) pn ∈ M&κn is a condition in col(ω, δ). (d) There exist some names a˙ and x, ˙ both members of M&δ + 1, so that un is ˙ a˙ , x, realized by A, ˙ ν in M&ν + 2. When I plays ln = “new” she is indicating that she wishes to start a fresh attempt at constructing h. This fresh attempt consists of some condition pn , the first in a decreasing chain which should form the generic h, and names a˙ , x˙ which will later on be forced to ˙ and to produce exactly a and x. Instead of directly playing the names, I belong to A, merely has to play their type. Note that in a sense this is easier for I; a single type may be realized by many different names, and I is not asked to pick among them. (3) (Rule for II) (a) wn is a (τn , 5)-type for some τn . If n > 0 we require further τn > κn−1 . (b) wn is a subtype of proj3 (un ). (Note this implies τn < κn .) Furthermore, wn must contain the following formulae: 1 We think of “new” as coded by some element of V . ω

22

1 Basic components

(c) “v3 is a dense set of conditions in the forcing col(ω, δ),” (d) “v4 is an ordinal,” and (e) “v4 + 4 is the largest ordinal.” ˙ a˙ , x, Thus, if un were realized by A, ˙ α in M&α + 2, then wn would be realized by ˙ ˙ A, a , x, ˙ D, β in M&β + 5 for some β < α (in fact β + 5 < α + 2) and some D which is dense in col(ω, δ). Note the two features of II’s response to un . First, II gives some dense set D. Secondly, II produces an ordinal β, smaller than the one had before. As with I’s moves, rather than asking II to play the dense set and the ordinal directly, we merely ask II to play their type. This gives II some extra freedom, since the type specifies a set of D-s and β-s (the set of realizations), rather than a single D and a single β. (4) (Rule for I) If ln ∈ N, i.e., ln = “new” then: (a) un is a (κn , 5)-type for some κn < δ. We require further κn > κn−1 . (b) proj3 (un ) is elastic. (c) pn ∈ M&κn is a condition in col(ω, δ), and extends pln . (d) un exceeds wln . Furthermore, un must contain the following formulae: (e) “ pn ∈ v3 ,” (f) “ pn col(ω, δ) ‘$v1 , v2 % ∈ v0 ,’” n col(ω, δ) ‘v1 (i) = ai ,’” (g) “For each i < ln , p n col(ω, δ) ‘v2 (i) = xi ,’” and (h) “For each i < ln , p (i) “v4 is an ordinal and v4 + 1 is the largest ordinal.” When playing ln ∈ N (rather than ln = “new”) I is indicating that in round n she wishes to continue the h she left off in round ln (note ln < n always). To do this I plays a condition pn which extends pln . To be fair we demand that this condition meets the dense set handed by II in round ln —this is the requirement (4e). To connect I’s moves with x and a we demand some agreement between these objects and the names played by I—rule (4g) demands that the first ln elements of the sequence named by v1 are in fact a0 , . . . , aln −1 , and rule (4h) demands that the first ln elements of the sequence named by v2 are in fact x0 , . . . , xln −1 . Rule (4f) says that $v1 , v2 % are forced to produce ˙ see rule (2d) above. So I is slowly an element of v0 . v0 in some sense stands for A; a , x%, and forcing this pair to constructing h, relating the pair named by $v1 , v2 % to $ ˙ belong to A[h]. As always, we ask the player to just play the types of the objects in questions. Note that the dense set given by II in round ln was not given with precision. II only played the τln -type of the set. In round n we ask I to play un which exceeds the type played by II in round ln . Thus, in a sense, I gets to pick one of the dense sets handed by II, and meet that one.

1B Generic runs

23

Remark 1A.17. With respect to rules (4g) and (4h) we note that for i < ln both ai and xi belong to M&κn , and so ai and xi are valid constants for the type un . The fact that ai belongs to M&κn traces back to rules (4a) and (2a) which certainly imply that κn > κi . The same rules imply that κn > rank(X), so that xi too belongs to M&κn . The rules above complete the description of the game A[x]. Observe that x only appears in rule (4h), and the only part of x relevant in round n is xln . Since ln < n, certainly xn suffices. Our description of A[x] therefore defines a map x → A[x], Lipschitz continuous in the sense that the rules for the first n rounds of A[x] depend only on xn. For s ∈ X 0. (2) For each i, pni is a condition in g. ˙ a˙ , x, ˙ ν in M&ν +2. (Recall that ν > δ is the lower (3) For i = 0, uni is realized by A, ordinal in the least pair of local indiscernibles of M relative to δ; see Section 1A.) ˙ a˙ , x, (4) For each i > 0, uni is realized by A, ˙ D, βi in M&βi + 2 for some dense set D. (5) β0 = ν and βi < βi−1 for all i > 0. Condition (1) simply says that n0 , n1 , . . . form a “branch” in I’s attempts in the run a of the game A[x]. Conditions (2)–(4) relate this branch to a˙ , x, ˙ g—our guaranteed attempt which cannot fail. Note the appearance of the ordinals βi in condition (4). These correspond to the ordinals β mentioned after rule (3) in Section 1A (2), the ordinals given by the variable v4 . These ordinals will decrease steadily, giving us the desired contradiction. Claim 1B.3. There exists n so that • ln = “new”, • pn is a condition in g, and ˙ a˙ , x, • un is realized by A, ˙ ν in M&ν + 2. Proof. Suppose not. The claim can be phrased inside M[g]. If it fails, this must be forced by some condition in g. So fix q ∈ g which forces the claim to fail. Strengthening q if needed, we may assume that q also forces “a˙ is a generic run of A[x] ˙ wrt ρ, ˇ g.” ˙ Let p be your favorite condition in g. Strengthening q further we may assume that q extends p. Let j < ω be the domain of q. Let λ < δ be large enough that ρ ◦ q(0), . . . , ρ ◦ q(j − 1), and p all belong to M&λ. Using Lemma 1A.12, fix ˙ a˙ , x˙ in M&ν + 2 is elastic. κ > max{λ, rank(X)}, below δ, so that the (κ, 3)-type of A, Let u be the (κ, 4)-type of the same parameters plus ν (again in M&ν + 2). Observe that $“new”, p, u% is a legal move for I in round n of A[y], for any y ∈ X ω , so long as κn−1 < λ. (This follows directly from rule (2) in Section 1A (2). Note particularly the fact that proj3 (u) is elastic, by choice of κ.) Let q ∗ be the condition q extended by j → ρ −1 $“new”, p, u%. We will show that ∗ q forces the claim to hold, contradicting our initial choice of q ∈ g which forces the claim to fail.

1B Generic runs

25

˙ ∗ ] and let a ∗ = a˙ [g ∗ ]. Fix some g ∗ , a generic which contains q ∗ . Let x ∗ = x[g Say a ∗ = $a0∗ , a1∗ , . . . %, and ai∗ = $li∗ , pi∗ , u∗i , wi∗ %. Our goal is to show that the claim holds for these objects. Certainly it is enough to find n < ω so that $ln∗ , pn∗ , u∗n % = $“new”, p, u%. Now g ∗ contains q, so we know that a ∗ is a generic run of A[x ∗ ] wrt ρ, g ∗ . Thus, for each n < ω, $ln∗ , pn∗ , u∗n % is the least legal move, wrt to the enumeration ρ◦g ∗ . To be more precise: For each n < ω let en∗ < ω be the first number so that $ln∗ , pn∗ , u∗n % = ρ ◦g ∗ (en∗ ). We know that en∗ is the least number valid (wrt g ∗ ) at n. Let n be least so that en∗ ≥ j . It is enough to show that en∗ = j . Then $ln∗ , pn∗ , u∗n % = ρ ◦ g ∗ (j ) = $“new”, p, u% as required. (Note ρ ◦ q ∗ (j ) = $“new”, p, u% by our definition of q ∗ .) To show that en∗ = j it is enough to check that ρ ◦ g ∗ (j ) is a legal move following ∗ a n. Then j is valid at n, and by the minimality condition in Definition 1B.1, en∗ > j is ruled out. Remember that $“new”, p, u% is a legal move in round n of A[y], for any y, so long as κn−1 < λ. So it is enough to check that u∗0 , u∗1 , . . . , u∗n−1 all belong to M&λ. Since ∗ , p ∗ , u∗ % = ρ ◦ g ∗ (e∗ ), certainly it is enough to check that ρ ◦ g ∗ (e∗ ) belongs to $lm m m m m ∗ < j, M&λ for all m < n. Now n is least such that en∗ ≥ j . So m < n implies that em ∗ ∗ ∗ and ρ ◦ g (em ) then equals ρ ◦ q(em ). We chose to begin with λ which is bigger than ∗ ) ∈ M&λ, as required. the ranks of ρ ◦ q(0), . . . , ρ ◦ q(j − 1). So ρ ◦ q(em # Claim 1B.4. Suppose m and α are such that • pm is a condition in g, and ˙ a˙ , x, ˙ α in M&α + 2. • um is realized by A, Then there exists an n > m, a dense set D, and an ordinal β so that • ln = m, • pn is a condition in g, ˙ a˙ , x, ˙ D, β in M&β + 2, and • un is realized by A, • β < α. Proof. Suppose not. The claim can be phrased inside M[g]. If it fails, this must be forced by some condition. Fix q ∈ g which forces the claim to fail. Strengthening q as needed we may assume that q forces “a˙ is a generic run of A[x] ˙ wrt ρ, ˇ g.” ˙ 3 By the rules of A[x], specifically rule (3), wm is a subtype of proj (um ). Since um ˙ a˙ , x, is realized by A, ˙ α in M&α + 2, it follows that there is an ordinal β and a set D ˙ a˙ , x, ˙ D, β in M&β + 5, and β + 5 < α + 2. In particular so that wm is realized by A, β < α as required for our current claim. Find a condition p ∈ g ∩ D, extending pm , which forces the following: ˙ (i) “$a˙ , x% ˙ ∈ A,”

26

1 Basic components

(ii) for each i ≤ m, “a˙ (i) = aˇ i ,” and (iii) for each i < m, “x(i) ˙ = xˇi .” ˙ (remember our initial assumption This is possible since $a˙ [g], x[g]% ˙ does belong to A[g] ˙ for contradiction at the start of the proof of Lemma 1B.2: $ a , x% ∈ A[g]), a˙ [g](i) does equal ai for each i < m, and x[g](i) ˙ does equal xi for each i < m. By strengthening q again, we may assume that q extends p. Let j be the domain of q. Let λ < δ be large enough that ρ ◦ q(0), . . . , ρ ◦ q(j − 1), p, and a0 , . . . , am all belong to M&λ. Using Lemma 1A.12, find κ > λ, below δ, so that the (κ, 3)-type of ˙ a˙ , x˙ in M&β +2 is elastic. Let u be the (κ, 5)-type of the same parameters plus D, β. A, ˙ a˙ , x, (Precisely, u is the (κ, 5)-type of A, ˙ D, β in M&β + 2.) Observe that u exceeds wm . This follows directly from our choice of u, the realization of wm in M&β + 5 indicated above, and Definition 1A.7. (Note λ, and hence κ, is greater than τm since am belongs to M&λ.) Observe further that $m, p, u% is a legal move in A[y] for any y ∈ X ω which extends xm, and following any position b0 , . . . , bk in A[y] which extends a m + 1, as long as the types played in b0 , . . . , bk have domains below λ. This follows directly from the rules of A[y], specifically rule (4). (For rule (4c) note that p was chosen extending pm . For rule (4e) remember that p was taken from g ∩ D. Rules (4f)–(4h) hold because of conditions (i)–(iii) above. In the case of rule (4g) we use also the fact that bi = ai for i < m.) Let q ∗ be the condition q extended by j → ρ −1 $m, p, u%. An argument similar to the one used in Claim 1B.3 now shows that q ∗ forces $m, p, u% to be played in any generic run which extends a m + 1. By condition (ii), p and hence q ∗ force a˙ to extend # a m + 1. Hence q ∗ forces Claim 1B.4 to hold. Claim 1B.5. Suppose m, C, and α are such that • pm is a condition in g, and ˙ a˙ , x, • um is realized by A, ˙ C, α in M&α + 2. Then there exists an n > m, a dense set D, and an ordinal β so that • ln = m, • pn is a condition in g, ˙ a˙ , x, • un is realized by A, ˙ D, β in M&β + 2, and • β < α. ˙ a˙ , x, Proof. This is similar to Claim 1B.4. We start with um realized by A, ˙ C, α in 3 ˙ ˙ a , x˙ in M&α + 2. Since wm is M&α + 2. This means that proj (um ) is realized by A, a subtype of proj3 (um ) it follows that there is an ordinal β and a set D so that wm is ˙ a˙ , x, realized by A, ˙ D, β in M&β + 2, and β + 5 < α + 2. The rest of the proof is identical to that of Claim 1B.4. #

