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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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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135 135 141 155 155 157 158 162 162 165 166 166 167 167 168 169 172 173 174 174
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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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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
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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) . . . . . . . . .
$x(i) | i < ω% ∈ C
(2)
where again the quantifiers range over natural numbers. Determinacy then is the statement that one of the equations holds, in other words that equation (2) is the negation of equation (1). This would simply be a matter of logic if the quantifier strings were finite. But the strings here are of length ω, and determinacy is far from trivial. Indeed, one cannot expect determinacy to hold for all sets: using the axiom of choice, more specifically using a wellordering of the real line, a straightforward construction produces a non-determined set C. Surprisingly, it has turned out that one can expect determinacy to hold for “concrete” sets of reals, meaning sets which are definable, over the real line and in fact more.
2
Introduction
Following standard usage let ω<ω denote the set of finite sequences of natural numbers. For s ∈ ω<ω let Ns = {x ∈ ωω | x extends s}. The sets Ns (s ∈ ω<ω ) are the basic open neighborhoods in ωω , and a set C ⊂ ωω is open if it is a union of basic open neighborhoods. Similar definitions can be made for the product spaces (ωω )k (though strictly speaking there is no need for this, since these spaces are isomorphic to ωω ). A set D ⊂ ωω is analytic if it is the projection of the complement of an open set. In other words D ⊂ ωω is analytic if there is an open C ⊂ (ωω )2 so that x ∈ D ⇐⇒ (∃y ∈ ωω )$x, y% ∈ C. The class of analytic sets begins a hierarchy of definability, from open sets, through applications of complementations and real quantifiers. (A real quantifier here means a quantifier over ωω . The reason for the terminology is that ωω is isomorphic to the real line minus the rational numbers. In descriptive set theory R often means ωω , for the same reason.) In general a set D ⊂ ωω is said to be 1n if there is an open set C ⊂ (ωω )n+1 so that: (∃y1 )(∀y2 ) . . . . . . (∃yn )$x, y1 , . . . , yn % ∈ C if n is odd; and x ∈ D ⇐⇒ (∃y1 )(∀y2 ) . . . . . . (∀yn )$x, y1 , . . . , yn % ∈ C if n is even, where the quantifiers in both cases range over ωω . The analytic sets are thus the 11 sets. A set is said to be 1n if its complement is 1n , and said to be 1n if it is both 1n and 1n . (The lightface versions n1 , 1n , and 1n are defined similarly, only replacing open, which means an arbitrary union of basic neighborhoods, by its lightface parallel, namely a recursively enumerable union of basic neighborhoods.) A set is projective if it is 1n for some n < ω, in other words if it can be obtained from open and closed sets through the application of finitely many real quantifiers. The point of introducing here the definitions of open, analytic, and projective sets is that these sets are somehow more concrete than the general sets one can obtain from the axiom of choice, and the kind of argument that uses a wellordering of the reals to contradict determinacy should not apply to them. These sets could, conceivably, be determined. At the start of the projective hierarchy stand the open sets, and in 1953 Gale–Stewart [8] proved that they are determined. The proof uses the kind of quantifier reasoning that works for finite games, and adds to it a convergence argument. Fourteen years after its publication, Blackwell [2] used the determinacy of open games to give a new proof of Kuratowski’s reduction theorem for analytic sets [16], a theorem which states that the intersection of any two analytic sets A, B ⊂ R can be presented as the intersection of two analytic sets A ⊃ A and B ⊃ B, such that A ∪B = R. Inspired by the method of his proof, Martin [20] and Addison–Moschovakis [1] proved a similar theorem for 31 sets, assuming the determinacy of 12 sets. Moreover they obtained for the 13 level the same structural property of the 11 level which is used to prove Kuratowski’s theorem for 11 sets, and in fact propagated this property through all the odd projective levels assuming projective determinacy. There have been earlier results on sets of reals proved using infinite games, see Oxtoby [36], Davis [4], and Mycielski–Swierczkowski [28]. These papers derived regularity properties, Lebesgue measurability in the case of Mycielski–Swierczkowski
Introduction
3
[28], for sets of reals assuming determinacy. But the work of Martin and Addison– Moschovakis was the first to apply determinacy to a structural property of a class of definable sets. It gave a substantial push to the doctrine—an elaboration on the proposal in Mycielski–Steinhaus [27]—of assuming determinacy in the study of definable sets of reals. Additional research following this doctrine gradually established a very rich structure theory for classes of definable sets of reals, see for example Moschovakis’ book [26]. To actually bring this theory to bear on a particular level of definability one had to prove the relevant determinacy. The open games determinacy of Gale–Stewart [8] and the somewhat stronger results of Wolfe [43] and Davis [4] on low levels of the Borel hierarchy were only enough to bring the structure theory to bear on the level of analytic sets, where it was anyway known classically. Even full Borel determinacy (which would be proved in ZFC by Martin [22] some years later) would not lift the theory further. Applying the theory to levels beyond analytic required proving determinacy beyond Borel games. There were some vague indications at the time that determinacy may be connected to large cardinal axioms—axioms which state the existence of cardinals with various reflection properties not provable in ZFC. Solovay had shown that determinacy for all subsets of ωω (a statement which contradicts the axiom of choice) implies that ω1 carries a total, countably complete, 2-valued measure. Cardinals which carry such measures are called measurable, and in the context of the axiom of choice their existence is a large cardinal axiom. In the other direction Solovay had proposed that large cardinal axioms (the ones he had in mind were much stronger than measurable) should imply projective determinacy. It came as a very pleasant surprise when, not long afterwards, Martin [21] proved 11 determinacy from a measurable cardinal. This was the first, and at the time only, proof of determinacy from a large cardinal axiom. Its ideas gave rise to several important concepts and methods, including the notion of homogeneously Suslin sets which is now useful both in proofs of determinacy and in the study of consequences of determinacy. Moreover it turned out through work of Martin and Harrington that the proof is essentially optimal. A large cardinal axiom of roughly the magnitude of a measurable cardinal is necessary for a proof of 11 determinacy, see Harrington [9]. 11 determinacy allowed extending some of the regularity results previously known for analytic sets, to the level of 12 sets. But an extension of the deeper structural results to levels beyond analytic required at least the determinacy of games with 12 payoff—the kind of games used in the results of Martin [20] and Addison–Moschovakis [1] on the 13 level—and an extension to all projective levels required projective determinacy. The first proof of 12 , and in fact 12 determinacy was found in 1978 by Martin [23]. Some six years later Woodin (1984) proved projective determinacy. But there was something missing. Both proofs used extremely strong large cardinal axioms. It was expected, certainly at the time of Martin’s proof and for a few brief months after Woodin’s proof, that these axioms should prove necessary, just as the axioms used by Martin to prove 11 determinacy proved necessary through Harrington [9]. But no one was able to show
4
Introduction
this, and in fact it turned out that much weaker axioms suffice. The appropriate axioms for projective determinacy were discovered in 1984 through a now legendary process, involving a very fast succession of results obtained by Foreman, Magidor, Shelah, and Woodin during a time of close communications. The process was initiated by the work of Foreman–Magidor–Shelah [6], which among other things dealt with generic elementary embeddings. These embeddings are generated by forcing with the poset P (κ)/I , where κ is an infinite cardinal and I is a nontrivial ideal over κ. A generic filter for this poset gives rise to an ultrafilter over κ, which in turn gives rise to an ultrapower M of the universe V and an elementary embedding j : V → M. Properties of the ideal I translate to properties of the embedding j and its target M. For example if I is κ-complete then the critical point of j is precisely κ. If I is also κ + -saturated (meaning that P (κ)/I has no antichains of size κ + ) then the ultrapower M is closed under sequences of length κ in the generic extension, and in particular it is wellfounded. For the discussion below let “ideal” mean ω1 -complete, non-trivial ideal. The discussion centers on the additional property of saturation. Prior to the work of Foreman– Magidor–Shelah [6] it was known that the existence of an ω2 -saturated ideal over ω1 can be forced assuming a large cardinal axiom called “huge.” The forcing, due to Kunen [14], involved making a huge cardinal of V into ω1 of the forcing extension (in particular collapsing ω1V ), and extending a hugeness embedding from V to the kind of ultrapower embedding one would get from an ω2 -saturated ideal over the new ω1 . Foreman–Magidor–Shelah [6, §§1,2] improved on this in several ways. They reduced the large cardinal assumption substantially, from huge to the existence of a cardinal known as supercompact; they managed to deal specifically with NS ω1 , the ideal of nonstationary subsets of ω1 ; and most remarkably they found a way to force and make this ideal ω2 -saturated without collapsing ω1V . It followed that a supercompact cardinal outright implies the existence of generic elementary embeddings with critical point ω1V and wellfounded targets (reached by first passing to the Foreman–Magidor–Shelah extension, and then forcing with P (ω1 )/NS ω1 ). These were embeddings of a substantially new nature, obtained not as extensions of large cardinal embeddings from smaller models, but through a true use of forcing. There had been results, see Foreman [7] and Magidor [17], connecting generic elementary embeddings with regularity properties for definable sets of reals. Magidor [17] had proved that all 13 sets are Lebesgue measurable using large cardinals and a generic elementary embedding with critical point ω1V and a wellfounded target. His argument combined with the results described above to prove Lebesgue measurability for 13 sets, from a supercompact cardinal. The connection with higher levels of definability was made by Woodin. Through a modification of his earlier work [44, Theorems 5–8] he showed that all projective sets are Lebesgue measurable, assuming the existence of a supercompact cardinal, and using the Foreman–Magidor–Shelah theorem that one can force to obtain an ω2 -saturated ideal over ω1 without collapsing ω1 (in fact just the consequence that there are generic elementary embeddings with critical point ω1V and wellfounded targets). He also noted that an easier argument [6, pp. 27–28] would show that all subsets of ωω in L(R) are Lebesgue measurable, if only there was a forcing
Introduction
5
which obtains saturation without adding reals—a stronger demand than that of not collapsing ω1 . Shelah at the same time realized that techniques from his book [38] could be used to adjust the construction of [6, §§1,2] and produce such a forcing [6, Theorem 21, Proposition 22]. Thus it was shown from a supercompact cardinal that not only all projective sets, but moreover all subsets of ωω in L(R), are Lebesgue measurable. The fact that supercompact cardinals imply Lebesgue measurability for all projective sets hinted that the strength of projective determinacy need not pass a supercompact, much less than the axioms used by Martin and Woodin for 12 and projective determinacy. Moreover it was clear that even less than a supercompact should suffice for the proof of Lebesgue measurability. Woodin and Shelah began taking turns weakening the hypothesis. First Woodin noticed that a superstrong cardinal, rather than a supercompact, is enough. Then Shelah isolated a weaker concept now known as a Shelah cardinal, and showed that the existence of n + 1 Shelah cardinals (still much less than a single superstrong cardinal) implies that all 1n+2 sets are Lebesgue measurable. Shelah’s proof of Lebesgue measurability used a certain forcing property, rather than directly using a large cardinal property. This property followed from the existence of n + 1 Shelah cardinals, but it was clear that even less would do. The precise nature of the end assumption was discovered by Woodin. By inverting quantifiers in Shelah’s definition he isolated a weaker concept now known as a Woodin cardinal. He then showed that the weaker concept is still sufficient for a proof of Lebesgue measurability. The end result, published in Shelah–Woodin [37], was that the existence of n Woodin cardinals and a measurable cardinal above them implies that all 1n+2 sets are Lebesgue measurable. It was obtained during a conversation in Jerusalem, just a couple of weeks after the original impetus of [6]. The work of Foreman, Magidor, Shelah, and Woodin described above led to many important advances on forcing relative to large cardinals. From our perspective though the most crucial discoveries reached through this work were first the isolation of the concept of a Woodin cardinal (see Definition 1A.11), and second the Shelah–Woodin theorem that the existence of n Woodin cardinals and a measurable cardinal above them implies the kind of Lebesgue measurability that would follow from 1n+1 determinacy. Equipped with an expectation created by these two discoveries, Martin and Steel set out to prove 1n+1 determinacy from n Woodin cardinals and a measurable cardinal above them. Their approach was motivated by considerations from inner model theory. Inner models are generalizations of Gödel’s L, and like L they have an internal wellordering of their reals. In the case of L the wellordering is 12 , and from this it follows that in L not all 12 sets are Lebesgue measurable. In richer inner models the internal wellorderings are more complex, but in all inner models considered up to the work of Martin–Steel [18], [19] the complexity was at most 13 . In particular these inner models could not satisfy the statement “all 13 sets are Lebesgue measurable.” In light of the Shelah–Woodin result the inner models were limited to at most a Woodin cardinal, a much lower level than previously expected. Martin and Steel began their investigation trying to understand the reason for this limitation. They discovered (though this was done after the proof of projective determinacy) that it had to do with the linear nature of the iterated ultrapowers used in inner
6
Introduction
O
MO 6
MO 4 j2,4
MO 2 j0,2
M0
O MJ 7 M5 7 o o ooo ? M3 j1,3 M1 oo7 o o o j0,1
Diagram 2. A sample iteration tree, with the tree order 0 T 1, 0 T 2, 1 T 3, 1 T 7, . . . .
model theory at the time. Inner model theory at the level of a Woodin cardinal and beyond is now known to require non-linear iterated ultrapowers, which Martin and Steel called iteration trees. Such non-linear iterations involve some choices at limits, and it is the need for these choices that adds complexity to the internal wellorderings of the reals. The added complexity in the case of finitely many Woodin cardinals corresponds precisely to the Shelah–Woodin theorem: Martin–Steel [19] produced models with n Woodin cardinals and a 1n+2 wellordering of their reals; Shelah–Woodin [37] showed that if the models were to contain a bit more, namely an extra measurable cardinal above the n Woodin cardinals, they would not have a 1n+2 wellordering of their reals. A precise, though brief, definition of iteration trees is given in Appendix A, and more details may be found in Martin–Steel [18, §3]. For the sake of this introduction, only iteration trees of length ω are needed. Roughly speaking an iteration tree of length ω on a model M consists of: • a tree order T on ω (see Section 7B for the definition of a tree order); • a sequence of models $Mk | k < ω% with M0 = M; and • embeddings jk,l : Mk → Ml for k T l. Each individual step along an iteration tree is created through an ultrapower by an extender (see Appendix A for a brief definition and further references). This is illustrated in Diagram 3. Ml+1 is generated as an ultrapower by an extender El ∈ Ml . El is applied to a model Mk , for a k ≤ l. The T -predecessor of l + 1 is set equal to k, and jk,l+1 : Mk → Ml+1 is set equal to the ultrapower embedding by El . In a linear iteration El would be applied to Ml , in other words k would have to always equal l.