1C Pivots

27

Equipped with Claims 1B.3–1B.5 we can complete the proof of Lemma 1B.2. Our goal is to construct a sequence ni , βi , i < ω, satisfying conditions (1)–(5) on page 24. Let n0 be some n witnessing Claim 1B.3, and let β0 = ν. Let n1 , D1 , and β1 witness Claim 1B.4 applied with m = n0 and α = β0 . Working inductively, let ni+1 , Di+1 , and βi+1 for i ≥ 1 witness Claim 1B.5 applied with m = ni , C = Di , and α = βi . It is easy to verify that conditions (1)–(5) hold for these objects. In particular $βi | i < ω% forms a decreasing sequence of ordinals, giving the desired contradiction and completing the proof of Lemma 1B.2. # We have so far defined generic runs and obtained Property 2. Let us end by noting that, with the crucial help of a strategy for II in A[x], one can easily construct a generic run. The construction is simply a matter of ascribing the first legal move for player I in each round. The following claim guarantees the existence of legal moves. Claim 1B.6. Suppose a0 , . . . , an−1 is a position in A[x]. Then there is a move $l, p, u% which is legal for I following a0 , . . . , an−1 . Proof. Take your favorite names a˙ and x˙ in M&δ + 1. Using Lemma 1A.12 find κ < δ, large enough that a0 , . . . , an−1 ∈ M&κ, larger than rank(X), and so that the κ-type ˙ a˙ , x˙ in M&ν + 2 is elastic. Let u be the (κ, 4)-type of the same objects plus ν, of A, again in M&ν + 2. It is easy to see that $“new”, ∅, u% is a legal move for I following # a0 , . . . , an−1 . Lemma 1B.7 (ρ : δ ←→ M&δ, g col(ω, δ)-generic/M). For every x ∈ Xω there exists a strategy σgen [x] for I in A[x] so that every run according to σgen [x] is generic. Moreover, the association x → σgen [x] is Lipschitz continuous, induced by some map σgen = (s → σgen [s]). The map σgen belongs to M[g]. Proof. σgen [x] simply plays the first (in the enumeration given by ρ ◦ g) legal move in each round. Such a move exists by Lemma 1B.6. The dependence of σgen [x] on x is Lipschitz continuous because the notion of a legal move in round n of A[x] depends only on xn. Finally, to define the map σgen one needs the map A and the enumeration ρ ◦ g. Both exist in M[g]. # Lemma 1B.7 states, in a precise way, that generic runs of A[x] can be constructed with the aid of a strategy for II. Given some strategy σ for II in A[x] simply pit σ against σgen [x]. The result is a generic run. We refer to the map σgen of Lemma 1B.7 as the generic strategies map associated ˙ δ, and X. There is a dependence on g in the definition of σgen . g will always be to A, clear from the context, so we suppress the dependence in our notation.

1C Pivots A length ω iteration tree T is called nice if (2n) T (2n+2) for every n. 0 T 2 T 4 T · · · then forms a branch, called the even branch, through T . We use Mn = MnT to denote

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1 Basic components

T : M → M for n T m to denote the embeddings the models of T , and jn,m = jn,m n m T to denote the direct limit along the even branch of of the tree. We use Meven = Meven T to denote the direct limit embeddings. jeven stands for T and use j2n,even = j2n,even j0,even . We refer to branches b other than the even branch as odd. We use Mb = MbT T to denote their direct limits, and jn,b = jn,b for n ∈ b to denote the direct limit maps. jb stands for j0,b .

Definition 1C.1. A pivot for x ∈ Xω is a pair T , a , where (1) T is a nice iteration tree on M; (2) a is a run of jeven (A)[x]; and (3) for every odd branch b of T there exists h so that (a) h is col(ω, jb (δ))-generic/Mb , and ˙ (b) $ a , x% ∈ jb (A)[h]. ˙ and A = AX,δ . This definition is made with reference to particular M, δ, X, A, When there is danger of ambiguity we say A-pivot rather than just pivot. Remark 1C.2. Condition (3) applies to all odd branches b, including ones where Mb is illfounded. In the previous section we saw that a generic run a of A witnesses that $ a , x% does ˙ not belong to A[g]. A pivot is an attempt to witness the opposite, that $ a , x% does ˙ ˙ for an odd belong, not to A[g] but at least to some shift of this set, namely jb (A)[h] branch embedding jb and a generic h. Our goal in this section is to formulate a result on pivots similar to Lemma 1B.7. We first phrase the construction of a pivot for x as a game, Apiv [x]. We then describe a strategy σpiv [, x] which plays for II in this game. Just as σgen [x] plays for I in A[x] and always produces generic runs, σpiv [, x] will play for II in Apiv [x] and produce pivots. From the point of view of player I the new game Apiv [x] will be nothing more than a shift of the original auxiliary game A[x]. A strategy σ for I in the original game A[x] could thus be used in the new game, and pitted against σpiv [, x]. The resulting run will form a pivot. The reader should contrast this with the discussion following Lemma 1B.7, where a strategy for II was pitted against σgen [x] to form a generic run. 1C (1) The game Apiv [x]. The game Apiv [x] is played according to Diagram 1.2 and rules (1)–(6) below. Player II has the task of playing extenders defining an iteration tree T . In addition, the players play the game A[x], but shifted along the even branch of T . I starts with l0 , p0 , u0 , a legal move in A[x]. Then II plays E0 , E1 , which are used to determine the first models M1 , M2 of T , and the embedding j0,2 . We then shift A[x] to M2 , and player II plays w0 , a legal move in j0,2 (A)[x] following j0,2 (l0 , p0 , u0 ). The

1C Pivots

I II

l0 , p0 , u0

l1 , p1 , u1 E0 , E1 , w0

···

l2 , p2 , u2 E2 , E3 , w1

29

...

Diagram 1.2. The game Apiv [x].

game continues in this way. Player I plays l1 , p1 , u1 which must be a legal move in j0,2 (A)[x]. Player II replies in M4 , etc. We list the exact rules, rules (1)–(6) below, interspersed with some helpful terminology. The terminology, and the rules, apply to the run of Apiv [x] displayed in Diagram 1.2. Let T be the unique tree order which satisfies: • the T -predecessor of 2n + 2 is 2n for each n < ω, • if ln = “new” then the T -predecessor of 2n + 1 is 2n, and • if ln ∈ N then the T -predecessor of 2n + 1 is 2ln + 1. Let T be the iteration tree on M determined by $En | n < ω% and the tree order T . (1) (Rule for II) The extenders En must be played in a way which makes this definition of T work. Precisely: (a) If ln ∈ N then E2n ∈ M2n must be an extender with critical point within the level of agreement between M2n and M2ln +1 . We set M2n+1 = Ult(M2ln +1 , E2n ). (b) E2n+1 ∈ M2n+1 must be an extender with critical point within the level of agreement between M2n+1 and M2n . We set M2n+2 = Ult(M2n , E2n+1 ). (2) (Rule for II) If ln = “new” then E2n must equal “pad” so that M2n+1 = M2n and j2n,2n+1 = id. (3) (Rule for II) T must be normal, and must only use extenders taken from below δ. (An iteration tree is normal if {StrengthMn (En )}n 0, κn is greater than j2n−2,2n (κn−1 ). (This follows from rules (2a) and (4a) in Section 1A (2).) Thus, for n > 0, the critical point of j2n,2n+2 is greater than j2n−2,2n (κn−1 ). It follows that an−1 , and so certainly a0 , . . . , an−2 , are not moved by j2n,2n+2 . Using the fact that κ0 > rank(X) and condition (E) we see that the critical point of j2n,2n+2 is also higher than rank(X). This secures rule (5) of Apiv [x]. The objects E2n , E2n+1 , wn defined above thus form a legal move for II in round n of Apiv [x]. This completes the construction in case 1 of round n. Conditions (A)–(F) are easy to verify. # (Case 1) Case 2. If ln ∈ N. We know by the rules of A[x], specifically rule (4d) in Section 1A (2), that un exceeds wln . We know by condition (A) that wln is elastic. Applying the One Step Lemma inside M2n we obtain E2n ∈ M2n &j0,2n (δ), a κn + ω-strong extender 2n with critical point τln , so that un < StretchE κn +ω (wln ). To satisfy rule (3), make sure

1C Pivots

33

that E2n is stronger than all previous extenders on the tree. This can be done using Remark 1A.14. Set M2n+1 = Ult(M2ln +1 , E2n ) and set j2ln +1,2n+1 to be the ultrapower embeddings. Note how this corresponds to the definition of T given in the previous subsection. Note further that M2n and M2ln +1 agree to τln + ω, so this makes sense and complies with rule (1a) in Section 1C(1). This agreement between M2n and M2ln +1 follows from the inductive conditions (E) and (F) in our construction. 2n By Lemma 1A.9, StretchE κn +ω (wln ) is equal to projκn +ω (j2ln +1,2n+1 (wln )). So un is a subtype of the latter. It follows that un < j2ln +1,2n+1 (wln ). ˙ and νn = j0,2n+1 (ν). Set a˙ n = j2ln +1,2n+1 (a˙ ln ) and x˙n = Set A˙ n = j0,2n+1 (A) j2ln +1,2n+1 (x˙ln ) as required by condition (C). Let Cn = j2ln +1,2n+1 (Dln ). Using condition (B) we see that j2ln +1,2n+1 (wln ) is realized by A˙ n , a˙ n , x˙n , Cn , νn in M2n+1 &νn +5. Since un is a subtype of j2ln +1,2n+1 (wln ) we conclude that un is realized by the same objects, in M2n+1 &νn + 2. pn ∈ v3 ”; see rule (4e) Before proceedings, let’s recall that un contain the formula “ in Section 1A (2). Our realization of un in M2n+1 has Cn standing for v3 . Thus pn ∈ Cn = j2ln +1,2n+1 (Dln ). Our assignments so far therefore satisfy the inductive condition (D) at n. We now continue very much as we did in case 1. Switching between the local indiscernibles νn and νn∗ = j0,2n+1 (ν ∗ ) we see that un is realized by A˙ n , a˙ n , x˙n , Cn , νn∗ in M2n+1 &νn∗ + 2. It follows that (∗) proj3 (un ) is realized by A˙ n , a˙ n , x˙n in M2n+1 &νn∗ + 2. Pick some Dn ∈ M2n+1 &δn + 1, a dense set in col(ω, δn ). The precise way we pick Dn uses the enumeration and will be explained later. Using Lemma 1A.12 find τn < δn , greater than κn = dom(un ), so that the τn -type of A˙ n , a˙ n , x˙n , Dn , νn in M2n+1 &νn + 5 is elastic. Let wn be this type. Observe that wn exceeds proj3 (un ) in the model M2n+1 . This follows as in case 1, using (∗) and noting that νn + 5 (the level of realization of wn ) is smaller than νn∗ + 2. Applying the One Step Lemma inside M2n+1 find E2n+1 ∈ M2n+1 &j0,2n+1 (δ), a τn + ω-strong extender with critical point κn , so that wn E 3 is a subtype of Stretchτn2n+1 +ω (proj (un )). As usual make sure that E2n+1 is stronger than all previous extenders on the tree. Let M2n+2 = Ult(M2n , E2n+1 ) and let j2n,2n+2 be the ultrapower embedding. As in case 1 we get wn < proj3 (j2n,2n+2 (un )). Using this it is again easy to check that wn is a legal move for II in j0,2n+2 (A)[x] following the position Qn . It follows that E2n , E2n+1 , wn as defined above form a legal move for II in round n of Apiv [x]. This completes the construction in round n. Conditions (A)–(C), (E), and (F) are easy to verify. Condition (D) was verified above. # (Case 2)

34

1 Basic components

Let us take note of what we have so far. We defined a strategy for II in the game Apiv [x], modulo some method of picking the dense sets Dn . Every run according to our strategy satisfies conditions (A)–(F) and (∗) of cases 1 and 2. Now un is required to include certain formulae, listed in the rules of A[x], specifically rules (4f)–(4h) in Section 1A (2). Applying these formulae to the realization given by (∗) we get in the case ln ∈ N: 2n+1 (1) pn col(ω,δ “$a˙ n , x˙n % ∈ A˙ n .” n)

M

2n+1 “a˙ n (i) = aˇ i .” (2) For each i < ln , pn col(ω,δ n)

M M

2n+1 “x˙n (i) = xˇi .” (3) For each i < ln , pn col(ω,δ n)

Moreover, using rule (4c) we get: (4) pn extends pln . (We also use here the fact that j2ln ,2ln +2 (pln ) = pln so that pln is not affected by the shift of aln . This fact follows from the rules in Section 1A (2), which tell us that pln ∈ M2ln &κln . Remember that crit(j2ln ,2ln +2 ) = κln by construction.) For each odd branch b of T , let hb be the filter generated by the increasing conditions ˙ and let δb = jb (δ). Let a˙ b = j2n+1,b (a˙ n ) for {pn | 2n + 1 ∈ b}. Let A˙ b = jb (A) some n so that 2n + 1 ∈ b. Which n we take does not matter; this follows from condition (C). Similarly let x˙b = j2n+1,b (x˙n ). Transferring conditions (1)–(3) to Mb using the embedding j2n+1,b we get: b (1 ) hb M ˙ b , x˙b % ∈ A˙ b .” col(ω,δb ) “$a b ˙ b (i) = aˇ i .” (2 ) For each i < ln , hb M col(ω,δb ) “a b (3 ) For each i < ln , hb M col(ω,δb ) “x˙ b (i) = xˇ i .”

Note that we use here the fact that pn is not moved by the embedding j2n+1,b . To see that pn is not moved by j2n+1,b , observe that pn ∈ M2n+1 &κn by rule (4c) in Section 1A (2), κn < τn by condition (E), and τn = crit(j2n+1,b ) by construction. A similar argument shows that xi and ai for i < ln are not moved. Let Db = {j2l+1,b (Dl ) | 2l + 1 ∈ b} = {j2n+1,b (Cn ) | 2n + 1 ∈ b ∧ ln ∈ N}. Using condition (D), and again using the fact that pn is not moved by j2n+1,b we see that: (5) hb intersects all the dense sets in Db . Claim 1C.3. Assume that for each odd branch b, Db contains all the dense sets in col(ω, δb ) which belong to Mb . Then T , a is a pivot. Proof. This is immediate using condition (5), which under the assumption of the claim says that hb is col(ω, δb )-generic/Mb , and conditions (1 )–(3 ) which say that $ a , x% ∈ A˙ b [hb ]. #

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35

Remember that our goal is to make sure that all runs according to σpiv [, x] are pivots. Claim 1C.3 says that the strategy we constructed achieves this, provided we pick the sets Dn so as to satisfy the assumption of the claim. The assumption of the claim is not hard to satisfy. Each dense set in Mb has a pre-image in M2n+1 for some 2n + 1 ∈ b. Each dense set in M2n+1 in turn belongs to M&δ + 1. (M2n+1 is a finite iterate of M, and therefore contained in M. δ is not moved by a finite iteration tree using only extenders from below δ, so M2n+1 &δn + 1 is a subset of M&δ + 1.) So certainly j2n+1,b (M&δ + 1 ∩ M2n+1 ). Db ⊂ 2n+1∈b

Using this inclusion it is easy to devise a method of picking sets Dn during the construction, which uses the map : ω → M&δ + 1, and which secures the assumption of Claim 1C.3 whenever is onto. Devising the precise method is a simple matter of book-keeping, which we leave to the pleasure of the reader. Let us only say that the book-keeping can be arranged so that the choice of Dn in round n depends only on n + 1. (If n + 1 does not give a suitable set, simply “delay” by taking the trivial dense set, Dn = {all conditions}.) Remark 1C.4. It was not necessary during the course of the construction to pick dense sets Dn . All we needed were the types wn of these dense sets. Types, unlike the dense sets, are elements of M&δ. Taking advantage of this observation, one can revise the construction to use a generic enumeration of M&δ instead of the enumeration of M&δ + 1. We need a generic enumeration of M&δ because we cannot just play all types (for one thing, {τn }n 0 then f (0) belongs to [0, f (n))T . This matter is discussed in Section 1F (1). The reference to the largest m < n so that f (m) belongs to [0, f (n))T in Definition 6C.3 therefore makes sense. ˙ Definition 6C.4. Let δ belong to λ ∩ W , and let κ belong to δ ∩ L. Define K(κ, δ, P) ˙ to be equal to j0,f (n) (C)(κ, δ, P , γ (P)). P in Definition 6C.4 is the sequence $a0 , . . . , an−1 % given by P. The name ˙ . . ) is the one given by Definition 4C.14 shifted to Mf (n) . We saw above j0,f (n) (C)(. that the critical point of j0,f (n) is larger than λ. So W and L in the sense of M are the ˙ δ, . . . ) in same as W and L in the sense of Mf (n) , up to λ. The reference to j0,f (n) (C)(κ, ˙ δ, . . . ) is a δ-name in the sense of Definition 6C.4 therefore makes sense. j0,f (n) (C)(κ, Mf (n) . Since the critical point of j0,f (n) is greater than λ which in turn is greater than δ, ˙ ˙ δ, . . . ) is also a δ-name in the sense of M. It follows that K(κ, δ, P) is a j0,f (n) (C)(κ, δ-name in M. ˙ ˙ Remark 6C.5. If n = 0 and f (0) = 0 then K(κ, δ, P) is simply equal to C(κ, δ, ∅, γL ). Note that the conditions n = 0 and f (0) = 0 determine P completely: T f (0) + 1 must be the trivial tree on M with first and last model equal to M, and there are no subsequent moves in P. We refer to the unique P determined by the conditions n = 0 and f (0) = 0 as the trivial position of length 0.2. Fix δ ∈ λ ∩ W and κ ∈ δ ∩ L for the rest of Section 6C (1). Let g be col(ω, δ)generic/M. Fix a δ-sequence t over M. Suppose that: ˙ (C1) t belongs to K(κ, δ, P)[g]. Claim 6C.6 (under assumption (C1) above). n is not used in t (κ). Proof. Recall that n is equal to the length of the sequence P given as part of the position ˙ P. Assumption (C1) says that t belongs to an interpretation of j0,f (n) (C)(κ, δ, P , γ (P)). The fact that n is not used in t (κ) now follows directly from condition (2) in the definition ˙ Definition 4C.14. of C, #

224

6 Along a single branch

Let w = t (κ) + $n, δ%. This is a witness for δ + 1 = rdm(t). Let x = t, w. Suppose that: (C2) x agrees with x¯ to n. ¯ Since x and x¯ agree Recall that P was fixed a position of length n + 0.2 in Rmix [x]. to n we may regard P also as a position in Rmix [x]. Let σmix be the mixed pivot ˙ τ , and X = M&λ. (See Section 1F (2) for details strategies map associated to R, regarding this map.) Fix a surjection : ω → M&τ + 1. Suppose that: (C3) P is played according to σmix [, x]. Suppose finally that: (C4) P is useful (see Definition 1F.8). Let κ ∗ denote δ + 1. Let δ † denote the first Woodin cardinal of M above δ. Claim 6C.7 (under assumptions (C1)–(C4) above). There exist n∗ , δ ∗ , and P∗ so that: (1) n∗ is a natural number greater than n; (2) δ ∗ is an element of λ ∩ W , greater than δ † ; (3) P∗ is a position of length n∗ + 0.2 in Rmix [x], extending P; (4) P∗ is played according to σmix [, x]; (5) P∗ is useful; and (6) ϕsuc (t, w, C˙ † ) holds true with the assignment C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ), where ˙ ∗ , δ ∗ , P∗ ). C˙ ∗ = K(κ Proof. We use G∗lim (. . . ) to refer to the games defined in Section 4C(3), and played ac˙ cording to Diagram 4.2. The assumption that t belongs to an interpretation of K(κ, δ, P) ∗ ∗ tells us that I wins the game j0,f (n) (Glim )(t, κ, δ, P , γ (P)). Let σ be a winning strategy for I in this game. Since the game is clopen and belongs to Mf (n) [g] we may fix σ ∗ inside Mf (n) [g]. σ ∗ and σmix [, x] together give rise to n∗ , δ ∗ , and P∗ : • σ ∗ plays δ ∗ and n∗ ; • σmix [, x] plays the moves corresponding to II in P∗ ; • we play f (m) = f (m − 1) + 2 for I in the “first fifth” of each round m > n in P∗ , so that T f (m) + 1 is simply the tree constructed up to round m, unextended; and • σ ∗ and its shifts to the models Mf (m) play I’s moves outside the first fifth in each round m ≥ n of P∗ .

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We leave it to the reader to verify that the objects δ ∗ , n∗ , and P∗ constructed through this use of σ ∗ and σmix [, x] satisfy the conditions of the claim. Let us just note that: condition (6) follows from the shift of the payoff condition (P) in Section 4C (3) to the model Mf (n∗ ) ; and condition (5) follows from the fact that P is useful, the definition of # γ (P), and the demands in Section 4C(3) which force the ordinals γi to descend. 6C (2) Infinite. Let t be an annotated M-position of relative domain λ or possibly less. Let w be a witness for rdm(t). Set x = t, w. Let be a surjection of ω onto M&τ + 1. Let P be an infinite run of Rmix [x]. Suppose that: (D1) P is useful and played according to σmix [, x]. P gives rise to: • a sequence of ordinals γ = $γn | n < ω%; • an infinite increasing list of natural numbers f = $f (n) | n < ω%; • a length ω iteration tree T on M, with models Mn and embeddings jm,n : Mm → Mn for m < n < ω. Let b be a cofinal branch through T . Let Q be the direct limit along b and let jk,b : Mk → Q for k ∈ b be the direct limit embeddings. The run P also gives rise to: • a sequence a = $an | n < ω% in (Q&j0,b (τ ))ω . We follow the notation of Section 1F(1) when discussing the objects γ , f , T , and a given by P. Suppose that: (D2) Q is wellfounded. We work under the assumptions (D1) and (D2) for the rest of Section 6C (2). Claim 6C.8. b is an odd branch. Proof. Recall that a cofinal branch of T is even if it contains arbitrarily large nodes in {f (n) | n < ω}. Otherwise the branch is odd. The fact that P is useful implies that all cofinal even branches of T lead to illfounded direct limits, see Definition 1F.8. Since Q is wellfounded b must be odd. # Claim 6C.9. The embeddings jk,b have critical points greater than λ. Proof. The critical point of jk,b is greater than rank(X) by Claim 1F.12, and X in our context is equal to M&λ by Remark 4C.12. # Let I = {$κ, δ% ∈ (λ ∩ L) × (λ ∩ W ) | κ < δ}. Recall that γ (P, b) denotes jf (m),b (γm ) where m < ω is largest so that f (m) ∈ b, see Definition 1F.6.