Introduction
7
Ml+1 = Ult(Mk , El ) O jk,l+1
El ∈ Ml
Mk Diagram 3. Forming Ml+1 .
But in an iteration tree k < l is allowed too, provided that Mk and Ml are in sufficient agreement that the ultrapower Ult(Mk , El ) makes sense. A cofinal branch through an iteration tree of length ω is an infinite set b ⊂ ω which is linearly ordered by T . For example the sample iteration tree of Diagram 2 has an even branch—the branch consisting of {0, 2, 4, 6, . . .}. Given a cofinal branch b through T let Mb denote the direct limit of the models and embeddings of T along b, more precisely the direct limit of the system $Mk , jk,l | k T l ∈ b%. Martin and Steel had not yet developed the theory of iteration trees in the spring of 1985. But Steel had realized that non-linear iterations may help in the comparison process for inner models, and began to use them in a limited way. Among the various forms of non-linearity one should be prepared to tackle in developing inner model theory, perhaps the simplest is that of an alternating chains: an iteration tree with precisely two branches, an even branch consisting of {0, 2, 4, 6, . . .}, and an odd branch consisting of {0, 1, 3, 5, . . .}. It seemed natural to conjecture that in an alternating chain on V at least one of the branches should have a wellfounded direct limit (in fact something of the sort seemed necessary for any development of inner model theory). During a conversation on this in the department lounge at UCLA, Martin and Steel came to suspect that this matter of getting the wellfoundedness of one direct limit from the illfoundedness of the other may potentially be related to the problem of finding a homogeneous Suslin representation for the complement of a given homogeneously Suslin set. They then focused their attention on this problem. Homogeneous Suslin representations were abstracted in the late ’70s by Kechris [13] and independently Martin, and trace back to Martin’s earlier proof of 11 determinacy [21]. That proof can be broken into two parts: (1) a proof that homogeneously Suslin sets are determined; and (2) a proof from measurable cardinals that 11 sets are homogeneously Suslin. If one could somehow show that all projective sets are homogenously Suslin then by (1) projective determinacy would follow. The intuition that alternating chains could potentially be useful for this proved accurate, and by the end of the summer Martin and Steel constructed homogeneous Suslin representations for 11 sets, namely for the complements of the sets covered by (2), using alternating chains. More generally they found a construction which propagated homogeneous Suslin representations along the projective hierarchy, using iteration trees. At each level of the hierarchy propagation
8
Introduction
to the next level required a Woodin cardinal, used to supply enough extenders for the creation of the iteration trees. Combining the propagation with (1) and (2) Martin and Steel then obtained 1n+1 determinacy from n Woodin cardinals and a measurable cardinal above them, precisely the result they hoped for given the Shelah–Woodin theorem. The proof of projective determinacy was published in Martin–Steel [18], and the subsequent development of the general theory of iteration trees was published in Martin– Steel [19]. The two papers had a dramatic effect on the study of large cardinals, opening the doors to new avenues of research. Particularly notable for the purpose of this book is a result of Woodin, reached as part of a proof of the consistency of 12 determinacy from the consistency of one Woodin cardinal (avoiding the measurable cardinal used by Martin–Steel [18]). It was motivated partly by descriptive set theory, partly by the developing study of iteration trees and inner models for Woodin cardinals, and, curiously, by a classical theorem of Vopˇenka. Recall that HOD denotes the class of sets which are hereditarily ordinal definable. Vopˇenka’s theorem states that every real of V is generic over HOD. Relativized to L[x], where x is a real, the theorem implies in particular that x is generic over HODL[x] . Assuming 12 determinacy Woodin expected that on a cone of x, meaning for all x of sufficiently large Turing degree, HODL[x] should be a model with one Woodin cardinal, equal to ω2 of L[x]. (This he later proved true, see Burke–Schimmerling [3].) In fact it seemed reasonable to expect that HODL[x] should be an iterate of M1 on a cone, where M1 is the minimal iterable class model for one Woodin cardinal. This would explain the fact that under 12 determinacy the theory of HODL[x] in parameter ω2 L[x] is constant on a cone (it would be equal always to the theory of M1 in parameter δ, where δ is the Woodin cardinal of M1 ) and also allow proving 12 determinacy from the existence of M1 , since having the theory of HODL[x] in parameter ω2 L[x] constant on a cone implies 12 determinacy, assuming ω1 is inaccessible to reals. If this appealing connection between HODL[x] and M1 were to hold, then by Vopˇenka, x would be generic over an iterate of M1 . Inspired by this chain of thoughts, Woodin [45] made a surprisingly general discovery. Given any model M with a Woodin cardinal, he defined a poset P ∈ M which can, through an iteration, be made to admit any real as generic. The poset is now known as Woodin’s extender algebra, and the process by which it is made to admit a given real as generic is known as Woodin’s genericity iteration. The genericity iteration shares some qualities with the comparison process for inner models, and just like the development of inner models, it requires non-linearity. Woodin’s extender algebra and genericity iterations proved extremely useful in the study of large cardinals. Their original use in [45] helped settle the consistency strength of 12 determinacy (though not quite through the route described above, and the precise connection between HODL[x] and M1 is still open). Later uses involved proofs of generic absoluteness, results connecting large cardinals with descriptive set theory, and most recently Woodin’s core model induction, an intricate method for extracting models with large cardinals from various strong statements about V. In fact Woodin’s extender algebra is used also in this text, as we shall see below.
Introduction
9
Let us return now to the matter of proofs of determinacy. There are two possible directions for extensions of the projective determinacy of Martin–Steel [18]. The first involves keeping the length of the games at ω, and allowing payoff sets more complicated than projective. Working in this direction Woodin (1985) proved from ω Woodin cardinals and a measurable cardinal above them that all subsets of ωω in L(R) are determined. He had already proved earlier that these, and even more complicated subsets of ωω , are weakly homogeneously Suslin (see Woodin [46] for the relevant definition and the proof) assuming the existence of a supercompact cardinal. For sets in L(R) he had reduced the assumption to infinitely many Woodin cardinals and a measurable cardinal above them. As for the conclusion, a direct application of the Martin–Steel propagation [18] allows obtaining homogeneous Suslin representations, and hence determinacy, from weakly homogenous Suslin representations. Woodin’s theorem then gives determinacy for all subsets of ωω in L(R). More generally, to each model M with infinitely many Woodin cardinals, and each limit of Woodin cardinals δ in M, Woodin associated a certain pointclass and showed that all subsets of ωω in L(R∗ , ) are determined, where R∗ are the reals of a symmetric collapse of δ over M. This is the definitive result in the direction of games of length ω with increasing payoff complexity. The second way to progress beyond projective determinacy involves keeping the payoff at the level of projective sets (11 sets even), and allowing games of lengths greater than ω. It is this second direction, the direction of long games, that is treated in this book. The current methods for proofs of determinacy of long games trace back to the work of Neeman [29], [32] on optimal proofs of projective determinacy. The result of Martin–Steel [18], that 1n+1 determinacy follows from the existence of n Woodin cardinals and a measurable cardinal above them, is in spirit optimal; the construction of models with n Woodin cardinals and a 1n+2 wellordering of their reals in Martin– Steel [19] shows that n Woodin cardinals by themselves are not sufficient for a proof of 1n+1 determinacy. Still, the extra measurable cardinal used in Martin–Steel [18] is an overkill, and one should be able to do with less. Let (∗)n denote the statement that over every real there exists a countable, iterable model with a sharp for n Woodin cardinals. (Sharps are weakenings of measures, introduced by Silver [39], [40]. Iterability has to do with iteration trees, see below for more details.) (∗)n is the natural weakening of the assumption used in Martin–Steel [18], to a statement that should be outright equivalent to 1n+1 determinacy. Woodin (1989) proved the equivalence for odd n. For even n > 0 the direction 1n+1 determinacy implies (∗)n was proved by Woodin (1995). The direction (∗)n implies 1n+1 determinacy was established in 1994 by Neeman [29], [32] (the work there applies to all n > 0, both even and odd) using techniques which build on the determinacy proof of Martin–Steel [18]. Neeman [29], [32] showed that (∗)n in fact implies determinacy for a pointclass larger than 1n+1 . Combined with Woodin’s result that 1n+1 implies (∗)n this led to the Neeman–Woodin transfer theorem on determinacy (see [32]), generalizing the transfer theorem [12, Theorem 4] of Kechris–Woodin. But for the purpose of this book most important is the fact that Neeman [29], [32] began to connect moves in long games on ω with moves in iteration games.
10
Introduction
Iteration games, introduced by Martin–Steel [19], are games in which two players, “good” and “bad,” collaborate to construct iteration trees and branches through them, starting from a given model M. “Bad” plays the trees, and “good” picks the branches. The payoff condition involves wellfoundedness. If ever “good” picks a branch which leads to an illfounded direct limit, she loses. If she manages to always maintain wellfoundedness, she wins. M is said to be iterable if “good” has a winning strategy in the iteration game starting from M. The precise definitions of the iteration games relevant to the developments in this book are included in Appendix A. For now a very restricted version suffices. Given a model M and a number n < ω define W Gn (M) to be the following game between the players “good” and “bad”: In mega-round k “bad” plays a length ω iteration tree Tk on Mk (on M0 = M if k = 0). “Good” then plays a cofinal branch bk through Tk . Mk+1 is set equal to the direct limit along this branch, allowing the players to proceed to mega-round k + 1. They continue this way through mega-round n − 1, producing an iteration of the kind displayed in Diagram 4. If Mn , the final model they reach, is illfounded, then “bad” wins; if Mn is wellfounded, “good” wins. M0
kk b0 kk b1 Sk Sk k Sk Sk SSS / M1 k SSS / M2 T0
T1
k / Mn−1 kSkSkSkS bn−1 / Mn S Tn−1
Diagram 4. A weak iteration of length n.
Define M to be n-iterable if “good” has a winning strategy in W Gn (M). n-iterability is then a restricted version of the full iterability informally discussed above. Let (†)n be the statement that over every real there exists a countable, n-iterable model with a sharp for n Woodin cardinals. Recall that the statement of (∗)n above is identical except that it demands full iterability. Certainly then (∗)n implies (†)n . Working with n < ω and a set C ⊂ (ωω )n define Gω·n (C) to be played as follows: Players I and II alternate playing natural numbers according to the format in Diagram 5, producing reals yk = $yk (i) | i < ω% for k < n. If $yk | k < n% ∈ C then player I wins; and otherwise player II wins. I II
y0 (0)
y0 (2) . . . . . . . . . . . . yn−1 (0) yn−1 (2) . . . y0 (1) ... ......... yn−1 (1) ... ω moves to produce y0 ω moves to produce yn−1 n repetitions Diagram 5. The game Gω·n (C).
It is easy to see that the determinacy of Gω·(n+1) (C) for all C in 11 implies the determinacy of Gω (D) for all D in 1n+1 ; this is because the n outermost quantifiers
Introduction
11
used in the 1n+1 definition of D can be simulated through the moves for y1 , . . . , yn in Gω·(n+1) (C). Neeman [29], [32] reaches 1n+1 determinacy from (∗)n essentially by proving the determinacy of Gω·(n+1) (C) for every C in 11 , from (†)n . The proof is structured as an induction on n, and never literally handles Gω·(n+1) (C); rather it handles the first ω moves in this game and then appeals to induction for the part involving the remaining ω · n moves. But by joining the inductive steps together one can see that in essence Gω·(n+1) (C) is reduced to the iteration game W Gn (M) on the model M given by (†)n . The reduction either matches player I in Gω·(n+1) (C) to “good” in W Gn (M), and in this case player I ends up winning Gω·(n+1) (C), or else it matches II to “good,” and in this case II ends up winning. In either case it is ultimately a strategy for “good” in the iteration game that is converted into a strategy—either for I or for II—in the game Gω·(n+1) (C). There are thus two factors that go into the determinacy proof. First is a construction which reduces Gω·(n+1) (C) to the iteration game W Gn (M), and this is where the n Woodin cardinals of M are used. Second is the appeal to a strategy for “good” in W Gn (M), and this is where the n-iterability of M comes in. The iterability is derived from the results of Martin–Steel [19]. It is shown there that if M embeds into a rank initial segment of V then it is n-iterable for every n, and in fact more. (In particular then, the existence of a sharp for n Woodin cardinals in V is certainly enough to secure (†)n .) The main tool for proving iterability in Martin–Steel [19] is quoted as Theorem 12 in Appendix A, and the appendix draws two specific consequences of this theorem: weak iterability and mild iterability (both defined precisely in the appendix) for elementary substructures of rank initial segments of V. The appendix also quotes a theorem of Neeman [31] which secures the existence of a fully iterable model with a sharp for a Woodin limit of Woodin cardinals, provided that in V there is a sharp for such a cardinal. These forms of iterability are sufficient for all the determinacy proofs in this book, and the matter of proving iterability is therefore not discussed any further. It is the first factor mentioned above, the construction which allows reducing Gω·(n+1) (C) to W Gn (M), that is the starting point for this book. The book begins by breaking this construction into its essential blocks. Subsequent developments reassemble the blocks in various ways. These developments again lead to reductions of long games to iteration games, but apply to much stronger long games than Gω·(n+1) (C). The matter of breaking the construction of Neeman [29], [32] into basic blocks is treated in Chapter 1, and a sample application there shows how these blocks can be reassembled to prove 12 determinacy from (†)1 (or more precisely the lightface version of this). Chapter 2 begins the exploration of longer games. It defines games of fixed countable length, and proves the determinacy of these games from weakly iterable models with an appropriate number of Woodin cardinals. Games of fixed countable length are direct generalizations of the games Gω·n (C) (n < ω) of Diagram 5, allowing any fixed countable number of repetitions instead of the finitely many repetitions in Gω·n (C).