226


Definition 6C.10. For each n < ω let uP,b (n) : I → Q&λ be the function defined by ˙ uP,b (n)(κ, δ) = j0,b (C)(κ, δ, a n, γ (P, b)). Let uP,b denote the sequence $uP,b (n) | n < ω%. uP,b is simply the sequence ua,γ given by the shift of Definition 4C.17 to Q, applied with γ = γ (P, b). Recall that t, fixed as the outset of Section 6C (2), is assumed to have relative domain λ or possibly less. The next claim shows among other things that t must in fact have relative domain equal to λ. Like all claims here it is made under assumptions (D1) and (D2). Claim 6C.11. The relative domain of t is precisely equal to λ. Moreover there exists some v ∈ (Q&λ + 1)ω so that one of the following conditions holds with the settings y = uP,b ⊕ v and s = t−−, w, y: (1) s is I-acceptably obstructed over Q; or (2) s is obstruction free over Q and there exists some h so that: (a) h is col(ω, j0,b (τ ))-generic/Q, and (b) s ∈ j0,b (Y˙ )[h]. Proof. The proof of Claim 6C.11 is a straightforward application of the methods of Section 1F. The key point is the correspondence between the conclusions of the claim and the definition of R˙ in Section 4C(3). Let us go over this quickly. Lemma 1F.11 tells us that P is a mixed R-pivot for x. Since b is an odd branch of T it follows that there exists some h so that: (i) h is col(ω, j0,b (τ ))-generic/Q; and ˙ b ][h], where γb stands for γ (P, b). (ii) $ a , x% ∈ j0,b (R)[γ Combining the second condition with the shift of Definition 4C.18 to Q we see that: (iii) x is a λ-code; and (iv) there exists some v ∈ (Q&λ + 1)ω so that x (uP,b ⊕ v) is either I-acceptably obstructed over Q, or an element of j0,b (Y˙ )[h]. The conclusion of the current claim follows immediately from conditions (iii) and (iv). It is in passing from condition (ii) to conditions (iii) and (iv) that we make use of the definition of R˙ in Section 4C(3). # Fix some n < ω. We work with this fixed n for the rest of Section 6C (2). Let m ¯ < ω be largest so that f (m) ¯ belongs to b. γ (P, b) by definition is equal to (γ ). Recall that e(n), in the notation of Section 1F, is equal to 0 if n = 0 and jf (m),b m ¯ ¯ to f (n − 1) + 2 if n > 0.

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Let k be the first element of the odd branch b which is larger than e(n), and larger than f (m). ¯ Let P∗ be the position of length n + 0.2 in Rmix [x] which follows P for rounds 0 through n−1, and contains the moves f ∗ (n) = k and T ∗ f ∗ (n)+1 = T k+1 for the first fifth of round n. We use f ∗ (i), γi∗ , Mi∗ , etc. when discussing the objects which form P∗ . Diagram 6.5 displays the connection between P and round n of P∗ . We leave it to the reader to check that the moves f ∗ (n) = k and T ∗ f ∗ (n) + 1 = T k + 1 satisfy the requirements of rule (1) in Section 1F (1). II

#

P

Me(n)

_ _I _ Mf (n)

P∗

Me(n)

_ _ _ _ _ _ _ _ _ _I _ _ _ _ _ _ _ _ _ _ Mk

Mf (n)+1

Mf (n)+2

_ I _ Mf (n+1)

Mk

(Mf∗ ∗ (n) )

(Me∗∗ (n) )

Diagram 6.5. Round n (the first fifth) in P∗ .

Definition 6C.12. We refer to the position P∗ defined above as trunc(P, b, n). Let c = uP,b (n). c is a function which assigns a col(ω, δ)-name c(κ, δ) to each pair $κ, δ% ∈ I . Claim 6C.13. Let $κ, δ% belong to I . Let g be col(ω, δ)-generic/M. Then c(κ, δ)[g] ⊂ ˙ K(κ, δ, P∗ )[g]. ˙ Proof. c(κ, δ) is by definition equal to j0,b (C)(κ, δ, a n, jf (m),b ¯ )). Pulling this ¯ (γm equality back to Mk via jk,b we see that: ˙ (i) c(κ, δ) = j0,k (C)(κ, δ, a n, jf (m),k ¯ )). ¯ (γm Note that a n is not affected by jk,b , because of Claim 1F.13 and because k is larger than e(n). κ, δ, and c(κ, δ) are also not affected by jk,b , since they lie below λ. Let m < n be largest so that f (m) belongs to [0, k)T . Then: ˙ ˙ (ii) K(κ, δ, P∗ ) = j0,k (C)(κ, δ, a n, jf (m),k (γm )). This can be seen directly by applying the definition of K˙ to P∗ . ¯ is largest so that f (m) ¯ belongs to [0, k)T . m Both m and m ¯ belong to [0, k)T . m on the other hand is just the largest number below n so that f (m) belongs to [0, k)T . So m ≤ m. ¯ Using the fact that P is useful (see Definition 1F.8) it follows from this that ¯ > m strictly). Using jf (m),k to shift γm¯ ≤ jf (m),f (m) ¯ (γm ) (with strict inequality iff m ¯ the inequality to Mk we get: (iii) jf (m),k ¯ ) ≤ jf (m),k (γm ). ¯ (γm ˙ The dependence of j0,k (C)(κ, δ, a n, γ )[g] on γ is monotone increasing by Remark 4C.15. It follows from this and from conditions (i)–(iii) that c(κ, δ)[g] ⊂ ˙ # K(κ, δ, P∗ )[g].

228


6C (3) Construction. We can now phrase and prove the main result of Section 6C. skip (M, t, τ )(Y˙ ), We demonstrate the existence of winning strategies for player I in G for annotated positions t which belong to interpretations of pullbacks of Y˙ . Lemma 6C.14 (for a relative limit τ and a τ -name Y˙ in M). Let δ < τ be an element of W . Let t be a δ-sequence over M. Suppose that M&τ +1 is countable in V. Suppose that t belongs to an interpretation of Back I (δ, τ )(Y˙ ), where the pullback is computed in M. Then player I has a winning skip (M, t, τ )(Y˙ ). strategy in G Proof. Let g witness that t belongs to an interpretation of Back I (δ, τ )(Y˙ ). g is col(ω, δ)generic/M, and t belongs to Back I (δ, τ )(Y˙ )[g]. Let $γL , γH % be the least pair of local indiscernibles of M relative to τ . Definition 4C.20 tells us that Back I (δ, τ )(Y˙ ) is equal ˙ to C(0, δ, ∅, γL ). So: ˙ (i) t ∈ C(0, δ, ∅, γL )[g]. Remember that we aim to demonstrate that player I has a winning strategy in skip (M, t, τ )(Y˙ ). Fix an imaginary opponent willing to play for II in this game. G We describe how to play for I, and win. We work in mega-rounds subject to the rules in Section 6A. At the start of mega-round β for β a successor or zero we will have the following objects: (A) a wellfounded model Mβ ; (B) an elementary embedding j0,β ; (C) δβ ∈ Wβ and a δβ -sequence tβ over Mβ ; (D) a generic gβ for col(ω, δβ ) over Mβ ; (E) κβ ∈ Lβ and nβ < ω; (F) a map φβ : aβ × ω → Mβ &j0,β (τ ) + 1 where aβ ⊂ ω is the set of numbers used in the witness tβ (κβ ); and (G) a position Pβ of length nβ + 0.2 in j0,β (Rmix )[x¯β ], where x¯β equals tβ κβ , tβ (κβ ). Wβ and Lβ above are the classes W and L of Section 4A, computed in Mβ . Fix for the entirety of the proof a bijection r : ω → ω × ω, with the following property: (ii) For each n < ω, r n ⊂ n × ω. Using φβ and the bijection r define a map ¯ β : ω → Mβ &j0,β (τ ) + 1 by: φβ (r(i)) if r(i) ∈ aβ × ω; and ¯ β (i) = ∅ otherwise.

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We will make sure that the objects listed in conditions (A)–(G) satisfy the following conditions at the start of mega-round β, for β a successor or zero: (1) Pβ is played according to j0,β (σmix )[x¯β , ¯ β ]. (2) Pβ is useful. ˙ β , δβ , Pβ )[gβ ]. (3) tβ belongs to j0,β (K)(κ (4) tβ (κβ ) is precisely equal to {$nξ , δξ % | ξ < β and ξ is a successor or zero}. (5) φβ extends jξ,β ◦ φξ for each ξ < β. (6) Let ζ < β be a successor or zero. Suppose that nξ ≥ nζ for each successor ξ ∈ (ζ, β]. Then Pβ extends jζ,β (Pζ ). Only the first three conditions will actually be used in the successor (or zero) case of the construction. The remaining three conditions are maintained as preparation for the limit case. Set to begin with: M0 = M, δ0 = δ, t0 = t, g0 = g, κ0 = 0, and n0 = 0. Let φ0 : ∅ × ω → M&τ + 1 be the empty function. Let P0 be the trivial position of length 0.2, see Remark 6C.5. Conditions (1) and (2) for β = 0 hold trivially with these assignments. Condition (3) follows from condition (i) above, and Remark 6C.5. Condition (4) holds trivially, since ∅ is the only witness for κ0 = 0. Conditions (5) and (6) are vacuous for β = 0. Let us now describe mega-round β. We divide into three main cases: successor (or zero); limit with an early end; and limit with a leap. Successor (or zero). We start with the case that β is a successor or zero. Using condition (3) and Claim 6C.6 we see that: (iii) nβ is not used in tβ (κβ ). (In other words nβ ∈ aβ .) Let wβ = tβ (κβ )+$nβ , δβ %. This is a witness for δβ +1 = rdm(tβ ). Let xβ = tβ , wβ . Using Claim 4A.16 we see that: (iv) xβ and x¯β = tβ κβ , tβ (κβ ) agree to nβ . Note that j0,β preserves countability. This is trivial if β = 0, and follows from Remark 6A.16 if β is a successor. We assume in Lemma 6C.14 that M&τ + 1 is countable in V. Combining this with the fact that j0,β preserves countability it follows that Mβ &j0,β (τ )+1 is countable in V. Fix a surjection ψβ : {nβ }×ω → Mβ &j0,β (τ )+1. Define β : ω → Mβ &j0,β (τ ) + 1 by:   φβ (r(i)) if r(i) ∈ aβ × ω; β (i) = ψβ (r(i)) if r(i) ∈ {nβ } × ω; and   ∅ otherwise.

230


There are no conflicts between the first two clauses in the definition of β , since nβ ∈ aβ . β : ω → M&j0,β (τ ) + 1 is surjective, since its range contains the range of ψβ . Using condition (ii) above it is easy to check that β and ¯ β agree to nβ . We already saw that xβ agrees with x¯β to nβ . Combining these agreements with condition (1) we get: (v) Pβ is played according to j0,β (σmix )[β , xβ ]. Conditions (3), (iv), (v), and (2) allow for an application of Claim 6C.7 over Mβ . Let n∗β , δβ∗ , and P∗β be the objects given by that claim. These objects satisfy the conditions listed in Claim 6C.7. In particular: (vi) δβ∗ is greater than the first Woodin cardinal of M above δβ , and smaller than j0,β (λ). Let κβ∗ = δβ + 1 and let δβ† be the first Woodin cardinal of Mβ above δβ . Let C˙ β∗ denote ˙ ∗ , δ ∗ , P∗ ). Let C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ). Condition (6) of Claim 6C.7 j0,β (K)(κ β

β

β

β

β

β

β

tells us that ϕsuc (tβ , wβ , C˙ β† ) holds true. ϕsuc (tβ , wβ , C˙ β† ) is absolute between generic extensions of M which have t and w as elements. So we don’t have to specify exactly in which extension it holds. We will use the fact that it holds in Mβ [gβ ]. The fact that ϕsuc (tβ , wβ , C˙ β† ) holds in Mβ [gβ ] puts us in a position to apply Lemma suc (Mβ , tβ , wβ , C˙ † ). 6B.2. The lemma states that player I has a winning strategy in G β Let β be such a strategy. Remember that we are working with an imaginary opponent to construct a run of skip (M, t, τ )(Y˙ ). We are currently working on mega-round β, where β is a successor G or zero. The mega-round is played according to rules (S1)–(S6) in Section 6A. The assignment of wβ above covers the move corresponding to rule (S1). Let β (playing for I) and the imaginary opponent (playing for II) cover the moves corresponding to rules (S2)–(S4). They produce a real yβ , an iteration tree Tβ , and a cofinal branch bβ through Tβ . Following the notation of Section 6A let Qβ be the direct limit along the branch bβ of Tβ . Let kβ : Mβ → Qβ be the direct limit embedding. Let tβ† = tβ −−, wβ , yβ . We constructed Qβ , kβ , and tβ† using the strategy β , which is winning for I in suc (Mβ , tβ , wβ , C˙ † ). Copying from the payoff conditions (B1)–(B3) in Section 6B G β we see that at least one of the following conditions must hold: • Qβ is illfounded; • tβ† is I-acceptably obstructed over Qβ ; or • Qβ is wellfounded, tβ† is obstruction free over Qβ , and tβ† belongs to an interpretation of kβ (C˙ † ) over Qβ . β

skip (M, t, τ )(Y˙ ) ends and player I wins through condition (I1) If Qβ is illfounded then G † skip (M, t, τ )(Y˙ ) in Section 6A. If tβ is I-acceptably obstructed over Qβ then again G ends, and player I wins through condition (I2) in Section 6A. So suppose that:

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(vii) Qβ is wellfounded, tβ† is obstruction free over Qβ , and tβ† belongs to an interpretation of kβ (C˙ † ) over Qβ . β

So far we constructed moves corresponding to rules (S1)–(S4) in Section 6A. We continue mega-round β with a skip subject to rules (S5) and (S6). The assignments of δβ∗ and C˙ β∗ above cover the moves described in rule (S5). We leave it to the reader to verify that these assignments satisfy the conditions of the rule. Let us just note that condition (4) in rule (S5) follows from the final clause in condition (vii) above. Let the imaginary opponent play the moves hβ , Mβ+1 , and tβ+1 corresponding to rule (S6). Let jβ,β+1 = hβ ◦ kβ . The demands placed on the imaginary opponent through rule (S6) are such that: (viii) tβ+1 extends tβ† ; and (ix) tβ+1 belongs to an interpretation of jβ,β+1 (C˙ β∗ ). Let κβ+1 = δβ + 1. It follows from condition (viii) and from the definition of tβ† that tβ+1 (κβ+1 ) equals wβ . Combining this with the definition of wβ we get: (x) tβ+1 (κβ+1 ) equals tβ (κβ ) + $nβ , δβ %. It follows from this that aβ+1 , the set of numbers used in tβ+1 (κβ+1 ), is precisely equal to aβ ∪ {nβ }. Define φβ+1 : aβ+1 × ω → Mβ+1 &j0,β+1 (τ ) + 1 by setting: (xi) φβ+1 = (jβ,β+1 ◦ φβ ) ∪ (jβ,β+1 ◦ ψβ ). The union is a function since the domains of φβ and ψβ are disjoint. The former has domain aβ × ω, the latter has domain {nβ } × ω, and nβ ∈ aβ by condition (iii). Let nβ+1 = n∗β , let Pβ+1 = jβ,β+1 (P∗β ), and let δβ+1 = jβ,β+1 (δβ∗ ). (Recall that ∗ nβ , P∗β , and δβ∗ are the objects obtained above through an application of Claim 6C.7.) Combining condition (ix) with the definition of C˙ β∗ we see that tβ+1 belongs to an ˙ β+1 , δβ+1 , Pβ+1 ). Let gβ+1 be a generic object which interpretation of j0,β+1 (K)(κ witnesses this. We have now defined all the objects listed in conditions (A)–(G) for β + 1. It is easy to check that conditions (1)–(6) hold for β + 1 with the definitions we made. Let us only make the following comments: Condition (1) for β + 1 follows from the properties of P∗β obtained through the application of Claim 6C.7, specifically the fact that P∗β is played according to j0,β (σmix )[β , xβ ]. Condition (6) for β + 1 follows from the same # (Successor case) condition for β and the fact that P∗β extends Pβ . Recall that we are working with the imaginary opponent to construct a run of skip (M, t, τ )(Y˙ ). So far we described the construction for mega-round β in the case G that β is a successor or zero. Let us now handle limit mega-rounds. Let α be a limit ordinal. Suppose the construction reached mega-round α. Suppose that conditions (1)–(6) hold for β = 0 and for all successor β < α. We describe how to play in mega-round α.

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Let Mα be the direct limit of the system $Mβ , jβ,β ¯ | β¯ ≤ β < α%. Let jβ,α be the skip (M, t, τ )(Y˙ ) ends and player I direct limit embeddings. If Mα is illfounded then G wins through condition (I3) in Section 6A. So suppose that Mα is wellfounded. Let tα = β β (β) and δβ∗ < j0,β (θ). ˙ ∗ , δ ∗ , n∗ , j0,β (γL )) and let C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ). ApplyLet C˙ β∗ = j0,β (C)(κ β β β β β β β ing condition (iv) with γ ∗ = j0,β (γL ) we see that: (vi) ϕsuc (tβ , wβ , C˙ β† ) holds. (This is just the shift to Mβ of the payoff condition (P) in Section 4D (1).) From suc (Mβ , tβ , wβ , C˙ † ). Let β condition (vi) and Lemma 6B.2 it follows that I wins G β be a winning strategy for I in this game. Remember that we are working with an imaginary opponent to construct a run of skip (M, s, θ ). We are currently working on mega-round β, which is played according G to rules (S1)–(S6) in Section 6A. The assignment of wβ above covers the move described in rule (S1). Let β (playing for I) and the imaginary opponent (playing for II) cover the moves described in rules (S2)–(S4). They produce a real yβ , an iteration tree Tβ , and a cofinal branch bβ through Tβ . Following the notation in Section 6A let Qβ be the direct limit along the branch bβ of Tβ . Let kβ be the direct limit embedding. Let tβ† = tβ −−, wβ , yβ . suc (Mβ , tβ , wβ , C˙ † ), guarantees that at The use of β , a winning strategy for I in G β least one of the following conditions must hold: • Qβ is illfounded; • tβ† is I-acceptably obstructed over Qβ ; or • Qβ is wellfounded, tβ† is obstruction free over Qβ , and tβ† belongs to an interpretation of kβ (C˙ † ) over Qβ . β

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(These conditions are simply the payoff conditions (B1)–(B3) in Section 6B, translated skip (M, s, θ)(Y˙ ) through to the current context.) If Qβ is illfounded then player I wins G † condition (I1) in Section 6A, and our job is done. If tβ is I-acceptably obstructed over skip (M, s, θ)(Y˙ ) through condition (I2) in Section 6A, and Qβ then player I wins G again our job is done. So suppose that: (vii) Qβ is wellfounded, tβ† is obstruction free over Qβ , and tβ† belongs to an interpretation of kβ (C˙ † ) over Qβ . β

We continue mega-round β with a skip following rules (S5) and (S6) in Section 6A. The assignments of δβ∗ and C˙ β∗ above cover the moves described in rule (S5). The reader may easily verify that these assignments satisfy the conditions in that rule. Note specifically how condition (4) in rule (S5) follows from the last clause in condition (vii) above. Let the imaginary opponent cover the moves hβ , Mβ+1 , and tβ+1 described in rule (S6). Let jβ,β+1 = hβ ◦ kβ . Let δβ+1 = jβ,β+1 (δβ∗ ). The demands placed on the imaginary opponent through rule (S6) are such that: (viii) tβ+1 extends tβ† ; and (ix) tβ+1 belongs to an interpretation of jβ,β+1 (C˙ β∗ ). j0,β+1 preserves countability by Remark 6A.16. Using the initial assumption that θ is countable in V it follows that j0,β+1 (θ) is countable in V, so certainly j0,β+1 (θ ) has cofinality ω in V. Pick in V a cofinal map β+1 : ω → j0,β+1 (θ ). By increasing the values taken by β+1 as needed, make sure that it also satisfies: (x) β+1 (i) ≥ (jβ,β+1 ◦ β )(i) for all i < ω. The choice of β+1 above secures the inductive condition (1) for β + 1. The inductive condition (2) for β + 1 follows from the same condition for β (but shifted to Mβ+1 ), the fact that tβ+1 extends tβ , and Claim 4D.3 applied in Mβ+1 . In shifting the hopefulness of tβ from Mβ to Mβ+1 we use the fact that jβ,β+1 has critical point greater than rdm(tβ ). This fact follows from the restrictions on critical points in rules (S3) and (S6) in Section 6A. Let κβ+1 = κβ∗ and let nβ+1 = n∗β . Using condition (ix), pick gβ+1 so as to satisfy the inductive condition (3) for β + 1. Our work above completes mega-round β of the construction (where β is smaller than ω), and puts us in the position necessary to start mega-round β + 1. Let us now fastforward to the start of mega-round ω. Let Mω be the direct limit of the system $Mβ , jβ,β ¯ | β¯ ≤ β < ω%. Let jβ,ω : Mβ → skip (M, s, θ)(Y˙ ) Mω be the direct limit maps. If Mω is illfounded then player I wins G through condition (I3) in Section 6A, and our job is done. So suppose that Mω is wellfounded. Translating the limit case condition (e) in Section 6A to our context we get:

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(xi) crit(jβ,ω ) ≥ δβ + 1 for each β < ω. Let tω = β (jβ,β+1 ◦ β )(β) and δβ+1 < j0,β+1 (θ ) for each β < ω. Applying the embedding jβ+1,ω to these inequalities we see that δβ+1 > (jβ,ω ◦ β )(β) and δβ+1 < j0,ω (θ) for each β < ω. (Note that δβ+1 is not moved by jβ+1,ω , because of condition (xi).) For each β < ω let ω (β) = (jβ,ω ◦ β )(β). By the argument in the previous paragraph δβ+1 is trapped between ω (β) and j0,ω (θ ) for each β < ω. Using conditions (1) and (x) it is easy to see that ω : ω → j0,ω (θ ) is cofinal in j0,ω (θ ). So # supβ α. The argument is similar to the one in the proof of Lemma 6D.2. The inductive condition (1) holds for β + 1 through the choice of β+1 . Our work above completes mega-round β of the construction (where β is smaller than ω) and puts us in the position necessary to start mega-round β + 1. Suppose now that we reached mega-round ω. Let Mω be the direct limit of the system $Mβ , jβ,β | ¯ ¯ β ≤ β < ω%. By condition (iv) we have: • γβ+1 < jβ,β+1 (γβ ) for each β < ω. skip (M, s, θ)(Y˙ ) ends through condiIf follows from this that Mω is illfounded. So G tion (I3) in Section 6A, and player I wins. # (Lemma 6D.8) The reader may wish to compare closely the proofs of Lemmas 6D.2 and 6D.8. Note how here we argue for victory not through the payoff condition (P2) in Section 6A, but through the “snag” of illfoundedness in condition (I3). The initial assumption here that s is not hopeful eliminates any hope for victory through condition (P2). But the same assumption gives rise to the ordinals γβ∗ which end up witnessing victory through condition (I3). 6D (3) Combined. We continue to work with a transitive model M of ZFC∗ , a Woodin limit of Woodin cardinals θ in M, and a θ-name Y˙ in M. γL is the lower ordinal in the least pair of local indiscernibles of M relative to θ . α < θ is the least ordinal which seals Y˙ . C˙ ∈ M is the function which associates to each δ ∈ [α, θ ) ∩ W , each ˙ δ, n, γ ) κ ∈ δ ∩ L, each n < ω, and each ordinal γ < jumpM (θ ), the δ-name C(κ, given by Definition 4D.5. ˙ δ, 0, γL ). For each δ ∈ [α, θ) ∩ W let C˙ δ denote the name C(0,

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Claim 6D.10. Suppose that θ is countable in V. Let δ belong to [α, θ ) ∩ W . Let s be a δ-sequence over M. Suppose that s belongs to an interpretation of C˙ δ . Then player I skip (M, s, θ)(Y˙ ). has a winning strategy in G Proof. s is a δ-sequence and δ is larger than α. So we may ask whether or not s is hopeful with respect to Y˙ . If s is hopeful with respect to Y˙ then the claim follows from Lemma 6D.2. If s is not hopeful with respect to Y˙ then the claim follows from Lemma 6D.8. # Lemma 6D.11 (for a Woodin limit of Woodin cardinals θ and a θ-name Y˙ in M). Let χ < θ be an element of W . Let r be a χ -sequence over M. Suppose that θ is countable in V. Suppose that r belongs to an interpretation of Back I (χ , θ )(Y˙ ), where the pullback is computed in M. Then player I has a winning skip (M, r, θ)(Y˙ ). strategy in G Proof. Let χ † be the first Woodin cardinal of M above χ . Remember that C˙ δ , defined ˙ for each δ ∈ [α, θ) ∩ W , denotes the name C(0, δ, 0, γL ). For each δ > χ † in [α, θ ) ∩ I † W let A˙ δ = Back (χ , δ)(C˙ δ ). Let q witness that r belongs to an interpretation of Back I (χ , θ )(Y˙ ). q is col(ω, χ)-generic/M, and r belongs to Back I (χ , θ )(Y˙ )[q]. The fact that r ∈ Back I (χ, θ)(Y˙ )[q] allows us, directly by Definition 4D.6, to fix some δ > χ † in [α, θ ) ∩ W , and to fix in M[q] some witness w for χ + 1, so that: (i) ϕsuc (r, w, A˙ δ ) holds. From this and Lemma 6B.2 it follows that player I has a winning strategy in suc (M, r, w, A˙ δ ). Let be a winning strategy for I in this game. G skip (M, r, θ)(Y˙ ). We Fix now an imaginary opponent willing to play for II in G describe how to play for I, and win. Let M0 = M and let t0 = t. We start with mega-round 0. We play according to the rules (S1)–(S6) in Section 6A. Let I play the witness w of condition (i) above for the move w0 described in rule (S1). Then let I follow for the moves described in rules (S2)–(S4). Player I and the imaginary opponent together create y0 , T0 , and b0 . Following the notation of Section 6A let Q0 be the direct limit along the branch b0 of T0 , and let k0 be the direct limit embedding. Let t0† = r−−, w, y0 . The use of guarantees that at least one of the following conditions holds: • Q0 is illfounded; • t0† is I-acceptably obstructed over Q0 ; or • Q0 is wellfounded, t0† is obstruction free over Q0 , and t0† belongs to an interpretation of k0 (A˙ δ ) over Q0 . (These conditions are simply the winning conditions listed in Section 6B, translated skip (M, r, θ)(Y˙ ) ends and I wins to the current context.) If Q0 is illfounded then G † through condition (I1) in Section 6A. If t0 is I-acceptably obstructed over Q0 then skip (M, r, θ )(Y˙ ) ends and I wins through condition (I2). So suppose that: G

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(ii) Q0 is wellfounded, t0† is obstruction free over Q0 , and t0† belongs to an interpretation of k0 (A˙ δ ) over Q0 . We continue mega-round 0 with a skip according to rules (S5) and (S6) in Section 6A. Let I play δ0∗ = δ and C˙ 0∗ = C˙ δ for the moves described in rule (S5). It is easy to check that these moves satisfy the conditions of rule (S5). Let us just note that condition (4) in rule (S5) follows from the last clause in condition (ii) above. The imaginary opponent responds with moves h0 , M1 , and t1 subject to the conditions in rule (S6). Let j0,1 = h0 ◦ k0 . The conditions of rule (S6) are such that: (iii) t1 belongs to an interpretation of j0,1 (C˙ δ ) over M1 . j0,1 preserves countability by Remark 6A.16. Since θ is assumed countable in V it follows that j0,1 (θ) is countable in V. Using this and condition (iii) we may apply Claim 6D.10 over M1 , with s = t1 . The claim states that player I has a winning strategy skip (M1 , t1 , j0,1 (θ))(j0,1 (Y˙ )). Continue to play for I by following this strategy. in G skip (M1 , t1 , j0,1 (θ )) are precisely This is possible since the rules for mega-round β in G skip (M, r, θ). Following the same as the rules for mega-round 1 + β in our run of G the strategy given by Claim 6D.10 may lead to victory for I through one of the “snags” described in conditions (I1)–(I7) in Section 6A, and in this case we are done. Otherwise it leads to: (iv) a wellfounded model Mω ; (v) an elementary embedding j1,ω : M1 → Mω ; and (vi) a j1,ω (j0,1 (θ))-sequence tω over Mω , which belongs to an interpretation of j1,ω (j0,1 (Y˙ )). (These conditions are simply the end conditions in the construction for Lemma 6D.2, translated to apply over M1 .) The position described by conditions (iv)–(vi) is a victory skip (M, r, θ)(Y˙ ), so again we are done. for player I in G skip (M, r, θ)(Y˙ ) we Working with an imaginary opponent who plays for II in G described how to play for I, and win. The description can be formalized to give a skip (M, r, θ)(Y˙ ). # winning strategy for I in G

6E Relative successors and compositions Let M be a transitive model of ZFC∗ . Let δ † be a relative successor in W (computed in M). Let δ be the largest Woodin cardinal of M below δ † . We work with these fixed objects throughout Section 6E. Recall that the pullback operation is defined in five cases, listed in Section 4D (3). Our work so far handled situations which correspond to cases (I) and (II). We handle here the situations which correspond to cases (III)–(V).

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Lemma 6E.1. Let t be a δ-sequence over M. Let C˙ † be a δ † -name in M. Suppose that M&δ † +1 is countable in V. Suppose that t belongs to an interpretation of Back I (δ, δ † )(C˙ † ), where the pullback is computed in M. Then player I has a winning skip (M, t, δ † )(C˙ † ). strategy in G Proof. We divide the proof into two cases, depending on whether δ is a Woodin limit of Woodin cardinals or an element of W . In both cases we follow the notation of Section 4D (2). Case 1. Suppose first that δ is an element of W . Using the assumption that t belongs to an interpretation of Back I (δ, δ † )(C˙ † ) fix some g so that: (i) g is col(ω, δ)-generic/M; and (ii) t belongs to Back I (δ, δ † )(C˙ † )[g]. Using the last condition and looking at the definitions in Section 4D (2) we see that: (iii) In M[g] there exists a witness w for δ + 1 so that ϕsuc (t, w, C˙ † ) holds. Fix such a witness w. Since M&δ † +1 is countable in V we may appeal to Lemma 6B.2. Applying the lemma to the objects of condition (iii) we get: suc (M, t, w, C˙ † ). (iv) Player I has a winning strategy in G skip (M, t, δ † )(C˙ † ). The It is easy from this to see that I has a winning strategy in G game ends after mega-round 0, and I can win by playing w for rule (S1), following her suc (M, t, w, C˙ † ) for rules (S2)–(S4), and declaring an early end winning strategy in G so as to avoid rules (S5) and (S6). # (Case 1) Case 2. Suppose next that δ is a Woodin limit of Woodin cardinals in M. In this case Back I (δ, δ † )(C˙ † ) is a name in the forcing Wδ . Using the assumption that t belongs to an interpretation of Back I (δ, δ † )(C˙ † ) fix some G so that: (i) G is Wδ -generic/M; and (ii) t belongs to Back I (δ, δ † )(C˙ † )[G]. Using the last condition and looking at the definitions in Section 4D (2) we see that: (iii) The statement “there exists some witness w for δ so that ϕsuc (t, w, C˙ † ) holds” is forced by some condition in col(ω, δ) over M[G]. We’re assuming in Lemma 6E.1 that M&δ † + 1 is countable in V. Since δ < δ † this assumption certainly allows finding in V generics for col(ω, δ) over M[G]. Using condition (iii) fix some g and some w so that: (iv) g is col(ω, δ)-generic/M[G]; (v) w ∈ M[G][g] is a witness for δ; and

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(vi) ϕsuc (t, w, C˙ † ) holds true (in M[G][g]). Applying Lemma 6B.2 we see that: suc (M, t, w, C˙ † ). (vii) Player I has a winning strategy in G As in case 1 it is now easy to check that player I has a winning strategy in skip (M, t, δ † )(C˙ † ). # (Case 2) G The two cases above complete the proof of Lemma 6E.1.

#

Lemma 6E.2 (for a relative successor δ † ∈ W ). Let χ < δ † be an element of W . Let r be a χ-sequence over M. Let C˙ † be a δ † -name in M. Suppose that M&δ † +1 is countable in V. Suppose that r belongs to an interpretation of Back I (χ , δ † )(C˙ † ), where the pullback is computed in M. Then player I has a winning skip (M, r, δ † )(C˙ † ). strategy in G Proof. We work by induction on δ † . (The use of the inductive assumption is made in Claim 6E.3 below.) The assumption that χ is an element of W smaller than δ † implies that χ is smaller than or equal to δ. If χ is equal to δ then the current lemma follows from Lemma 6E.1. So suppose that χ < δ. Let C˙ = Back I (δ, δ † )(C˙ † ). The definition of the pullback in ˙ So: case (IV) in Section 4D(3) is such that Back I (χ , δ † )(C˙ † ) = Back I (χ , δ)(C). ˙ (i) r belongs to an interpretation of Back I (χ , δ)(C). skip (M, r, δ)(C). ˙ Claim 6E.3. Player I has a winning strategy in G Proof. The claim follows from condition (i) using: Lemma 6D.11 if δ is a Woodin limit of Woodin cardinals in M; Lemma 6C.14 if δ is a relative limit in W ; and an inductive application of Lemma 6E.2 if δ is a relative successor in W . # Remember that our goal is to show that player I has a winning strategy in skip (M, r, δ † )(C˙ † ). One can obtain such a strategy by composing Claim 6E.3 with G an application of Lemma 6E.1. Let us quickly describe this composition. We describe skip (M, r, δ † )(C˙ † ), and win. how to play for I in G skip (M, r, δ)(C), ˙ given by Claim 6E.3. Start by following a winning strategy for I in G This may lead to a victory for I through one of the “snags” described in conditions (I1)– (I7) in Section 6A, and in this case we are done. Otherwise it leads to a position which skip (M, r, δ)(C), ˙ through one of the payoff conditions (P1) and is won by player I in G (P2). In other words it leads to: (ii) a wellfounded model Mβ+1 ; (iii) an elementary embedding j0,β+1 : M → Mβ+1 ; and (iv) a j0,β+1 (δ)-sequence tβ+1 over Mβ+1 , which belongs to an interpretation of ˙ j0,β+1 (C).