12
Introduction
The results of Chapter 2 are thus direct generalizations of projective determinacy. The additional complexity which separates these results from projective determinacy is the matter of propagation through limits. Every determinacy proof for long games involves propagating a construction of a run of the game. The planning for this propagation starts from the end really, since the construction has a definite goal, namely the payoff set. In the case of Gω·n (C) (n < ω) every stage of the construction except the initial one has an immediate predecessor, and one can easily plan from the end backwards. But in the general case of fixed countable length one has to somehow deal with limits. The problem is solved in Chapter 2 by breaking the construction in a limit stage θ into ω parts—this can be done naturally since the methods of Chapter 1 give rise to constructions carried in ω rounds—and spreading these parts over stages cofinal in θ. Beyond games of fixed countable length, yet still below games of fixed length ω1 , there lies a very rich hierarchy of games of variable countable length. These are long games where the length of a run, while always countable, depends on the moves played during that run. Chapter 3 deals with one particular class of games of this kind, the class of games of continuously coded length. The notion of continuously coded length was first introduced by Steel [41]. Roughly speaking his games can be described as follows: Players I and II alternate playing natural numbers to produce reals yξ = $yξ (i) | i < ω%. The reals are produced in the manner of Diagram 5, except that there is no set limit on the number of repetitions. Instead, player I is asked to play a natural number nξ at the start of the ξ -th repetition, with the proviso that she is not allowed to play the same number twice. Sooner or later she must run out of natural numbers, and at that point the game ends. Since there are just countably many natural numbers I can play, the end of the game must arrive at a countable stage. But player I can stretch the game to make it as long as she wishes below ω1 . Determinacy for these games is therefore stronger than determinacy for games of fixed length ω · θ for every θ < ω1 , yet weaker than determinacy for games of length ω1 . Our own games in Chapter 3 are somewhat stronger than those of Steel described above. Instead of having nξ played at the start of the ξ -th repetition, we fix some function f : R → ω and set nξ = f (yξ ) for each ξ , so that nξ depends uniformly on the moves made during the ξ -th repetition. Chapter 3 proves determinacy for games of this kind. Remember that a determinacy proof for long games always must involve advance planning for propagation through limits. The planning in Chapter 3 is more difficult than that in Chapter 2, since it is not known from the start how many limits to plan for. Roughly speaking this problem is solved by recycling: taking moves that are part of the current limit stage, and pushing them upward using some extender which overlaps a Woodin cardinal, so that they can be used again in a future limit stage. Without going into the details of the construction it is worthwhile looking at the kind of iterations involved in such a process. A typical iteration for Chapter 3 is illustrated in Diagram 6. jξ,ξ +1 in the diagram is the ultrapower embedding by the extender Fξ , which is taken from Qξ and applied to Mξ . Notice that the iteration in Diagram 6 is similar to that in Diagram 4, except for the addition of the extenders Fξ . It is these extenders which do the pushing mentioned above, shifting the moves of stage ξ upward, so that they can be used again in later stages.
13
Introduction
j0,1
M0
j1,2
kkk Sk k Sk SSSS k / Q0 0 T0
F0
#
M1
kkk Sk k Sk SSSS k / Q1 1
#
F1
/
M2
T1
jξ,ξ +1
kkk / Mξ S kk Sk SSSS k / ξ
Qξ
Fξ
#
/
Mξ +1
Tξ
Diagram 6. Illustration of the iterations used in Chapter 3.
This idea of pushing moves upward using extenders which overlap Woodin cardinals turns out to be the seed for a method of advance planning that allows for proofs of substantially stronger determinacy than that in Chapter 3. The method makes runs of long games generic for Woodin’s extender algebra. It uses a particular format of the extender algebra, see Section 4B, involving only extenders which overlap Woodin cardinals. The advance planning for the determinacy construction is made to fit with the extender algebra, in such a way that the use of extenders overlapping Woodin cardinals in the determinacy construction doubles as a use of these extenders in a genericity iteration. A typical stage of the resulting iteration is illustrated in Diagram 7. It is similar to stage ξ in Diagram 6, except that Fξ is now applied to Mζ , where ζ may be smaller than ξ . This is in line with the non-linear dictates of Woodin’s genericity iteration, where extenders are always applied to the earliest possible model. Fξ continues also to serve for pushing moves upward so that they can be used in later stages. In the settings of Diagram 7 it takes moves from stage ζ of the construction, and shifts them upward for recycled use. jζ,ξ +1
/
Mζ
kkk / _ _ _ _ _ _ _ Mξ S kk SkSSS S kξ
Qξ
Fξ
'
Mξ +1
/
Tξ
Diagram 7. Stage ξ , with non-linearity.
The precise definitions involved in the fitting of the advance planning for determinacy proofs with Woodin’s extender algebra are given in Chapter 4. The relevant constructions are then presented in Chapters 5, 6, and 7. It is in Chapter 7 that iterations of the kind mentioned in the previous paragraph, and illustrated partially in Diagram 7, show up. To facilitate the transition from the linear construction in Chapter 3 to the non-linear construction in Chapter 7, the linear strands of the latter construction are iso-
14
Introduction
lated and dealt with in Chapter 6. Typical stages on such a linear strand have the format illustrated in Diagram 8, and at least in one part (Section 6C) this makes the construction of Chapter 6 similar to the constructions of Chapter 3 (illustrated in Diagram 6). jζ,ξ +1
/
Mζ
kkk Sk k Sk SSSS k / Qζ _ _ _ _ _ _ _ Qξ ζ
Fξ
'
Mξ +1
/
Tζ
Diagram 8. From the perspective of stage ζ .
There is, alas, much more to ponder in the development of Chapters 4 through 7 than the structure of the relevant iterations, discussed above. The bulk of the work by far goes into the definitions and constructions which reduce long games to iteration games. It is too early to say anything on this now, except for mentioning that it is there that the large cardinals of the given model are used. The method of fitting advance planning for determinacy constructions with Woodin’s extender algebra, developed in Chapters 4 through 7, is really the pinnacle of this book. It brings determinacy proofs to the level of long games which run up to a “locally uncountable” ordinal. These are games where players I and II produce reals yξ in the standard manner, continuing through transfinitely many repetitions until they reach the first α which cannot be collapsed to ω by any function which is definable, in some given level of complexity, from the play. (The word “locally” in the terminology refers to the fact that α is only required to be uncountable with respect to some collection of definable functions. Typically α is still smaller than ω1V .) In the first determinacy result we give in Chapter 7, the restriction of definability is to functions which are constructible from the play. Later we formulate a more general restriction. Finally we discuss a couple of applications of our work, due to Woodin. One, which we only mention briefly, leads to a connection between determinacy and completeness in !-logic. The other leads to the consistency of determinacy for all ordinal definable games of length ω1 . It turns out that the tools developed in Chapters 4 through 7 can be used further (see Neeman [33]) to prove determinacy for games which outright run to ω1V . But this, already, is another story.
Chapter 1
Basic components Fix throughout this chapter a ZFC∗ model M, a Woodin cardinal δ of M, and some set X ∈ M&δ. We work with elements of the space X ω . Our goal is to develop some Lipschitz continuous method for witnessing that a sequence x ∈ Xω has a given property. Continuity is of the essence here. Ultimately we shall want to use this method in proofs of determinacy, where x is presented by installments as the game proceeds. Let us be more precise. Fix some g which is col(ω, δ)-generic/M. We work in M and in the generic extension M[g]. Fix A˙ ∈ M, a col(ω, δ) name for a subset of (M&δ)ω × X ω . We intend to associate to A˙ a certain family of auxiliary games AX,δ [x], defined for each x ∈ Xω . Since X and δ are fixed throughout the chapter, we omit the subscript and refer to the games as A[x]. By a game we often—and in particular here—mean a game tree, that is a set of rules for valid moves, without any payoff at the end. A[x] will be a length ω game with moves taken from M&δ. The map x → A[x] will be Lipschitz continuous; the rules for the first n rounds of A[x] will only depend on xn. For s ∈ X <ω we will thus talk about A[s], a game of lh(s) many rounds. We will have: Property 1. Each A[s] belongs to M. Moreover, the map s → A[s] belongs to M. We refer to this map as A. We will define the concept of a generic run of A[x] and show that: ˙ Property 2. If a ∈ (M&δ)ω is a generic run of A[x], then $ a , x% ∈ A[g]. We will define further the concept of a pivot for x. A pivot will be a pair T , a where: • T is an iteration tree of length ω on M; • T has a distinguished branch—called the even branch—leading to a direct limit T T : M →M model Meven = Meven and a direct limit embedding jeven = jeven even ; and • a is a run of jeven (A)[x]. (For the last item, jeven (A)[x] is simply the union of the games jeven (A)[xn] over n < ω.) We refer to the branches of T other than the even branch as odd branches. Given an odd branch b we use Mb = MbT to denote the direct limit along b, and jb = jbT : M → Mb to denote the direct limit embedding. Most importantly a pivot for x will satisfy:
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Property 3. For every odd branch b of T , there exists some h so that: (1) h is col(ω, jb (δ))-generic/Mb ; and ˙ (2) $ a , x% ∈ jb (A)[h]. Condition (2) of Property 3 should be contrasted with Property 2. The connection between the two properties comes through the game A[x]. With the aid of a strategy for II in this game we will be able to construct a generic run; with the aid of a strategy for I we will be able to construct a pivot. It is thus fair to characterize A[x] as a ˙ ˙ ˙ for some “shifts” jb (A)[h] of A; game where I tries to witness that $ a , x% ∈ jb (A)[h] ˙ while II tries to witness that $ a , x% ∈ A[g]. Intuitively, A = (s → A[s]) serves as a Lipschitz continuous method for witnessing a particular property of x ∈ X ω . The ˙ property addressed by a run a of A[x] is membership in the a -section of A[g] and its shifts.
1A The auxiliary games map We work here to define the map A associated to the name A˙ and obtain Property 1. Our definition of the games A[x] is an adaptation of definitions from Neeman [32], which builds on Martin–Steel [18]. We start with the basic definitions of types. These definitions follow the exposition of [32, Section 3], with some cosmetic changes. The changes are pointed out in Remark 1A.4. Then in Section 1A (2) we describe the rules of the game A[x]. In that part there is a more serious change from Neeman [32], since A˙ in [32] only involved Xω , while here it involves (M&δ)ω × X ω . 1A (1) Types. The definitions here form the rudiments for the later definition of the auxiliary games A[x]. We phrase the definitions working in V, with a distinguished ordinal δ. Later on we shall apply them in M with the ordinal δ, and in various other models of ZFC∗ . We work here, as we do throughout the book, in the language L∗ of Appendix A. By a “formula” we mean a formula of L∗ . Definition 1A.1. u is called a (κ, n)-type, where κ is a limit ordinal and n < ω, if u δ, and additional is a set of formulae involving n free variables v0 . . . vn−1 , a constant constants c for each c ∈ Vκ ∪ {κ}. Since κ is a limit ordinal, finite tuples of elements of Vκ belong to Vκ . It follows that (κ, n)-types can be coded as subsets of Vκ . We may therefore think of (κ, n)-types as (coded by) elements of Vκ+1 . Definition 1A.2. • If u is a (κ, n)-type, we call κ the domain of u (denoted dom(u)).
1A The auxiliary games map
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• For τ ≤ κ, and m ≤ n, we let c0 , . . . , ck , v0 , . . . , vm−1 ) | k ∈ N, c0 , . . . , ck ∈ Vτ ∪ {τ }, projm τ (u) = { φ(δ, ck , v0 , . . . , vn−1 ) ∈ u, and φ makes no menφ( δ, c0 , . . . , tion of vm , . . . , vn−1 }. • We use projτ (u) to denote projnτ (u), and projm (u) to denote projm κ (u). Definition 1A.3. We say that the (κ, n)-type u is realized (relative to δ) by x0 , . . . , xn−1 in Vη just in case that • x0 , . . . , xn−1 and δ are elements of Vη ; and δ, c0 , . . . , ck , v0 , . . . , vn−1 ), • for any c0 , . . . , ck ∈ Vκ ∪ {κ} and any formula φ( φ(δ, c0 , . . . , ck , v0 , . . . , vn−1 ) ∈ u ⇐⇒ Vη |= φ[δ, c0 , . . . , ck , x0 , . . . , xn−1 ]. (Implicitly we must have η > κ and η > 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
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• 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
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• “ 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 <δ-strong wrt H if it is λ-strong wrt H for each λ < δ. • δ is a Woodin cardinal if for every H ⊂ Vδ there exists κ < δ which is <δ-strong wrt H . Lemma 1A.12 (Martin–Steel, see [18], [30]). Assume that δ is a Woodin cardinal. Let η > δ be an ordinal, and let x0 , . . . , xn−1 be elements of Vη . Then there exist unboundedly many κ < δ such that the κ-type (relative to δ) of x0 , . . . , xn−1 in Vη+1 is elastic. Proof sketch. Apply Definition 1A.11 to H = $Hα | α < δ% where Hα is the α-type of # x0 , . . . , xn−1 in Vη . Finally we state the One Step Lemma, which essentially says that if w exceeds u and u is elastic, then one can stretch u to obtain a supertype of w.
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Lemma 1A.13 (One Step Lemma, Martin–Steel [18]). Assume that u is an elastic type, and that w exceeds u. Let τ = dom(w) and κ = dom(u). Assume τ < δ. Then there exists a τ + ω-strong extender E ∈ V&δ with critical point κ and such that w < StretchE τ +ω (u). Proof. Let ν, η, x0 , . . . , xm−1 witness that w exceeds u. Note, since u− exists, η is a − successor ordinal. Say η = η¯ + 1. Pick E ∈ V&δ so that StretchE τ +ω (u ) is realized by x0 , . . . , xn−1 in Vη¯ , and relative to δ. (This is possible since u is elastic.) Then − w is a subtype of StretchE τ +ω (u ), as it is realized by x0 , . . . , xn−1 , xn , . . . , xm in Vν and ν < η. ¯ Simple properties of realizable types now imply that w is a subtype of StretchE # τ +ω (u). Remark 1A.14. Given λ < δ we can add the clause “E is λ-strong” to the conclusion of E − − Lemma 1A.13; simply replace StretchE τ +ω (u ) in the proof by Stretchmax{τ +ω,λ} (u ). 1A (2) The rules of the game. Recall that δ is a Woodin cardinal of M, X is an element of M&δ, and A˙ is a name for a subset of (M&δ)ω × X ω in M col(ω,δ) . Fix x ∈ Xω . Our immediate goal is to define the auxiliary game AX,δ [x] = A[x] and obtain Property 1. Moves in A[x] will include types taken from M&δ. We shall need some “local indiscernible” above δ as a starting point for the realizations of our types. We make the following definition: Definition 1A.15. Let νL < νH be ordinals greater than δ. We say that $νL , νH % is a pair of local indiscernibles of M relative to δ just in case that: (M&νL + ω) |= φ[νL , c0 , . . . , ck−1 ] ⇐⇒ (M&νH + ω) |= φ[νH , c0 , . . . , ck−1 ] for any k < ω, any formula φ with k+1 free variables, and any c0 , . . . , ck−1 ∈ M&δ+ω. Local indiscernibles are thus indiscernible in a limited way for formulae with parameters in M&δ + ω. Exactly why we need a local indiscernible in the definition of A[x] will only become clear in Section 1C. For the time being let us note that there are local indiscernibles in M. In fact one needn’t go very far above δ to encounter them, as the next claim demonstrates: Claim 1A.16. Let jumpM (δ) denote the successor in M of the cardinality in M of M&(δ + ω + 1). Then there are νL < νH which are indiscernible in the sense of Definition 1A.15, and are both smaller than jumpM (δ). Proof. This is a simple cardinality argument. For each ν < jumpM (δ) let Tν be the set of tuples $k, φ, c0 , . . . , ck−1 % so that k is a natural number, φ is a formula with k +1 free variables, c0 , . . . , ck−1 belong to M&δ + ω, and (M&ν + ω) |= φ[ν, c0 , . . . , ck−1 ]. Tν is a subset of M&δ +ω. So there can only be cardM (M&δ +ω +1) many possible values for Tν . It follows that there exist some νL < νH below jumpM (δ) so that TνL = TνH . # $νL , νH % is then a pair of local indiscernibles of M relative to δ.
1A The auxiliary games map
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Let $ν, ν ∗ % be the least (lexicographically, minimizing first over ν ∗ ) pair of local indiscernibles of M relative to δ. ν will appear in our definition of A[x], specifically in rule (2d). ν ∗ will be used in Section 1C. Define the game A[x] to be played according to Diagram 1.1 and rules (1)–(4) below. We refer to a run of the game as a = $a0 , a1 , . . . % where a0 = $l0 , p0 , u0 , w0 %, a1 = $l1 , p1 , u1 , w1 %, etc. It is helpful to think of A[x] as a game where I tries to ˙ witness that there exists some h so that $ a , x% belongs A[h]. II tries to keep I honest by playing dense sets which I must meet. In addition II plays rank functions on I’s attempts to create h, in effect trying to witness that no such h-s exist. I II
l0 , p0 , u0
l1 , p1 , u1 w0
···
l2 , p2 , u2 w1
w2
...
Diagram 1.1. The game A[x].
The moves in A[x] take place inside M, and rules (1)–(4) should be read relativized to M. In particular all the types mentioned are elements of M. Realizations are in M and relative to δ. (1) (Rule for I) ln is a natural number smaller than n, or ln = “new”.1 (2) (Rule for I) If ln = “new” then: (a) un is a (κn , 4)-type for some κn < δ. If n > 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
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(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
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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<ω our description gives a game A[s] of lh(s) many rounds (the first lh(s) rounds of A[x] for any x which extends s), and it is clear that the map A = (s → A[s]) belongs to M. We call this map the auxiliary games map associated ˙ δ, and X. to A,
1B Generic runs We work with some g which is col(ω, δ) generic over M. Our goal is twofold. Given x ∈ Xω we wish to define the concept of a generic run of A[x] and verify Property 2. In addition we wish to demonstrate how generic runs can be constructed, with the aid of some strategy for II in A[x]. We shall have to convert g to a generic enumeration of M&δ. Fix, in M, some bijection ρ : δ ←→ M&δ. We regard g as a surjection g : ω → δ, and will talk about the composed surjection ρ ◦ g : ω → M&δ. Fix x ∈ X ω , not necessarily in M. Let a be a run of A[x], again not necessarily in M. Say a = $a0 , a1 , . . . % where an = $ln , pn , un , wn %. We say that e < ω is valid at n if ρ ◦ g(e) is a legal move for I in A[x] following a0 , . . . , an−1 . (In particular ρ ◦ g(e) has the form $l, p, u% for some l ∈ N ∪ {“new”}, some condition p, and some type u.) Definition 1B.1. a is a generic run of A[x] (wrt ρ, g) if for each n < ω, $ln , pn , un % is exactly equal to ρ ◦ g(e) for the least e which is valid at n. Thus, a run a is generic if I’s moves in the run are guided by a generic enumeration; I simply plays the first legal move in each round. Note that the definition places no restriction on II’s moves in a . The only restrictions placed are on I. ˙ Lemma 1B.2. Suppose that a is a generic run of A[x]. Then $ a , x% ∈ A[g]. (Note, this is only useful if x and a belong to M[g].) ˙ Proof. Assume for contradiction that $ a , x% ∈ A[g]. In particular a and x both belong ˙ to M[g]. Fix names a and x˙ in M&δ + 1 so that a˙ [g] = a and x[g] ˙ = x. (We can find such names in M&δ + 1 since both x and a are ω-sequences from M&δ.) Observe that a˙ , x, ˙ and g can meet any challenges posed by II in the game A[x]; the ˙ a˙ is forced by g to equal a , and x˙ is forced by pair $a˙ , x% ˙ is forced by g to belong to A,
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g to equal x. Our plan is to show that among her many attempts in the generic run of A, player I also tried a˙ , x, ˙ g. But the attempt a˙ , x, ˙ g must actually succeed, and this is a contradiction since II in addition to dense sets, plays “rank functions” to make sure all of I’s attempts fail. To be more precise, we intend to produce sequences n0 , n1 , n2 , . . . and β0 , β1 , . . . so that: (1) ln0 = “new” and lni = ni−1 for i > 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
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(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<ω, En =“pad” forms an increasing sequence. An iteration tree uses only extenders taken from below δ if for every n, En ∈ Mn &j0,n (δ).) For each n < ω let an = j2n,2n+2 (ln , pn , un )−−, wn .2 (Note that ln , pn , un are shifted from the 2n-th model to the 2n + 2-nd model.) Let a = $a0 , a1 , . . . %. We intend to split the following demand between the two players: 2 We treat sequences informally in this book, often dropping the enclosing brackets. (For otherwise brackets of all kinds would quickly pile up, to the point of obscuring the objects involved.) To avoid confusion we sometimes use the symbol “−−” to indicate a sequence, as opposed to a singleton element.
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• a is a run of jeven (A)[x]. We do this with the following three rules: (4) (Rule for I) ln , pn , un is a legal move for I in j0,2n (A)[x] following the position $a0 , . . . , an−1 %. (5) (Rule for II) The critical point of j2n,2n+2 is larger than rank(X) and large enough that none of a0 , . . . , an−1 is moved by this embedding. (6) (Rule for II) wn is a legal move for II in j0,2n+2 (A)[x] following the position $a0 , . . . , an−1 %−−, j2n,2n+2 (ln , pn , un ). Let Pn denote $a0 , . . . , an−1 %. Let Qn denote Pn −−, j2n,2n+2 (ln , pn , un ), which by rule (5) is equal to j2n,2n+2 (Pn −−, ln , pn , un ). Using rules (4) and (6) one can verify by induction on n that Pn is a position in the game j0,2n (A)[x]. The case of n = 0 is clear. Suppose inductively that Pn is a position in the game j0,2n (A)[x]. By rule (4) so is Pn −−, ln , pn , un . Applying j2n,2n+2 we see that Qn = j2n,2n+2 (Pn −−, ln , pn , un ) is a position in j0,2n+2 (A)[x]. By rule (6) so is Qn −−, wn . But this is Pn+1 . Thus we have indeed that a is a run of jeven (A)[x]. This completes the rules of Apiv [x]. Note that a run of Apiv [x] produces T and a which satisfy conditions (1) and (2) of Definition 1C.1. To make T , a a pivot we must further satisfy condition (3). Observe further that the association x → Apiv [x] is Lipschitz continuous. To figure out the rules for the first n rounds of Apiv [x] we need only know the rules for the first n rounds of A[x] (so that we can shift those rules along the even branch), and for this it is enough to know xn. In effect we described a map Apiv = (s → Apiv [s]) defined ˙ δ, and X. It is clear on s ∈ X<ω . We refer to it as the pivot games map associated to A, that the map Apiv belongs to M. 1C (2) Constructing σpiv [, x]. Fix x ∈ Xω , and fix some map : ω → M&δ + 1. We work to define σpiv [, x], a strategy for II in Apiv [x]. (We use “σpiv [, x]” in our notation to emphasize that our construction depends not only on x, but also on the enumeration .) Assuming that is onto we shall later on show that all runs according to σpiv [, x] are pivots. Historical Remark. Our construction here builds closely on the work of Martin–Steel [18]. In particular the iteration tree which stands at the heart of the construction is similar to the one in Martin–Steel [18]. For more on the connections between our construction and the earlier construction of Martin–Steel [18] see Neeman [32, p. 248]. We define σpiv [, x] by describing below the course of a run according to the strategy, played against some imaginary opponent as I. We then check that the run described is indeed a pivot. In round n of the construction our imaginary opponent will play the move ln , pn , un . It will be our task to play E2n , E2n+1 , wn . We use the notation established before:
1C Pivots
31
Our construction produces a nice iteration tree T , and a as already described in Section 1C (1). Recall that Pn = $a0 , . . . , an−1 % is a position in j0,2n (A)[x], and Qn = $a0 , . . . , an−1 %−−, j2n,2n+2 (ln , pn , un ) is a position in j0,2n+2 (A)[x]. Recall further that an = j2n,2n+2 (ln , pn , un )−−, wn . Of course ln is not moved by j2n,2n+2 . We shall make sure that pn is not moved either. So in fact we shall have an = $ln , pn , j2n,2n+2 (un ), wn %. In addition to constructing the run of Apiv [x], we construct the following objects: • a˙ n , x˙n ∈ M2n+1 &δn + 1 where δn = j0,2n+1 (δ), and • Dn ∈ M2n+1 &δn + 1. a˙ n , x˙n , Dn will be constructed in round n, together with E2n , E2n+1 , wn . a˙ n and x˙n will be col(ω, δn )-names in M2n+1 , and Dn will be a dense set. We intend to maintain all the requirements set by the rules of Apiv [x]. Note that this includes requirements from A[x] shifted to various models along the even branch. In addition we intend to maintain: (A) wn is elastic. ˙ (B) wn is realized by A˙ n , a˙ n , x˙n , Dn , and νn in M2n+1 &νn +5, where A˙ n = j0,2n+1 (A) and νn = j0,2n+1 (ν). (C) If ln ∈ N then a˙ n = j2ln +1,2n+1 (a˙ ln ) and x˙n = j2ln +1,2n+1 (x˙ln ). (D) If ln ∈ N then pn belongs to j2ln +1,2n+1 (Dln ). (E) κ0 < τ0 < κ1 < τ1 < · · · , where κn = dom(un ) and τn = dom(wn ). (F) M2n and M2n+1 agree to κn + ω. M2n+1 and M2n+2 agree to τn + ω. With regard to condition (B) we point out that realizations in M2n+1 use δn = j0,2n+1 (δ) to interpret the constant δ. We remind the reader that ν > δ is the lower ordinal in the least pair of local indiscernibles of M relative to δ. ν ∗ is the higher ordinal in that pair. We shall use the indiscernibility properties of ν and ν ∗ , given by Definition 1A.15, during the construction. With regard to conditions (C) and (D) let us remind the reader that 2ln + 1 is the T -predecessor of 2n + 1 when ln ∈ N. So j2ln +1,2n+1 makes sense. Remember that A˙ is a name for a subset of (M&δ)ω × X ω . The canonical name for A˙ therefore certainly belongs to M&δ + ω. Without loss of generality let us assume that A˙ itself belongs to M&δ + ω. Let us now start round n of the construction. We assume inductively that conditions (A)–(F) hold for m < n. Our opponent opens round n by playing $ln , pn , un %, a legal move for I in j0,2n (A)[x] following the position Pn . Case 1. If ln = “new”. Set E2n = “pad” as required, so that M2n+1 = M2n , j2n,2n+1 = ˙ νn = j0,2n (ν), and δn = j0,2n (δ). id, and j0,2n+1 = j0,2n . We have A˙ n = j0,2n (A), The rules of j0,2n (A)[x], specifically rule (2) in Section 1A (2), tell us that there exist
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names a˙ n and x˙n , both elements of M2n &δn + 1 = M2n+1 &δn + 1, so that un is realized by A˙ n , a˙ n , x˙n , νn in M2n &νn + 2 = M2n+1 &νn + 2. Let νn∗ = j0,2n+1 (ν ∗ ). Using the indiscernibility of ν and ν ∗ given by Definition 1A.15 we see that un is realized by A˙ n , a˙ n , x˙n , 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 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 on. For the time being let us just take Dn as given. 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 directly from Definition 1A.7, noting that νn + 5 (the level of realization of wn ) is smaller than νn∗ + 2 (the level of realization of proj3 (un ) given by (∗)). Applying the One Step Lemma (Lemma 1A.13) inside M2n+1 find E2n+1 ∈ M2n+1 , a τn + ω-strong extender E 3 with critical point κn , so that wn is a subtype of Stretchτn2n+1 +ω (proj (un )). To satisfy the normality clause in rule (3), make sure that E2n+1 is stronger than all previous extenders on the tree. This can be done using Remark 1A.14. Note that the One Step Lemma gives an E which belongs to M2n+1 &j0,2n+1 (δ). So the second clause in rule (3) is satisfied. Let M2n+2 = Ult(M2n , E2n+1 ) and let j2n,2n+2 be the ultrapower embedding. By E 3 3 Lemma 1A.9, Stretchτn2n+1 +ω (proj (un )) equals projτn +ω (j2n,2n+2 (un )). Thus wn is a 3 subtype of projτn +ω (j2n,2n+2 (un )). In particular this means that wn < proj3 (j2n,2n+2 (un )). Using this it is easy to check that wn is a legal move for II in j0,2n+2 (A)[x] following the position Qn . This secures rule (6) of Apiv [x]. Note that the critical point of j2n,2n+2 is equal to κn . For 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 ]. #
1C Pivots
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<ω must be increasing). The use of genericity here is similar to its use in Section 1B. We refer the reader to the final stages of case I in [32, Section 4] for a detailed construction of this kind. 1C (3) Properties of σpiv . Given x ∈ X ω and : ω → M&δ + 1 our construction of the previous subsection produces a strategy σpiv [, x] playing for II in Apiv [x]. We have: Lemma 1C.5. Suppose that is onto M&δ + 1. Then every run according to σpiv [, x] is a pivot for x. Proof. This follows from our work in the previous subsection, particularly Claim 1C.3 # and the book-keeping for picking the Dn -s modulo . The construction of σpiv [, x] is Lipschitz continuous. To be precise, for ϑ : n → M&δ + 1 and s ∈ Xn the construction produces a strategy σpiv [ϑ, s], playing for II in Apiv [s]. We refer to the map σpiv = (ϑ, s → σpiv [ϑ, s]) as the pivot strategies map ˙ δ, and X. We have, for x ∈ X ω and : ω → M&δ + 1, associated to A, σpiv [, x] = σpiv [n, xn]. n<ω
36
1 Basic components
It is clear that the map σpiv belongs to M. Our description in Section 1C (2) in fact defines σpiv inside M. Remark 1C.6. The description in Section 1C (2) leaves the precise choice of E2n , τn , and E2n+1 open. To literally obtain the map σpiv we must fix, inside M, some wellordering of M&δ, and take, in each stage of the construction where the description leaves room for choice, the first suitable object. We say that a pivot T , a resides above an ordinal λ < δ if all the extenders used in T have critical points higher than λ. Our construction in Section 1C (2) clearly yields: Lemma 1C.7. All runs according to σpiv [, x] reside above rank(X).