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Note that Mβ+1 &j0,β+1 (δ † ) + 1 is countable in V. This follows from Remark 6A.16 and the initial assumption that M&δ † + 1 is countable in V. Using condition (iv) an application of Lemma 6E.1 over Mβ+1 shows that I has a winning strategy in skip (Mβ+1 , tβ+1 , j0,β+1 (δ † ))(j0,β+1 (C˙ † )). Continue by following this strategy. This G may lead to a victory for I through one of the “snags” described in conditions (I1) and (I2) in Section 6A, and in this case we are done. Otherwise it leads to: (v) a wellfounded model Mβ+2 ; (vi) an elementary embedding jβ+1,β+2 : Mβ+1 → Mβ+2 ; and (vii) a j0,β+2 (δ † )-sequence tβ+2 over Mβ+2 , which belongs to an interpretation of j0,β+2 (C˙ † ). skip (M, r, δ † )(C˙ † ). This is a victory for I in G

# (Lemma 6E.2)

Let us continue to work under the assumption that δ † is a relative successor in W , and δ is the largest Woodin cardinal of M below δ † . The next lemma handles a situation which corresponds to case (V) in the definition of the pullback operation in Section 4D (3). Lemma 6E.4. Suppose that δ is a Woodin limit of Woodin cardinals in M. Let t be a δ-sequence over M. Let δ ∗ > δ be a Woodin cardinal in M. Let C˙ ∗ be a δ ∗ -name in M. Suppose that M&δ ∗ +1 is countable in V. Suppose that t belongs to an interpretation of Back I (δ, δ ∗ )(C˙ ∗ ), where the pullback is computed in M. Then player I has a winning skip (M, t, δ ∗ )(C˙ ∗ ). strategy in G Proof. The assumption that δ ∗ > δ implies that δ ∗ is greater than or equal to δ † . If δ ∗ = δ † then the current lemma follows from Lemma 6E.1. So suppose that δ ∗ > δ † . Let C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ). The definition of the pullback operation in case (V) in Section 4D (3) is such that Back I (δ, δ ∗ )(C˙ ∗ ) = Back I (δ, δ † )(C˙ † ). So: (i) t belongs to an interpretation of Back I (δ, δ † )(C˙ † ). Using Lemma 6E.1 it follows that: skip (M, t, δ † )(C˙ † ). (ii) Player I has a winning strategy in G skip (M, t, δ ∗ )(C˙ ∗ ). One Our goal is to show that player I has a winning strategy in G can obtain such a strategy by composing condition (ii) above with a follow-up strategy given by Claim 6E.5 below. Let us quickly describe this composition. We describe how skip (M, t, δ ∗ )(C˙ ∗ ), and win. to play for I in G skip (M, t, δ † )(C˙ † ), given We start by letting player I follow her winning strategy in G by condition (ii). This may lead to a victory for I through one of the “snags” described in conditions (I1) and (I2) in Section 6A, and in this case we are done. Otherwise it leads to:

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(iii) a wellfounded model M1 ; (iv) an elementary embedding j0,1 : M → M1 ; and (v) a j0,1 (δ † )-sequence t1 over M1 , which belongs to an interpretation of j0,1 (C˙ † ). skip (M1 , t1 , j0,1 (δ ∗ ))(j0,1 (C˙ ∗ )). Claim 6E.5. I has a winning strategy in G Proof. Note that M1 &j0,1 (δ ∗ ) + 1 is countable in V. This can be seen using Remark 6A.16 and the initial assumption that M&δ ∗ + 1 is countable in V. The claim now follows from condition (v) through an application over M1 of the appropriate lemma in the following list: Lemma 6C.14 if δ ∗ is a relative limit in W ; Lemma 6D.11 if δ ∗ is a Woodin limit of Woodin cardinals in M; and Lemma 6E.2 if δ ∗ is a relative successor in W . # Remember that we are working to describe a winning strategy for player I in Gskip (M, t, δ ∗ )(C˙ ∗ ). So far we played mega-round 0 and reached the position described in conditions (iii)–(v). We continue by letting player I follow her winning skip (M1 , t1 , j0,1 (δ ∗ ))(j0,1 (C˙ ∗ )), given by Claim 6E.5. This is possible strategy in G skip (M1 , t1 , j0,1 (δ ∗ )) are precisely the same as the since the rules for mega-round β in G skip (M, t, δ ∗ ). Following the strategy rules for mega-round 1 + β in the current run of G given by Claim 6E.5 may lead to a victory for I through one of the “snags” described in conditions (I1)–(I7) in Section 6A, and in this case we are done. Otherwise it leads to skip (M1 , t1 , j0,1 (δ ∗ ))(j0,1 (C˙ ∗ )), through one a position which is won by player I in G of the payoff conditions (P1) and (P2). In other words it leads to: (vii) a wellfounded model Mβ+1 ; (viii) an elementary embedding j1,β+1 ; and (ix) a j1,β+1 (j0,1 (δ ∗ ))-sequence tβ+1 over Mβ+1 , which belongs to an interpretation of j1,β+1 (j0,1 (C˙ ∗ )). skip (M, t, δ ∗ )(C˙ ∗ ), so again we are done. This position is a victory for I in G # (Lemma 6E.4)

6F Skips We work in this section to combine various strategies for I in skipping games into a branch . Let us start by consolidating our knowledge of strategies in strategy for I in G skipping games. Definition 6F.1. Let M be a transitive model of ZFC∗ , let δ < δ ∗ be Woodin cardinals of M, let C˙ ∗ be a δ ∗ -name in M, and let t be a δ-sequence over M. The tuple $M, t, δ ∗ , C˙ ∗ % is called promising just in case that:

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(1) M&δ ∗ + 1 is countable in V; and (2) t belongs to an interpretation of Back I (δ, δ ∗ )(C˙ ∗ ) (computed in M). We omit δ in the notation since it can be recovered from rdm(t). Lemma 6F.2. Let $M, t, δ ∗ , C˙ ∗ % be promising. Then player I has a winning strategy skip (M, t, δ ∗ )(C˙ ∗ ). in G Proof. Let δ be the unique ordinal so that t is a δ-sequence. The current lemma is simply: Lemma 6C.14 if δ ∗ is a relative limit in W and δ ∈ W ; Lemma 6D.11 if δ ∗ is a Woodin limit of Woodin cardinals in M and δ ∈ W ; Lemma 6E.2 if δ ∗ is a relative successor and δ ∈ W ; and Lemma 6E.4 if δ is a Woodin limit of Woodin cardinals in M. # skip be a function which associates to each promising tuple $M, t, δ ∗ , C˙ ∗ % a Let skip (M, t, δ ∗ )(C˙ ∗ ). Such skip (M, t, δ ∗ , C˙ ∗ ) which is winning for player I in G strategy a strategy exists by Lemma 6F.2. branch , with the goal skip into a strategy in G We intend to combine instances of ∗ ∗ ˙ of showing that I wins Gbranch (M, t, δ )(C ) whenever $M, t, δ ∗ , C˙ ∗ % is promising. branch Sections 6F (1) through 6F(3) establish terminology which connects positions in G with positions in Gskip . Section 6F(4) gives an informal idea of the argument which branch . Sections 6F (5) and skip to produce a strategy in G sews together instances of 6F (6) give the actual argument. 6F (1) Routes. Let $M, t, δ ∗ , C˙ ∗ % be promising. We work here to connect positions in skip (M, t, δ ∗ )(C˙ ∗ ). branch (M, t, δ ∗ )(C˙ ∗ ) with positions in G G skip (M, t, δ ∗ )(C˙ ∗ ) as being given by We think of positions of γ mega-rounds in G ∗ ∗ sequences $Tβ , bβ , C˙ β , hβ , Eβ , tβ+1 | β < γ %. Not all the objects listed are defined for each β. For example C˙ β∗ and hβ are only defined if β is a successor, and only if I called for a skip in mega-round β. Eβ∗ is only defined if β is a standard limit, and only if II elected a leap in mega-round β. skip (M, t, δ)(C˙ ∗ ) which are not listed among There are moves in mega-round β of G the objects Tβ , bβ , C˙ β∗ , hβ , Eβ∗ , and tβ+1 . But all the objects played can be recovered from the objects listed. For example wβ and yβ can be recovered from (tβ and) tβ+1 , and Mβ+1 in the case of a skip can be recovered from (Mβ and) hβ . So the sequence $Tβ , bβ , C˙ β∗ , hβ , Eβ∗ , tβ+1 | β < γ % gives a complete account of a position in skip (M, t, δ ∗ )(C˙ ∗ ). G skip (M, t, δ ∗ )(C˙ ∗ ) is a sequence Definition 6F.3. A reduced position in G R = $Tβ , bβ , hβ , Eβ∗ , tβ+1 | β < γ % which can be expanded, via the addition of objects $C˙ β∗ | β < γ %, to a position in skip (M, t, δ ∗ )(C˙ ∗ ). The combined sequence $Tβ , bβ , C˙ ∗ , hβ , E ∗ , tβ+1 | β < γ % is G β β called an expansion of R.

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Definition 6F.4. A reduced position R = $Tβ , bβ , hβ , Eβ∗ , tβ+1 | β < γ % is said to skip (M, t, δ ∗ , C˙ ∗ ) just in case that it can be expanded to a position be consistent with skip (M, t, δ ∗ , C˙ ∗ ). which is played according to Let R = $Tβ , bβ , hβ , Eβ∗ , tβ+1 | β < γ % be a reduced position consistent with skip . It follows skip (M, t, δ ∗ , C˙ ∗ ). Note that the moves C˙ ∗ correspond to player I in G β from this that there is a unique expansion of R to a position which is played accordskip (M, t, δ ∗ , C˙ ∗ ). We refer to this unique expansion as the position given ing to by R. (There is some abuse of notation here, since the notion depends on the tuple skip (M, t, δ ∗ , C˙ ∗ ). But this tuple is always $M, t, δ ∗ , C˙ ∗ % through the reference to clear from the context.) branch (M, t, δ ∗ )(C˙ ∗ ) as being given by sequences We think of positions in G ∗ $Tξ , bξ , Eξ , tξ +1 | ξ < η%. (This notation was already introduced in Section 6A.) Here again not all objects are defined for each ξ . Eξ∗ is only defined if ξ is a standard limit, and only if II elected a leap in mega-round ξ . Again there are moves in megaround ξ which are not among the objects listed, the objects Tξ , bξ , Eξ∗ , and tξ +1 that is. But all the moves in mega-round ξ can be recovered from these objects. Let η be an ordinal. Let P = $Tξ , bξ , Eξ∗ , tξ +1 | ξ < η% be a position of length branch (M, t, δ ∗ )(C˙ ∗ ). We work with this fixed P for the rest of Section 6F (1). η in G P gives rise to: • models Mξ for ξ ≤ η; and • embeddings jζ,ξ : Mζ → Mξ for ζ < ξ ≤ η. The definitions and claims below are made with reference to these models and embeddings. Definition 6F.5. A skip frame for P is a function f , from some ordinal γ + 1 into η + 1, satisfying the following conditions: (1) f : γ + 1 → η + 1 is increasing and continuous at limits; (2) f (0) = 0 and f (γ ) = η; (3) if β < γ is a limit then f (β + 1) = f (β) + 1; (4) if β < γ is a successor or zero then f (β + 1) is a successor, either equal to or greater than f (β) + 1. γ is called the length of f , denoted lh(f ). Claim 6F.6. Let f be a skip frame for P . Then f maps limit ordinals to limit ordinals, successor ordinals to successor ordinals, and zero to zero. Proof. That successors are mapped to successors follows from conditions (3) and (4) in Definition 6F.5. That limits are mapped to limits follows from condition (1). That zero is mapped to zero follows directly from condition (2). #