#
In applications we will often have to make sure that our pivots reside above a given ordinal λ < δ. We can achieve this indirectly by setting Y = X ∪ M&λ, and working ˙ δ, and Y , instead of the maps associated with the maps A, Apiv , and σpiv associated to A, ˙ δ, and X. Since A˙ is also a name for a subset of (M&δ)ω × Y ω , we are allowed to A, to do this. The change from X to Y has no effect on any of the previous arguments, except that it forces the κn -s to be greater than λ. Suppose we are handed some ZFC∗ model P and an elementary embedding π : M → P . The maps A, Apiv , and σpiv , being elements of M, can be shifted to P using the embedding π . The relevant notions (particularly that of a pivot) can be shifted similarly. For x ∗ ∈ π(X)ω , and a map ∗ : ω → P &π(δ) + 1, let π(σpiv )[∗ , x ∗ ] = π(σpiv )[∗ n, x ∗ n]. n<ω
Lemma 1C.8. Suppose ∗ is onto P &π(δ) + 1. Then all plays according to ˙ π(σpiv )[∗ , x ∗ ] are pivots for x ∗ , where “pivot” is interpreted over P and using π(A). ∗ ∗ Furthermore, all plays according to π(σpiv )[ , x ] reside above π(rank(X)). Proof. π(σpiv )[∗ , x ∗ ] is exactly the strategy we would get if we ran the construction of Section 1C (2) in P instead of M. Lemmas 1C.5 and 1C.7 would apply equally well in P . # This lemma will be used heavily throughout the book. In applications P and π will be handed to us in installments. We will be given π n : M n → M n+1 at a “stage n,” with M 0 = M. P and π : M → P will be the direct limit model and embedding. Our goal will be to build a pivot for x ∗ over P , also in installments. We will do this using Lemma 1C.8. Without going into details, let us only say that the construction will involve fixing ϑ n : n → M n &π 0,n (δ) + 1 so that ϑ n+1 extends π n (ϑ n ) and so that ϑ∞ = π n,∞ (ϑ n ) n<ω
is onto P &π(δ) + 1.
1D Mirror images
37
Remark 1C.9. For a fixed : ω → M&δ + 1, the map x → σpiv [, x] is Lipschitz continuous in x. Why did we not suppress , and use σpiv to denote the map s → σpiv [, s]? The reason is that this map (certainly when is onto) need not belong to M. This wasn’t a problem before. Remember that the map σgen only belonged to M[g], not to M. But here we do need a map which belongs to M, for otherwise we could not even phrase, let alone prove, Lemma 1C.8.
1D Mirror images So far we defined the auxiliary game map A = (s → A[x]), defined the notions of generic runs and pivots, and obtained the properties listed in the introduction to this chapter, Properties 1–3. Recall that Properties 2 and 3 are in some sense dual. The former places us outside ˙ ˙ There is however a qualitative A[g], while the latter places us inside some shift jb (A)[h]. difference between these two properties. The statement of Property 2 is only useful if x and a belong to M[g]. Part (2) of Property 3 on the other hand tells us that x and a belong to Mb [h]. Property 3 is thus substantially more potent, and we will always try to work with it, rather than with Property 2, in applications. To avoid the situation of Property 2 we ˙ and the auxiliary game associated to consider both the auxiliary game associated to A, ˙ the complement of A with the auxiliary roles of I and II reversed. We take this section to briefly explain the notation involved in reversing the roles of I and II. For an example of how we use this reversal to avoid the situation of Property 2 see Section 1E. We work as always with a ZFC∗ model M, its Woodin cardinal δ, an X ∈ M&δ, and some g which is col(ω, δ)-generic/M. Let B˙ be a name in M for a subset of (M&δ)ω × Xω in M[g]. For each x ∈ Xω we define a game B[x]. Our definition mirrors the definition of A[x] in Section 1A. B[x] is played according to Diagram 1.3. Note that this diagram is quite simply Diagram 1.1, with the roles of I and II reversed. The rules of B[x] are precisely the I II
w0 l0 , p0 , u0
w1 l1 , p1 , u1
w2 l2 , p2 , u2
... ···
Diagram 1.3. The game B[x].
rules (1)–(4) listed in Section 1A(2), but with “I” and “II” reversed, and with rule (2d) replaced by: (2) (d) There exist some names b˙ and x, ˙ both members of M&δ + 1, so that un is ˙ ˙ realized by B, b, x, ˙ ν in M&ν + 2. ˙ ˙ in the realization of un to “B.”) The dependence of B[x] (Note the change from “A” on x is Lipschitz continuous. We have therefore a map B = (s → B[s]). We refer
38
1 Basic components
˙ δ, and X. It is clear that to B as the mirrored auxiliary games map associated to B, this map belongs to M. We use b = $b0 , b1 , b2 , . . . % to denote runs of B[x], with bn = $ln , pn , un , wn %. Fix a bijection ρ : δ ←→ M&δ in M. Definition 1D.1. b is a generic run of B[x] (wrt ρ, g) if for each n < ω, $ln , pn , un % is equal to ρ ◦ g(e) where e is least so that ρ ◦ g(e) is a legal move for II in B[x] following bn. The next two lemmas are the mirror images of Lemmas 1B.2 and 1B.7. Note the change from “strategy for I” in Lemma 1B.7 to “strategy for II” in Lemma 1D.3. x% ∈ B[g]. ˙ Lemma 1D.2. Suppose that b is a generic run of B[x]. Then $b, (Note, this is only useful if x and b belong to M[g].) # Lemma 1D.3. For every real x there exists a strategy τgen [x] for II in B[x], so that every run according to τgen [x] is generic. Moreover, the association x → τgen [x] is Lipschitz continuous, induced by some map τgen = (s → τgen [s]). The map τgen belongs to M[g]. # We refer to the map τgen of Lemma 1D.3 as the mirrored generic strategies map ˙ δ, and X. associated to B, Let us now pass to the matter of pivots. We use the same word, pivot, in our definitions for the mirror image. When there is danger of ambiguity we distinguish between A-pivots and B-pivots. Definition 1D.4. A (B)-pivot for x ∈ Xω is a pair T , b where (1) T is a nice iteration tree on M, (2) b is a run of jeven (B)[x], (3) for every odd branch c of T there exists h so that (a) h is col(ω, jc (δ))-generic/Mc , and ˙ x% ∈ jc (B)[h]. (b) $b, The game Bpiv [x] is played according to Diagram 1.2 and the rules listed in Sec˙ and A is replaced tion 1C (1), except that “I” and “II” are reversed, A˙ is replaced by B, by B. We now use bn to denote j2n,2n+2 (ln , pn , un ), −−wn . We let b = $b0 , b1 , . . . %. A run of Bpiv [x] produces T , b which satisfy the first two conditions in Definition 1D.4. The construction in Section 1C(2) can be mirrored to produce τpiv [, x], a strategy for I (note as usual the reversal of “I” and “II”) in Bpiv [x]. τpiv [, x] is given by a Lipschitz continuous map τpiv = (ϑ, s → τpiv [ϑ, s]) which belongs to M. We have Lemma 1D.5. Suppose is onto M&δ + 1. Then every run according to τpiv [, x] is a B-pivot for x. # This parallels Lemma 1C.5. Lemma 1C.8 can be mirrored similarly.
1E Sample application
39
1E Sample application To illustrate the use of the techniques developed in the previous sections we include here a proof of the following theorem, which concerns standard games of length ω. The theorem is originally due to Woodin, proved by methods which predate the developments in this book. Our objective in this book is to handle games of lengths greater than ω. Theorem 1E.1 is only included as an illustration. More results are known on length ω games within the projective hierarchy. For details on the determinacy of games of length ω we refer the reader to Martin–Steel [18], Neeman [29], and Neeman [32]. Theorem 1E.1 (Woodin). Suppose that there exists a model M and a δ ∈ M so that: • M is a class model of ZFC; • M is weakly iterable (see Appendix A for the relevant definition); • δ is a Woodin cardinal of M; and • M&δ + 1 is countable in V. Then 12 determinacy holds. Remark 1E.2. The assumption in Theorem 1E.1 follows from the existence of a sharp for a Woodin cardinal in V (more precisely the existence of a Woodin cardinal δ in V and a sharp for Vδ ), see Appendix A. The proof of Theorem 1E.1 takes the rest of this section. Fix a 12 set C ⊂ R, say the set of all reals which satisfy a given 12 statement φ. Our goal is to prove that the standard length ω game Gω (C) is determined. Fix some g which is col(ω, δ)-generic over M. Such generics exist since we are assuming that M&δ + 1 is countable in V. Let A˙ ∈ M be the canonical name for the set of $ a , x% ∈ (M&δ)ω × ωω in M[g] so that φ holds of x in M[g]. Let B˙ ∈ M be the x% ∈ (M&δ)ω × ωω so that φ does not hold of x in canonical name for the set of $b, ˙ δ, and X = ω. Let B be the M[g]. Let A be the auxiliary games map associated to A, ˙ mirrored auxiliary games map associated to B, δ, and X = ω. Working in M, define a game G∗ played as follows: I and II alternate playing natural numbers xn , creating together the real x = $xn | n < ω%. In addition they play moves an−I = $ln , pn , un % and an−II = wn in the auxiliary game A[x] associated to x. The first player to violate these rules loses. If the rules are successfully followed for ω moves then II wins. I II
x0
a0−I
a1−I a0−II
x1
x2 a1−II
Diagram 1.4. The game G∗ .
... ...
40
1 Basic components
We point out that our definition of G∗ rests on the continuity of the map x → A[x]. The rules governing the moves ln , pn , un and wn depend only on xn, and are therefore available in round n of G∗ . We point out further that the game G∗ exists in M. This is true since the map A = (s → A[s]) exists in M. Mirroring G∗ , let H ∗ be the played as follows: Again I and II alternate playing natural numbers xn , creating together the real x = $xn | n < ω%. But now they play auxiliary moves in the game B[x]. The first player to violate these rules loses. Infinite plays here are won by I. I II
x0
b0−I b0−II
b1−I x1
b1−II
x2
... ...
Diagram 1.5. The game H ∗ .
G∗ ∈ M is an open game for player I, hence determined in M. Similarly H ∗ is open for II, and hence determined in M. We will use this determinacy as part of our proof of Theorem 1E.1. Case 1. If I wins G∗ in M. We intend to argue that I wins Gω (C) (in V). Fix σ ∗ ∈ M, a winning strategy for I in G∗ . Fix some imaginary opponent, playing for II in the standard game on natural numbers. We describe how to play against this imaginary opponent, and win. Our description takes the form of a construction of (among other things) the run x = $xn | n < ω%. Our opponent plays xn for odd n and it is our task to construct xn for even n. Once we complete the construction we will verify that x ∈ C, so that indeed the run constructed is won by I. ˙ δ, and X. Fix in V a surjection Let σpiv be the pivot strategies map associated to A, : ω → M&δ + 1. Surjections of this kind exist since we are assuming that M&δ + 1 is countable in V. We construct x = $xn | n < ω%, a run of the standard game on natural numbers, and T , a , a pivot for x. The participants in the construction are: • Our imaginary opponent, producing xn for odd n. • σpiv [, x], producing wn for all n and the extenders which give rise to T . • σ ∗ and its images along the even branch of T , producing xn for even n and ln , pn , un for all n. The time line of the construction is presented in Diagram 1.6. At the start of round n we have xn = $x0 , . . . , xn−1 %, the iteration tree T 2n + 1 ending with M2n , and the position Pn = $a0 , . . . , an−1 % in j0,2n (A)[xn]. If n is odd, our opponent opens the round producing xn . If n is even j0,2n (σ ∗ ) opens the round producing xn . (Remember σ ∗ is a strategy for I in G∗ , the game displayed in Diagram 1.4.) Then j0,2n (σ ∗ ) produces ln , pn , un which is a legal move for I in j0,2n (A)[xn + 1]. At this point we apply σpiv [, xn + 1]. σpiv extends the iteration tree producing M2n+1 and M2n+2 .
41
1E Sample application
M =M0
M1
M2
M3
M4
M5
M6
_ σ∗
x0
σ∗
l0 p0 /o /o /o /o /o /o /o / u0_
σpiv
w0
Oppnt
x1
j0,2 (σ ∗ )
l1 p1 /o /o /o /o /o /o /o / u1_
σpiv
w1
j0,4 (σ ∗ )
x2
j0,4 (σ ∗ )
l2 p2 /o /o /o /o /o /o /o / u2_
σpiv
w2 Diagram 1.6. The construction in Case 1.
We let Qn = j2n,2n+2 (Pn −−, ln , pn , un ). Note that by rule (5) in Section 1C (1), Qn is equal to Pn −−, j2n,2n+2 (ln , pn , un ). σpiv further produces wn , a legal move for II in j0,2n+2 (A)[xn + 1] following Qn . This completes round n of the construction. We set an = j2n,2n+2 (ln , pn , un )−−, wn , so that Pn+1 = Qn −−, wn = $a0 , . . . , an−1 , an %, and proceed to the next round.
42
1 Basic components
Remark 1E.3. We again point out the importance of continuity, this time the continuity of the map x → σpiv [, x]. In round n of the construction we use σpiv [, xn + 1] for round n of Apiv [xn + 1]. Note that x and a together form an infinite play of j0,even (G∗ ) which is according to j0,even (σ ∗ ). This follows from our use of σ ∗ and its images during the construction, and from rule (5) in Section 1C(1) which tells us that an = j2n+2,even (an ). Now σ ∗ ∈ M is a winning strategy for I, the open player. It follows that there are no infinite plays according to σ ∗ . On the other hand we just saw that there is an infinite play according to j0,even (σ ∗ ), the play x, a . If Meven were wellfounded, the existence of such a play could be reflected into Meven , and then pulled back by the elementarity of j0,even to give an infinite play according to σ ∗ . We thus conclude that Meven is illfounded. Using the iterability of M, pick a branch b of T so that Mb is wellfounded. We know that b is an odd branch. Our use of σpiv [, x] during the construction guarantees that T , a is a pivot for x; see Lemma 1C.5. In particular (see Definition 1C.1) there exists some h which is col(ω, jb (δ))-generic/Mb and so that: ˙ (∗) $ a , x% ∈ jb (A)[h]. (∗) and our definition of A˙ tell us that φ holds of x in Mb [h]. φ is a 12 statement, hence absolute between wellfounded class models of ZFC∗ . We conclude that φ holds of x in V, as required. # (Case 1) Case 2. If II wins H ∗ in M. Then an argument which mirrors that of case 1 (this time # (Case 2) using τpiv ) shows that II wins Gω (C) in V. Case 3. Since G∗ and H ∗ are both determined in M, the only remaining possibility is that II wins G∗ in M and I wins H ∗ in M. We intend to derive a contradiction in this case. This will show that either case 1 or case 2 must hold, completing the proof of Theorem 1E.1. Fix σ ∗ ∈ M, a winning strategy for II in G∗ . Fix similarly τ ∗ ∈ M, a winning strategy for I in H ∗ . ˙ δ, and X. Let τgen be the Let σgen be the generic strategies map associated to A, ˙ mirrored generic strategies map associated to B, δ, and X. Working inside M[g] we construct x, a , and b so that (1) x = $xn | n < ω% is a real; (2) a , x is a run of G∗ played according to both σ ∗ and σgen [x]; x is a run of H ∗ played according to both τ ∗ and τgen [x]. (3) b, The construction is straightforward: σgen produces the auxiliary moves an−I ; σ ∗ produces the auxiliary moves an−II and the numbers xn for odd n; τ ∗ produces the numbers xn for even n and the auxiliary moves bn−I ; and τgen produces the auxiliary moves bn−II .
1F Mixed pivots
43
Remark 1E.4. As usual the continuity of the maps x → σgen [x] and x → τgen [x] is important; when using σgen and τgen in round n we only have knowledge of xn + 1. ˙ Condition (2) tells us that $ a , x% ∈ A[g]; see Lemma 1B.2. Similarly condition (3) x% ∈ B[g]. ˙ x ∈ M[g] this means that on the one hand φ tells us that $b, Since a , b, does not hold of x in M[g], and on the other hand φ does not fail for x in M[g]. This is a contradiction. # (Case 3, Theorem 1E.1) Remark 1E.5. Running the proof above with C = R, or more precisely with the 12 statement “x = x,” we see that for every real x: there exists a length ω iteration tree T on M which has an illfounded even branch and such that for each odd branch b of T there is some h with (1) h is col(ω, jb (δ))-generic/Mb ; and (2) x ∈ Mb [h]. Thus every real can be absorbed into a generic extension of an iterate of M. This was noted in Neeman [29]. A different genericity iteration was previously discovered by Woodin [45].
1F Mixed pivots We work as usual with reference to the fixed M, δ, and X. But now we do not fix a ˙ Instead we fix a map A˙ = (γ → A[γ ˙ ]) in M, which assigns to each single name A. M ˙ ] for a subset of (M&δ)ω × X ω in M col(ω,δ) . ordinal γ < jump (δ), a name A[γ Remark 1F.1. The point of the restriction to γ < jumpM (δ) is to allow us to work with a set function A˙ which literally belongs to M, rather than a class function definable over M. There is nothing particularly important about jumpM (δ), except that it is larger than the ordinals γ we shall care about in applications later on. We work to describe games which are similar to the pivot games of Section 1C, but allow player I some greater flexibility. This greater flexibility is the result of two changes. We allow I to insert intervals to the iteration tree constructed; and we allow I ˙ ] to play on in each round of the game. Outcomes of to pick which of the names A[γ these new games which satisfy a condition similar to condition (3) of Definition 1C.1 we call mixed pivots. As in Section 1C we go on to describe a strategy for II which produces mixed pivots. 1F (1) The game. For each γ < jumpM (δ) let A[γ ] = (s → A[γ , s]) be the auxiliary ˙ ]. We use A to denote the map γ , s → A[γ , s]. games map associated to the name A[γ This map belongs to M. Fix x ∈ X ω . We define the game Amix [x] associated to x and to the map A˙ = ˙ ]). (γ → A[γ
44
1 Basic components
At the start of round n we have a natural number e(n) and an iteration tree T e(n)+1, ending with the model Me(n) . We have some position Pn = $a0 , . . . , an−1 % in Me(n) . For n = 0 we set e(0) = 0, M0 = M. I II
······
f (n), T f (n) + 1
γn
···
ln , pn , un Ef (n) , Ef (n)+1 , wn
...
Diagram 1.7. Round n of the game Amix [x].
The time line of round n is presented in Diagrams 1.7 and 1.8. At the start of round n player I plays some natural number f (n) ≥ e(n) and extends T e(n)+1 to T f (n)+1. We demand enough agreement between Me(n) and Mf (n) so that Pn belongs to Mf (n) . Player I then plays γn so that Pn is a legal position in j0,f (n) (A)[γn , x]. The rest of the round follows the rules in Section 1C: I plays a move in j0,f (n) (A)[γn , x] following the position Pn ; II shifts this move to the model Mf (n)+2 , as illustrated by the squiggly arrow in Diagram 1.8, and replies there. We let Pn+1 be the position obtained, let e(n + 1) = f (n) + 2, and proceed to the next round. Remark 1F.2. Suppose I fixes γ0 and always plays f (n) = e(n) and γn = j0,f (n) (γ0 ). ˙ 0 ]. Thus the Then the game degenerates into the pivot game associated to the name A[γ difference between our current game and the pivot games of Section 1C is in the extra flexibility accorded to I at the start of the round. I may play f (n) > e(n) and insert her own interval of models into the tree T . In addition, I may pick a new ordinal γn to work with. The exact rules of Amix [x] are as follows: (1) (Rule for I) f (n) ≥ e(n) and T f (n) + 1 extends T e(n) + 1. We make the following requirements on the extended iteration tree: (a) T f (n) + 1 is normal and uses only extenders taken from below δ; (b) Mf (n) and Me(n) are in sufficient agreement that Pn ∈ Mf (n) (more precisely they agree beyond the rank of Pn ); (c) the extenders Ek for k ∈ [e(n), f (n)) have critical points above rank(X); and (d) for k ∈ [e(n), f (n)), the T -predecessor of k + 1 does not belong to any of the intervals [e(m), f (m)), m < n. Read differently, rule (1d) states that none of the nodes in the interval of numbers [e(m), f (m)) can ever be used as an immediate predecessor in future moves by I. (2) (Rule for I) γn is an ordinal chosen so that Pn is a legal position in j0,f (n) (A)[γn , x].
1F Mixed pivots II
Me(n) _ _ I _ _ Mf (n)
Pn & I
I
6
A
Mf (n)+1
Mf (n)+2 _ _ I _ _ Mf (n+1)
γn K
_ R Y - Pn ln pn un_
_
/o /o /o /o /o /o /o /o /o /o /
wn _ Pn+1 #
II
& ) -
I
4
γn+1 @ '
Pn+1
Diagram 1.8. Round n of Amix [x] and the beginning of round n + 1.
(3) (Rule for I) ln , pn , un is a legal move for I in j0,f (n) (A)[γn , x] following Pn . As in Section 1C(1) we intend to extend the tree order T by setting: (i) the T -predecessor of f (n) + 2 is f (n); (ii) if ln = “new” then the T -predecessor of f (n) + 1 is f (n); and (iii) if ln ∈ N then the T -predecessor of f (n) + 1 is f (ln ) + 1. Continuing with the rules of Amix [x]:
45
46
1 Basic components
(4) (Rule for II) The extenders Ef (n) , Ef (n)+1 must be played in a way which agrees with conditions (i)–(iii) above. Precisely: (a) If ln ∈ N then Ef (n) ∈ Mf (n) must be an extender with critical point within the level of agreement between Mf (n) and Mf (ln )+1 . We then set Mf (n)+1 = Ult(Mf (ln )+1 , Ef (n) ). (b) Ef (n)+1 ∈ Mf (n)+1 must be an extender with critical point within the level of agreement between Mf (n)+1 and Mf (n) . We set Mf (n)+2 = Ult(Mf (n) , Ef (n)+1 ). We have now the extended tree T f (n) + 3, ending with Mf (n)+2 defined above. In line with the rules in Section 1C(1) we make the following additional restriction on the way II forms the models Mf (n)+1 and Mf (n)+2 : (5) (Rule for II) If ln = “new” then Ef (n) must equal “pad” so that Mf (n)+1 = Mf (n) and jf (n),f (n)+1 = id. (6) (Rule for II) T f (n) + 3 must be normal and use only extenders taken from below δ. Having specified Mf (n)+2 we let Qn be the position jf (n),f (n)+2 (Pn −−, ln , pn , un ). (7) (Rule for II) The critical point of jf (n),f (n)+2 must be larger than rank(X) and large enough that Pn is not moved by this embedding. (8) (Rule for II) wn must be a legal move for player II in the shifted game j0,f (n)+2 (A)[jf (n),f (n)+2 (γn ), x], following the position Qn . We set Pn+1 = Qn −−, wn , set e(n + 1) = f (n) + 2, and let T e(n + 1) + 1 be the tree defined above, ending with the model Me(n+1)=f (n)+2 . This completes the round. We are now in a position to start round n + 1. Remark 1F.3. As usual the dependence of our definition on x is Lipschitz continuous; x only comes in through rules (3) and (8). Thus we are as usual defining a map Amix = (s → Amix [s]). Definition 1F.4. We say that round n of a run of Amix [x] does not contain mixing if f (n) = e(n) and (when n > 0) γn = jf (n−1),e(n) (γn−1 ). Otherwise we say that round n contains mixing. A run of Amix [x] which does not contain mixing in any round is simply a run of ˙ 0 ] and x. This is a precise formulation of the pivot game associated to the name A[γ Remark 1F.2. For each n < ω let an = jf (n),f (n)+2 (ln , pn , un )−−, wn . Using rule (7) it is easy to verify inductively that Pn equals $a0 , . . . , an−1 %. Let a = $an | n < ω%.
1F Mixed pivots
47
Let T be the length ω iteration tree produced by the run of Amix [x] described above. An infinite branch of T is even if it contains arbitrarily large nodes in {f (n) | n < ω}. Otherwise the branch is odd. To illuminate this terminology we note that in the case of a run of Amix [x] which does not contain any mixing, {f (n) | n < ω} is precisely equal to {0, 2, 4, . . .}, and the only even branch is 0 T 2 T 4 T · · · . In the general case of a run which does contain mixing, the nodes f (n) need not all be even. Still the division of branches indicated above is useful, and the reference to the two kinds of branches as “even” and “odd” seems as good as any. Note that in the general case there may well be more than one even branch. This has to do with the fact that we do not require e(n) T f (n) in rule (1). Claim 1F.5. f (0) T l for each l > f (0). Proof. This follows from rule (1d) and conditions (i)–(iii), which together imply that for all k ≥ f (0), the T -predecessor of k + 1 is greater than or equal to f (0). # Suppose that b is an infinite odd branch of T . From Claim 1F.5 it follows that f (0) ∈ b. Let n < ω be largest so that f (n) ∈ b. We use root(b) to denote this n. We refer to root(b) as the root of b. By rule (1d) and conditions (i)–(iii), b avoids the intervals [e(m), f (m)) for all m > root(b). It follows that all nodes of b above f (root(b)) are of the form f (m) + 1. The behavior of T on nodes of the form f (m) + 1 was specified precisely in conditions (ii) and (iii). These conditions tell us that b corresponds to a branch in the tree order given by {ln }n<ω . More precisely, there is a sequence {nk }k<ω so that: • n0 = root(b); • ln0 = “new”; • lnk = nk−1 for k > 0; and • {f (n0 )} ∪ {f (nk ) + 1 | k < ω} forms a tail-end of b. We draw the reader’s attention to the similarity between the structure of odd branches here, and the structure of odd branches in Section 1C. Definition 1F.6. Let P be an infinite run of Amix [x], given by T , a , f , and γ say. We use γ (P, b) to denote jf (n0 ),b (γn0 ) where n0 stands for root(b). Definition 1F.7. Let P be an infinite run of Amix [x], given by T , a , f , and γ say. P is a mixed pivot for x just in case that for every odd branch b of T there exists some h so that: (1) h is col(ω, jb (δ))-generic/Mb ; and ˙ b ][h], where γb stands for γ (P, b). (2) $ a , x% ∈ jb (A)[γ
48
1 Basic components
Note the similarity between this and Property 3 in the introduction to this chapter. Of ˙ but with a function A˙ = (γ → course here we are working not with a specific name A, ˙ A[γ ]). We therefore have to specify which γ to use in condition (2) of Definition 1F.7. The γ we take is the one chosen by player I in the round corresponding to the root of the odd branch b. Definition 1F.8. A run of Amix [x] is said to be useful if for all n > 0, γn < jf (m),f (n) (γm ) where m < n is largest so that f (m) T f (n). We already saw that f (0) T l for all l > f (0). So the reference to the largest m < n so that f (m) T f (n) in Definition 1F.8 makes sense. If T is part of a useful run of Amix [x] then all the even branches of T lead to illfounded direct limits. An iteration strategy faced with the tree of a useful mixed pivot is thus forced to pick an odd branch. The branch chosen by the strategy is then subject to the conditions of Definition 1F.7. It is this observation which makes useful mixed pivots useful. We shall talk also of useful position (as opposed to infinite runs). A position of length i in Amix [x] is useful if it satisfies the condition of Definition 1F.8 for all n < i. The following claim is clear: Claim 1F.9. Suppose that P is an infinite run of Amix [x], and that Pi is useful for each i < ω. Then P is useful. # 1F (2) The strategy σmix [, x]. Fix a surjection : ω → M&δ + 1. Fix x ∈ Xω . Continuing the parallel with Section 1C we wish to describe a strategy σmix [, x] for II in Amix [x] which always produces mixed pivots. Fix some imaginary opponent. We describe how to construct a run of Amix [x], ourselves playing for II and letting the imaginary opponent play for I. The run we construct will be a mixed pivot, as required. The construction is similar to that of Section 1C (2), with only a few modifications. Rather than describe the whole construction we confine ourselves to highlighting the modifications. In addition to constructing the objects T , a , f , and γ which form a run of Amix [x], we construct the following objects: • a˙ n , x˙n ∈ Mf (n)+1 &δn + 1 where δn = j0,f (n)+1 (δ); • Dn ∈ Mf (n)+1 &δn + 1; and • γˆn , an ordinal in Mf (n)+1 . Note already the similarity with Section 1C(2). The main difference so far is in the addition of the ordinals γˆn . We intend to maintain all the requirements set by the rules of Amix [x]. In addition we intend to maintain: (A) wn is elastic.