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Remark 6F.7. As a special case of Claim 6F.6 we see that lh(f ) is a limit (respectively successor, zero) iff lh(P ) is a limit (respectively successor, zero). Definition 6F.8. Let f be a skip frame for P . For each β < lh(f ) let: jf (β)+1,f (β+1) if f (β + 1) > f (β) + 1; and hβ = undefined if f (β + 1) = f (β) + 1. The definition of hβ is made with reference to both f and P . (P is needed to give rise to the embeddings jζ,ξ .) But we suppress this dependence in the notation. Define red(P , f ) to be the sequence $Tf (β) , bf (β) , hβ , Ef∗ (β) , tf (β+1) | β < lh(f )%. skip (M, t, δ ∗ )(C˙ ∗ ). Notice that this sequence has the format of a reduced position in G The notation “red” stands for “reduced.” We are using the skip frame f to convert P into a reduced position R = red(P , f ) in the skipping game. If β is such that f (β + 1) = f (β) + 1 then we take mega-round f (β) in P and pass it without change to be mega-round β in R. If β is such that f (β + 1) > f (β) + 1 then we take megarounds ξ in P for all ξ ∈ [f (β), f (β + 1)) and lump them together into mega-round β in R. The “lumping,” which is illustrated in Diagram 6.6, is done by calling for a skip in mega-round β of R and using hβ = jf (β)+1,f (β+1) for that skip. Mf (β)

kkk Sk k Sk SSSS

/ Qf (β)

=

Mf (β)+1

/ Mf (β+1)

jf (β)+1,f (β+1)

Tf (β)

hβ Mf (β)

kkk Sk k Sk SSSS

/ Qf (β)

+

Mf (β+1)

Tf (β)

Diagram 6.6. Mega-rounds f (β) to f (β + 1) in P (upper line) lumped together into a single mega-round in R (lower line).

branch (M, t, δ ∗ )(C˙ ∗ ). Let f be a skip Definition 6F.9. Let P be a position in G frame for P . f is a route to P just in case that red(P , f ) is a reduced position in skip (M, t, δ ∗ )(C˙ ∗ ), consistent with skip (M, t, δ ∗ , C˙ ∗ ). G Intuitively a route to P is a way to generate P through a play of the skipping game, skip (M, t, δ ∗ , C˙ ∗ ). The definition of a route depends following I’s winning strategy ∗ ∗ skip (M, t, δ ∗ , C˙ ∗ ). But we suppress this on M, t, δ , and C˙ through the reference to dependence in the notation. Claim 6F.10. Let f be a route to P . Let β be a successor ordinal smaller than lh(f ). Then mega-round β in red(P , f ) contains a skip just in case that f (β + 1) > f (β) + 1.

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Proof. This is immediate from the definitions. Mega-round β in red(P , f ) contains a skip iff hβ is defined, and this by Definition 6F.8 happens precisely when f (β + 1) is greater than f (β) + 1. # By a P -route we mean a route either to P or to a proper initial segment of P . Lemma 6F.11. Let f be a P -route. Let γ = lh(f ). Then there is at most one P -route of length γ + 1 extending f . Proof. If f (γ ) = lh(P ) then there are no P -routes which extend f at all. So suppose that f (γ ) < lh(P ). Let ξ denote f (γ ). Let f ∗ and g ∗ be P -routes of length γ + 1, both extending f . Let ξ ∗ = f ∗ (γ + 1) and let ζ ∗ = g ∗ (γ + 1). We work to show that ξ ∗ = ζ ∗ . Since both f ∗ and g ∗ extend f this is enough to establish that f ∗ = g ∗ . If γ is a limit then both f ∗ (γ +1) and g ∗ (γ +1) must equal f (γ )+1 by condition (3) in Definition 6F.5, and in particular f ∗ (γ + 1) = g ∗ (γ + 1). So suppose that γ is a successor (or zero). Using Claim 6F.6 it follows that ξ too is a successor (or zero). Since ξ is a successor (or zero), mega-round ξ in P is played according to rules (S1)–(S4) in Section 6A. Let wξ , yξ , Tξ , and bξ be the moves played in mega-round ξ of P . Let kξ be the direct limit embedding along the branch bξ of Tξ . The settings in branch are such that: G (i) jξ,ξ +1 = kξ . Let R ∗ denote red(P ξ ∗ , f ∗ ) and let S ∗ denote red(P ζ ∗ , g ∗ ). Let R denote red(P ξ, f ). Then: (ii) R ∗ and S ∗ are reduced positions of length γ + 1; (iii) both R ∗ and S ∗ extend R; skip (M, t, δ ∗ , C˙ ∗ ); and (iv) both R ∗ and S ∗ are consistent with (v) in both R ∗ and S ∗ the moves corresponding to rules (S1)–(S4) in mega-round γ are precisely the moves wξ , yξ , Tξ , and bξ from mega-round ξ of P . The last condition follows from the fact that both f ∗ (γ ) and g ∗ (γ ) are equal to f (γ ), which is equal to ξ . skip (M, t, δ ∗ )(C˙ ∗ ) which consists of the γ megaLet K denote the position in G rounds given by R, followed by the moves wξ , yξ , Tξ , and bξ for rules (S1)–(S4) in mega-round γ . Both R ∗ and S ∗ extend K, and do so in a manner consistent with skip (M, t, δ ∗ , C˙ ∗ ). We shall use this to argue that in fact they are equal. We divide skip (M, t, δ ∗ , C˙ ∗ ) calls for a skip or the proof into two cases, depending on whether for an early end following K. skip (M, t, δ ∗ , C˙ ∗ ) elects an early end following the position given by K. In Case 1. If skip (M, t, δ ∗ , C˙ ∗ ), this case both R ∗ and S ∗ , which extend K and follow the dictates of must take an early end in mega-round γ . Using the appropriate instances of Claim 6F.10

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it follows that f ∗ (γ + 1) = f ∗ (γ ) + 1 and that g ∗ (γ + 1) = g ∗ (γ ) + 1. Since f ∗ (γ ) and g ∗ (γ ) are both equal to ξ = f (γ ) it follows from this that f ∗ (γ + 1) = g ∗ (γ + 1), as required. # (Case 1) skip (M, t, δ ∗ , C˙ ∗ ) calls for a skip following the position given by K. In Case 2. If this case both R ∗ and S ∗ must contain a skip in mega-round γ . The skips are made according to the format set in rules (S5) and (S6) in Section 6A. Let δγ∗a , C˙ γ∗a , haγ , Mγa +1 , and tγa +1 be the moves corresponding to the skip in megaround γ of R ∗ . Let δγ∗b , C˙ γ∗b , hbγ , Mγb +1 , and tγb +1 be the moves corresponding to the skip in mega-round γ of S ∗ . Since both R ∗ and S ∗ extend K in a manner consistent with the strategy skip (M, t, δ ∗ , C˙ ∗ ), the moves corresponding to player I are the same in R ∗ and in S ∗ . In particular this means that: (vi) δγ∗a = δγ∗b . This ultimately will allow us to show that ξ ∗ = ζ ∗ . Since R ∗ = red(P ξ ∗ , f ∗ ), the moves haγ and tγa +1 are the ones induced by P ξ ∗ and f ∗ through the definition of red(. . . ) above. So: (vii) haγ is equal to jf ∗ (γ )+1,f ∗ (γ +1) and tγa +1 = tf ∗ (γ +1) . Recall that ξ ∗ denotes f ∗ (γ + 1), and ξ denotes f (γ ) which is the same as f ∗ (γ ). Folding this into the last condition we get: (viii) haγ is equal to jξ +1,ξ ∗ and tγa +1 = tξ ∗ . The skip in mega-round γ of R ∗ is subject to rules (S5) and (S6) in Section 6A. Using condition (2) in rule (S6) and condition (1) in rule (S5) we see that: (ix) The relative domain of tγa +1 is equal to (haγ ◦ kξ )(δγ∗a ) + 1. Combining this with conditions (viii) and (i) above we get: (x) rdm(tξ ∗ ) = jξ,ξ ∗ (δγ∗a ) + 1. Recall that ζ ∗ denotes g ∗ (γ + 1). Working as we did above but with g ∗ and S ∗ we get the following condition, which is a parallel for g ∗ of condition (x). (xi) rdm(tζ ∗ ) = jξ,ζ ∗ (δγ∗b ) + 1. Remember that we are trying to prove that ξ ∗ = ζ ∗ . Suppose this is not the case. Assume for definitiveness that ζ ∗ < ξ ∗ . branch are such that: The rules of G (xii) the critical point of jζ ∗ ,ξ ∗ is greater than rdm(tζ ∗ ); and (xiii) tξ ∗ extends tζ ∗ strictly.

254


Combining condition (x), the shift of condition (xi) via jζ ∗ ,ξ ∗ , and most importantly condition (vi) we get: (xiv) jζ ∗ ,ξ ∗ (rdm(tζ ∗ )) is equal to rdm(tξ ∗ ). Conditions (xii) and (xiv) imply that rdm(tζ ∗ ) = rdm(tξ ∗ ), and this contradicts condition (xiii). # (Case 2) # (Lemma 6F.11) branch (M, t, δ ∗ )(C˙ ∗ ). Let f and f ∗ Corollary 6F.12. Let P and P ∗ be positions in G ∗ be routes to P and P respectively. Suppose that P ∗ extends P ( perhaps not strictly). Then f ∗ extends f ( perhaps not strictly). Proof. Suppose for contradiction that f ∗ does not extend f . It is easy to check directly from Definition 6F.5 that f ∗ cannot be a strict initial segment of f . So there must be some ordinal on which f and f ∗ take different values. Let α be the least such. Because of the continuity requirement in Definition 6F.5, α must be a successor ordinal. Let γ = α − 1. Let f¯ = f γ + 1. f¯ is a P ∗ -route of length γ . f γ + 2 and f ∗ γ + 2 are both ∗ P -routes of length γ + 1, extending f¯. Moreover f γ + 2 and f ∗ γ + 2 are distinct since f and f ∗ take different values on α = γ + 1. This contradicts Lemma 6F.11. # branch (M, t, δ ∗ )(C˙ ∗ ) just in case that there Definition 6F.13. P is said to be secure in G is a route to P . branch (M, t, δ ∗ )(C˙ ∗ ). Then there exists exactly Corollary 6F.14. Let P be secure in G one route to P . Proof. Suppose that f and g are two routes to P . Corollary 6F.12 implies that f extends g, and that g extends f . So f = g. # ρ% be a strictly increasing Corollary 6F.15. Let ρ be an ordinal. Let $Pι | ι < branch (M, t, δ ∗ )(C˙ ∗ ). Let P∞ = sequence of positions in G ι

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