1F Mixed pivots
49
(B) wn is realized by A˙ n , a˙ n , x˙n , Dn , νn in Mf (n)+1 &νn + 5, where ˙ γˆn ] and νn = j0,f (n)+1 (ν). (Recall that ν > δ is the lower A˙ n = j0,f (n)+1 (A)[ ordinal in the least pair of local indiscernibles of M relative to δ.) (C) If ln ∈ N then • a˙ n = jf (ln )+1,f (n)+1 (a˙ ln ), • x˙n = jf (ln )+1,f (n)+1 (x˙ln ), and • γˆn = jf (ln )+1,f (n)+1 (γˆln ). (D) If ln ∈ N then pn belongs to jf (ln )+1,f (n)+1 (Dln ). (E) κ0 < τ0 < κ1 < τ1 < · · · , where κn = dom(un ) and τn = dom(wn ). (F) Mf (n) and Mf (n)+1 agree to κn + ω. Mf (n)+1 and Mf (n)+2 agree to τn + ω. (G) If ln = “new” then Mf (n)+1 = Mf (n) , and γˆn is precisely the ordinal γn played by I (by our imaginary opponent that is) in round n of Amix [x]. We point out several differences between our conditions here and those maintained ˙ γˆn ]; note in Section 1C (2). First, in condition (B) we now have A˙ n = j0,f (n)+1 (A)[ the addition of γˆn . Secondly, in condition (C) there is an added clause that γˆn = jf (ln )+1,f (n)+1 (γˆln ). Finally, there is the new condition (G), relating the ordinals γˆn we construct to the ordinals γn played by I as part of the run of Amix [x]. Observe that condition (G) and the final clause in condition (C) precisely determine all the ordinals γˆn . The construction itself follows closely along the lines of the two cases in Section 1C (2). Just as in Section 1C(2) we obtain the following during the construction: (∗) proj3 (un ) is realized by A˙ n , a˙ n , x˙n in Mf (n)+1 &νn∗ +2. (νn∗ here is the indiscernible to ν, shifted to Mf (n)+1 .) This, together with the conditions in rule (4) in the Definition of A (Section 1A (2)), can be used to verify that T , a , f , and γ form a mixed pivot. The argument is similar to the final argument in Section 1C(2), and is left to the reader. We point out only that the argument uses the new condition (G), and the way it fits with Definition 1F.6. Remark 1F.10. For the record we note that if ln ∈ N then crit(Ef (n) ) = τln , see case 2 in Section 1C (2). Now τln > κln , Pln belongs to Mf (ln ) &κln , and Mf (ln ) and Mf (ln )+1 agree to κln . (All this is by construction, see for example case 1 in Section 1C (2).) It follows that Pln ∈ Mf (ln )+1 is not moved by jf (ln )+1,f (n)+1 . 1F (3) Properties of σmix . Let us recall the general framework of this section. We ˙ ]) in M which assigns to each ordinal γ < jumpM (δ) work with a map A˙ = (γ → A[γ ˙ ] for a subset of (M&δ)ω × Xω in M col(ω,δ) . In Section 1F (1) we some name A[γ worked with a specific x ∈ Xω and defined the game Amix [x]. As usual the definition was Lipschitz continuous in x, giving rise to a map Amix = (s → Amix [s]). It is clear
50
1 Basic components
that this map belongs to M. We refer to Amix as the mixed pivot games map associated ˙ δ, and X. to A, Working with some : ω → M&δ + 1 we went on and modified the construction of Section 1C (2) to construct σmix [, x], a strategy for II in Amix [x]. This construction too was Lipschitz continuous, both in x and in , giving rise to a map σmix = (ϑ, s → σmix [ϑ, s]). It is clear that this map belongs to M. We refer to σmix as the mixed pivot ˙ δ, and X. strategies map associated to A, We have: Lemma 1F.11. Suppose that is onto M&δ +1. Then every run according to σmix [, x] is a mixed pivot for x. # Continuing the parallel with Section 1C, let us say that a mixed pivot given by T , a , f , and γ resides above an ordinal λ < δ if all extenders used in T have critical points above λ. We get: Lemma 1F.12. All runs according to σmix [, x] reside above rank(X). Proof. The extenders Ef (n) and Ef (n)+1 have critical points above rank(X) by construction; see Section 1C(2) and Lemma 1C.7. The extenders Ek for k ∈ [e(n), f (n)) have critical points above rank(X) by rule (1c) in Section 1F (1). # For future use let us record the following consequence of Remark 1F.10: Lemma 1F.13. Let P be an infinite run of Amix [x], given by T , a , f , and γ say. Suppose that P is played according to σmix [, x]. Let b be an infinite odd branch of T . Fix a node k ∈ b larger than f (root(b)), say k = f (m) + 1. Then a m belongs to Mk and is not moved by jk,b . Proof. This follows by repeated applications of Remark 1F.10, going over n > m so that f (n) + 1 ∈ b. #
Chapter 2
Games of fixed countable length
The notions of the previous chapter provide a powerful tool for proving determinacy. Here we apply this tool to games of fixed countable length. Given a countable θ ≥ 1 and C ⊂ Rθ , define the game Gω·θ (C) to be played as follows: In mega-round ξ players I and II alternate natural numbers as in Diagram 2.1 to produce yξ = $yξ (n) | n < ω% ∈ R. Once θ mega-rounds have been played the game ends. If $yξ | ξ < θ % belongs to the payoff set C then I wins. Otherwise II wins. I II
y0 (0)
y0 (2) y0 (1)
......
yξ (0)
...
yξ (1)
yξ (2) ...
......
Diagram 2.1. The game Gω·θ (C).
We intend to prove the determinacy of Gω·θ (C) for countable θ > 1 and C in the pointclass <ω2 − 11 ,1 from a sharp for −1 + θ Woodin cardinals. The corresponding result for θ = 1 is a well-known theorem of Martin. The case of θ = 2 with C in 11 reduces to Theorem 1E.1, and proofs similar to the one in Section 1E can in fact handle all finite θ; see Neeman [29], [32] for details. The additional ingredient which we develop in this chapter is a method for handling limits. Historical Remark. For sufficiently closed limit ordinals θ < ω1 , ordinals θ = ω · β where β is additively closed, the determinacy we get had been proved earlier by Woodin. (It was a pleasant consequence of tight equiconsistency results he proved, connecting ω · β Woodin cardinals under choice to measures on [Pω1 (R)]β under AD.) In particular the determinacy of all <ω2 − 11 games of fixed countable lengths from ω1 Woodin cardinals is due to Woodin.
2A General games and iteration games Each determinacy proof in this book can be viewed as a construction which takes a general game, decides which player to root for, and reduces this player’s task to the task of winning an iteration game on a given model M. When M is iterable this yields the determinacy of the original game. 1 <ω2 − 1 is the pointclass of sets at levels below ω2 in the difference hierarchy on 1 sets. It is 1 1 somewhat larger than the simpler pointclass 11 , so its use makes our results stronger (to the point of being
optimal in fact).
52
2 Games of fixed countable length
The reduction itself can be viewed as a strategy in a long game. If rooting for player I, the game in the case of fixed countable length ω · θ consists of moves in Gω·θ , together with moves in the weak iteration game on M (see Appendix A) with I playing for “bad” and II playing for “good.” A strategy for I in such a game amounts to a reduction of I’s moves in Gω·θ to moves for the good player in the iteration game. Let us be more precise. Fix a countable ordinal θ . Let M be a ZFC∗ model with a Woodin cardinal δ∞ . Suppose that θ is countable also in M. Let A˙ be a name for fixed = G fixed (M, δ∞ , θ, A) ˙ which a subset of Rθ in M col(ω,δ∞ ) . We define a game G begins to formalize a reduction which roots for I. fixed serves to construct the following objects: A run of G • a sequence of reals y = $yξ | ξ < θ%; • a sequence of models $Mξ | 1 ≤ ξ ≤ θ + 1% starting with M1 = M; and • a sequence of embeddings jζ,ξ : Mζ → Mξ for ζ ≤ ξ ≤ θ + 1. The reals correspond to a run of Gω·θ ; the models and embeddings correspond to a run of the weak iteration game. fixed is played in −1 + θ + 1 mega-rounds, indexed 1 through θ. By the start of G mega-round ξ ≤ θ for ξ a successor ordinal we have y(ξ − 1) and the model Mξ . (i) To start mega-round ξ the two players alternate natural number moves in the usual fashion, producing together the real yξ −1 . (ii) Player I then plays an iteration tree Tξ of length ω on the model Mξ . Tξ must be normal, plus 2, and use only extenders taken from below the image of δ∞ . (iii) Player II ends mega-round ξ by playing a cofinal branch bξ through Tξ . We let Mξ +1 be the direct limit along bξ and let jξ,ξ +1 : Mξ → Mξ +1 be the direct limit map. The maps jζ,ξ +1 for ζ < ξ are defined by composition. By the start of mega-round ξ for a limit ordinal ξ ≤ θ we have yξ , the models $Mι | ι < ξ %, and the embeddings $jζ,ι | ζ ≤ ι < ξ %. We let Mξ be the direct limit of the system $Mι , jζ,ι | ζ ≤ ι < ξ % and let jζ,ξ be the direct limit embeddings. Mega-round ξ is then played according to the rules (ii) and (iii) above, but not rule (i). Once over the limit mega-round leaves us with yξ and the model Mξ +1 , precisely the position necessary to start mega-round ξ + 1. fixed results in a sequence of reals y = $yξ | ξ < θ%, a final model Mθ +1 , A run of G and a final embedding j = j1,θ +1 : M → Mθ+1 . If Mθ +1 , or any of the models Mξ for ξ ≤ θ, is illfounded, then I wins. Otherwise I wins just in case that there exists some h so that (1) h is col(ω, j (δ∞ ))-generic/Mθ+1 ; and ˙ (2) y = $yξ | ξ < θ% ∈ j (A)[h].
2A General games and iteration games
53
fixed is precisely a reduction of the kind mentioned Remark 2A.1. A strategy for I in G earlier. This will be clarified further below, through the proof of Theorem 2A.3. fixed = Given a name B˙ for a subset of Rθ in M col(ω,δ∞ ) we define next a game H ˙ Hfixed (M, δ∞ , θ, B) which begins to formalize a reduction rooting for II. Hfixed pre fixed it fixed . The rules of the two games are the same except that in H cisely mirrors G is player II who plays the iteration trees and player I who plays the branches. It is now player II who wins if Mθ+1 or any of the preceding models is illfounded. Otherwise II wins just in case that there exists some h which satisfies condition (1) above, and the mirrored condition: ˙ (2) y = $yξ | ξ < θ% ∈ j (B)[h]. Theorem 2A.2 (for θ ≥ 1 countable, both in V and in M). Suppose that there are −1+θ Woodin cardinals of M below δ∞ (where δ∞ too is a Woodin cardinal of M). Suppose further that M&δ∞ + 1 is countable in V. Fix g ∈ V which is col(ω, δ∞ )-generic/M. Then at least one of the following three cases holds: fixed ; (1) player I has a winning strategy in G fixed ; or (2) player II has a winning strategy in H (3) in M[g] there exists a sequence of reals y = $yξ | ξ < θ % which belongs to ˙ ˙ neither A[g] nor B[g]. Moreover M can distinguish this. To be precise, there are formulae φI and φII so that: ˙ then case (1) holds; if M |= φII [δ∞ , θ, B] ˙ then case (2) holds; if M |= φI [δ∞ , θ, A] and otherwise case (3) holds. In applications to determinacy we work with A˙ and B˙ which name complementary sets, so that case (3) is ruled out. Theorem 2A.2 then states that either the task of player I in Gω·θ can be reduced to playing for “good” in a weak iteration game on M, this is case (1); or else the task of player II in Gω·θ can be reduced to playing for “good” in a weak iteration game on M, this is case (2). We shall prove Theorem 2A.2 by induction on θ. The proof is delayed to Sections 2B (limits) and 2C (successors). We finish this section with an application that shows how the theorem can be used to prove determinacy. Theorem 2A.3. Let θ ≥ 1 be a countable ordinal. Suppose that there exists a model M and a cardinal δ∞ in M so that: • M is a class model of ZFC; • M is weakly iterable; • δ∞ is a Woodin cardinal of M; • there are −1 + θ Woodin cardinals of M below δ∞ (so that including δ∞ there are −1 + θ + 1 Woodin cardinals in M); and
54
2 Games of fixed countable length
• M&δ∞ + 1 is countable in V and has a sharp. Then the games Gω·(θ+1) (C) are determined for all C ⊂ Rθ +1 in the pointclass <ω2 − 11 . The assumption in Theorem 2A.3 follows from the existence in V of a sharp for −1 + θ + 1 Woodin cardinals, by the methods described at the end of Appendix A. Proof of Theorem 2A.3. By Remark 1E.5 every real can be made generic over an iterate of M. From this and the fact that M&δ∞ + 1 has a sharp it follows that every real has a sharp. By Martin (see Kanamori [11, Theorem 31.4]) we therefore have recourse to (boldface) <ω2 − 11 determinacy. Without loss of generality we may assume that θ is countable in M; if not, simply iterate the first measurable of M past θ, and pass to a generic extension where θ is collapsed. Fix C ⊂ Rθ +1 in <ω2 − 11 . For y ∈ Rθ let Cy = {yθ ∈ R | y−−, yθ ∈ C}. Note that Cy ⊂ R is <ω2 − 11 , and therefore determined. Define D ⊂ Rθ by: D = { y ∈ Rθ | I has a winning strategy in Gω (Cy )}. For y ∈ D we know that II has a winning strategy in Gω (Cy ). It is thus enough to prove that the game Gω·θ (D) is determined; whoever wins this game can then continue to win Gω·(θ+1) (C). D belongs to the pointclass ω (<ω2 − 11 ). By Martin [24] there is a number k < ω and a 1 formula φ so that y ∈ D ⇐⇒ L[ y ] |= φ[ y , c0 , . . . , ck−1 ] whenever c0 < · · · < ck−1 are Silver indiscernibles for y.
(2.1)
Let ψ(a, c0 , . . . , ck−1 ) be the formula “L[a] satisfies φ[a, c0 , . . . , ck−1 ].” Let u0 < · · · < uk−1 be uniform Silver indiscernibles. Let A˙ ∈ M be the canonical col(ω, δ∞ ) name for the set ˙ y , u0 , . . . , uk−1 ]}. A[g] = { y ∈ Rθ ∩ M[g] | M[g] |= ψ[ Let B˙ ∈ M be the canonical name for the set ˙ y , u0 , . . . , uk−1 ]}. B[g] = { y ∈ Rθ ∩ M[g] | M[g] |= ψ[ We apply Theorem 2A.2 using these names. Case 1. If case (1) of Theorem 2A.2 holds. Fix an imaginary opponent willing to play for II in Gω·θ (D). We describe how to play against the imaginary opponent, and win. The description as usual takes the form of a construction of, among other things, a run of Gω·θ (D). We verify at the end that the run constructed belongs to D, and is therefore won by player I.
2A General games and iteration games
55
fixed . Fix a winning strategy for I in the game G Using the case assumption fix , , a weak iteration strategy for M. Observe that , , and the imaginary opponent fixed . The imaginary opponent and collaborate to together cover all the moves in G produce the reals yξ ; plays the trees Tξ ; and plays the branches bξ . Letting , , and the imaginary opponent play each other we therefore obtain a complete run of fixed , consisting of a sequence y and a weak iteration leading to a model Mθ +1 and an G embedding j = j1,θ +1 : M → Mθ+1 . Since is an iteration strategy, Mθ+1 and the models leading to it are all well fixed , there exists some h so that: is winning for I in G founded. Since (1) h is col(ω, j (δ))-generic/Mθ+1 ; and ˙ (2) y ∈ j (A)[h]. Condition (2), combined with the definition of A˙ and the fact that Mθ +1 is a wellfounded class model, tells us that L[ y ] |= φ[ y , j (u0 ), . . . , j (uk−1 )]. It is easy to see that j (u0 ), . . . , j (uk−1 ) are Silver indiscernibles for all reals in Mθ +1 [h]. (In fact u0 , . . . , uk−1 are fixed points of j since that map belongs to L of a real— essentially a real coding M&δ∞ and the sequence $Tξ , bξ | 1 ≤ ξ ≤ θ%.) Using equation (2.1) it follows that y belongs to D, as required. # (Case 1) Case 2. If case (2) of Theorem 2A.2 holds. Then an argument which mirrors that of case 1 shows that II wins Gω·θ (D). # (Case 2) Case (3) of Theorem 2A.2 cannot occur, since A˙ and B˙ in the current application name complementary sets. Under case (1) of Theorem 2A.2 we saw that I wins Gω·θ (D). Under case (2) we saw that II wins Gω·θ (D). It follows that Gω·θ (D) is determined. # (Theorem 2A.3) ). Remark 2A.4. Notice that the winning strategy in Gω·(θ +1) (C) belongs to L(R, , This is clear from the proof of Theorem 2A.3, as the proof defines the winning strategy , and M. The strategy comes from Theorem 2A.2. Our proof of that theorem from , (certainly in a projective manner) from any real coding (M&δ∞ + 1) . will define Thus we show not only that Gω·(θ+1) (C) is determined, but that the player who wins the game has a winning strategy in L(R, ), where is a weak iteration strategy for a model satisfying the assumptions of Theorem 2A.3. For a fixed C one can do even with slightly less. We did not use the fact that iterates created using are wellfounded, but only the (strictly weaker) fact that iteration embeddings created using move the type of k indiscernibles correctly for a particular k < ω. This requirement on can be phrased precisely, and any which satisfies the requirement is sufficient to define a winning strategy, for the appropriate player, in Gω·(θ+1) (C).
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2B Limits Fix a countable limit ordinal θ . We prove Theorem 2A.2 for θ , assuming that it is known for θ¯ < θ. Our plan is to break the first θ mega-rounds of the relevant game—either fixed —into ω “blocks” of mega-rounds, reaching up to θ ; break mega-round fixed or H G θ into ω “parts”; and somehow distribute these parts of mega-round θ between the blocks of earlier mega-rounds. Induction will allow us to work through each of the blocks below θ , and the advance planning involved in distributing parts of mega-round θ between these blocks will allow us to work through mega-round θ. Fix a model M with a Woodin cardinal δ∞ . Fix names A˙ and B˙ in M for subsets of θ R in M col(ω,δ∞ ) . We work under the assumptions of Theorem 2A.2. Specifically, θ is countable in M, there are −1 + θ Woodin cardinals of M below δ∞ , and M&δ∞ + 1 is countable in V. Let {δξ }ξ ∈[1,θ) be the first −1 + θ Woodin cardinals of M listed in increasing order. We have δξ < δ∞ for all ξ ∈ [1, θ). Since θ is countable in M, sup{δξ | ξ ∈ [1, θ)} < δ∞ also. Let λ denote this supremum. Fix in M a strictly increasing sequence of ordinals ηk , k < ω, starting with η0 = 0 and cofinal in θ. This is possible since θ is seen to be countable in M. Let θk = ηk+1 −ηk , or more precisely let θk be the unique ordinal such that ηk+1 = ηk + θk . Let δk be δη k+1 . Each δk is a Woodin cardinal of M, and sup{δk | k < ω} = λ < δ∞ . Moreover, for each k < ω there are precisely −1 + θk Woodin cardinals of M below δk and (if k > 0) above δk−1 . We think of [ηk + 1, ηk+1 ] as a “block,” and later plan to handle this block by using Theorem 2A.2 with θk and δk over a generic extension of M which collapses δk−1 . Fix in M some continuous injection ϕ of Rθ into R. The precise injection used is not important, so long as knowledge of yηk suffices to determine ϕ( y )(k + 1). We think y ) and carelessly write y instead of ϕ( y ) throughout. of y ∈ Rθ as coded by the real ϕ( ˙ Fix some g∞ ∈ V which is col(ω, δ∞ )-generic/M. Let A ∈ M be the canonical ˙ ∞ ] in M[g∞ ]. Similarly let B˙ ∈ M be the canonical name for the set (M&δ∞ )ω ×ϕ A[g ω ˙ ∞ ]. Let A be the auxiliary games map associated name for the set (M&δ∞ ) × ϕ B[g ˙ to A , δ∞ , and X = M&λ. Let B be the mirrored auxiliary games map associated to B˙ , δ∞ , and X = M&λ. Intuitively A is a length ω game where I tries to witness ˙ B is a game where II tries to witness membership in a shift membership in a shift of A. ˙ of B. We carelessly write A[ y ] instead of A[ϕ( y )] and similarly with B. Note that knowledge of yηk suffices to determine the rules for the first k + 1 rounds of A[ y ] and B[ y ]. For x ∈ Rηk we shall talk about A[ x ] and B[ x ]. Both are games of k + 1 many rounds. Remark 2B.1. The use of the set X = M&λ instead of X = ω for the auxiliary games maps will later ensure that the pivots in our constructions reside above λ = sup{δk | k < ω}, see Lemma 1C.7 and the comment following it.
2B Limits
57
2B (1) The basic definitions. Working inside M[g∞ ] fix a sequence $gk | k < ω% so that gk is col(ω, δk )-generic/M[g k ] where for each k < ω, g k = [g0 ∗ · · · ∗ gk−1 ]. Below we informally talk about sets in M[g k ][gk ] where formally we should be talking about names in M[g k ]col(ω,δk ) , or better yet names in M col(ω,δ0 )∗···∗col(ω,δk ) . Similarly we talk about M[g k ] where formally we should be using the forcing language to talk about M col(ω,δ)∗···∗col(ω,δk−1 ) . Definition 2B.2 (for k < ω). A k-sequence over M[g k ] is a triplet $ x , P , γ % which belongs to M[g k ] and so that: • γ is an ordinal; • x belongs to Rηk ; and • P is a position of k rounds in A[ x ]. We think of a k-sequence as the result of progressing through k “blocks” of megarounds below θ , to reach an x ∈ Rηk , and progressing through k “parts” in the advance planning for mega-round θ , to reach a position P of k rounds in A[ x ]. Definition 2B.3 (for k < ω). An extended k-sequence over M[g k ] is a triplet $ x , P ∗ , γ ∗ % ∈ M[g k ] so that • γ ∗ is an ordinal; • x belongs to Rηk ; and • P ∗ is a position of k + 1 rounds in A[ x ]. The difference between Definitions 2B.2 and 2B.3 is in the last condition: P is a position of k rounds, while P ∗ is a position of k + 1 rounds. An extended k-sequence thus involves one more “part” in the planning toward mega-round θ. For each k < ω and each k-sequence $ x , P , γ % ∈ M[g k ] we plan to define: • A finite game Gk ( x , P , γ ), played in M[g k ]. The definition will result in a map $ x , P , γ % → Gk ( x , P , γ ), belonging to M[g k ]. k ˙ ˙ We use Gk = Gk [g ] to denote this map. Gk ∈ M names this map in the forcing col(ω, δ0 ) ∗ · · · ∗ col(ω, δk−1 ). For each k < ω and each extended k-sequence $ x , P ∗ , γ ∗ % ∈ M[g k ] we plan further to define: x , P ∗ , γ ∗ ) in M[g k ][gk ], subset of Rθk . We let A˙ k [g k ]( x, P ∗, γ ∗) ∈ • A set Ak ( k M[g ] be the canonical name for this set. Again the definition will result in a map $ x , P ∗ , γ ∗ % → A˙ k [g k ]( x , P ∗ , γ ∗ ), belonging k k to M[g ]. We use A˙ k [g ] to denote this map, and A˙ k ∈ M to denote its canonical name. Let us begin to define these objects. We work by induction on γ and γ ∗ . Everything is done inside M, though for expository simplicity we talk about M[g k ] and M[g k ][gk ] instead of working with the appropriate forcing languages.
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2 Games of fixed countable length
Definition 2B.4. The game Gk ( x , P , γ ) is played according to Diagram 2.2 and the following rules: (1) (Rule for I) γ ∗ must be an ordinal smaller than γ . x ] following the (2) (Rule for I and II) ak−I and ak−II must be legal moves in A[ position P . I II
γ∗
ak−I ak−II
Diagram 2.2. The game Gk ( x , P , γ ).
Once the game is concluded, let ak = $ak−I , ak−II % and P ∗ = P −−, ak . Player I wins just in case that (M&δk+1 )[g k ] |= φI [δk , θk , A˙ k [g k ]( x , P ∗ , γ ∗ )]. x , P , γ ) are actually elements of M. But the payoff condition is Moves in Gk ( phrased in M[g k ]. Note that the definition of Gk ( x , P , γ ) requires knowledge of the x , P ∗ , γ ∗ ), but only for γ ∗ < γ , because of rule (1). names A˙ k [g k ]( x , P , γ ) involves the Remark 2B.5. The payoff condition in the definition of Gk ( formula φI (. . . , θk , . . . ). We assume knowledge of this formula as part of the inductive assumption that Theorem 2A.2 holds for all θ¯ < θ. Remark 2B.6. Note the reference to (M&δk+1 )[g k ] in the payoff condition, rather than M[g k ]. This restriction to a rank initial segment of M[g k ] makes the definition more local; the definition does not depend on the whole of M, only on the relevant initial segment. The local definability will be used later on, see Claim 2B.8. There is nothing specifically important about the particular initial segment taken. We just need a level which satisfies ZFC∗ , and is large enough to include δk and hence the name x , P ∗ , γ ∗ ). A˙ k [g k ]( Note that if $ x , P ∗ , γ ∗ % is an extended k-sequence and z ∈ Rθk ∩ M[g k ][gk ], then ∗ ∗ $ x −− z, P , γ % is a k + 1-sequence. Definition 2B.7. For z ∈ Rθk ∩ M[g k ][gk ] put z ∈ Ak ( x , P ∗ , γ ∗ ) just in case that k ∗ ∗ x −− z, P , γ ).” M[g ][gk ] |=“I wins Gk+1 ( It should be stressed that Definitions 2B.4 and 2B.7 are carried out by simultaneous x , P , γ ) requires knowledge of the names induction on γ and γ ∗ . The definition of Gk ( A˙ k [g k ]( x , P ∗ , γ ∗ ), which in turn depend on the games Gk+1 ( x −− z, P ∗ , γ ∗ ), but only ∗ for γ < γ . The induction therefore makes sense. ˙ k and A˙ k The definitions produce maps Gk and A˙ k [g k ], or more precisely names G for these maps. Since the definitions, when formalized to talk about names, are phrased ˙ k | k < ω% and $A˙ k | k < ω% belong to M. This is important; we shall inside M,2 $G 2 Remember that the Lipschitz continuous map A, which is relevant to the definition of G ( k x , P , γ ), belongs to M.
2B Limits
59
later have to shift the definitions via elementary embedding, and re-interpret them using ˙ k )[hk ] for example, where j : M → Mk various generics. Thus we shall talk about j (G k ˙ k )[hk ] is is elementary and h is generic for col(ω, j (δ0 )) ∗ · · · ∗ col(ω, j (δk−1 )). j (G k the association of Definition 2B.4, but carried out over Mk [h ]. x , P , γ % so that I For the sake of a brief discussion let Sk be the set of k-sequences $ has a winning strategy in Gk ( x , P , γ ). Combining Definitions 2B.4 and 2B.7 we may intuitively think of Sk as the set of k-sequences from which I can win to produce: (1) an ordinal γ ∗ < γ , (2) a one round extension P ∗ of P , and (3) an extension of x by θk mega-rounds to some x∗ ∈ Rηk+1 , so that $ x ∗ , P ∗ , γ ∗ % belongs to a shift of Sk+1 . The moves of (1) and (2) are produced directly through I’s strategy in Gk . Then through the use of the formula φI [. . . θk . . . ] in the payoff for Gk an inductive application of Theorem 2A.2 with θk shows that I has a strategy to create a shift of Sk+1 and an extension x∗ of x which enters this shift. Thus from Sk player I can win to enter a shift of Sk+1 . Then from there I can win to enter a shift of Sk+2 , etc. This intuition will be made precise in Section 2B (2). Here let us end with a claim on the local definability of the set Sk alluded to above. The claim will be needed later on, for example in the proof of Claim 2B.16 in Section 2B (4). Claim 2B.8. There is a formula ψ, with parameters δ∞ , {θn }n<ω , {δn }n<ω , {gn }n<ω , and A, so that for all x, P , γ ∈ M[g∞ ], and for any limit ordinal µ > max{γ , δ∞ }: ˙ k [g k ]( $ x , P , γ % is a k-sequence and I wins G x , P , γ ) ⇐⇒ x, P ∈ (M&µ)[g∞ ] and (M&µ)[g∞ ] |= ψ[k, x, P , γ ]. Proof. Definitions 2B.4 and 2B.7 essentially provide the formula ψ. Note that the only parameters used in the definitions are the ones listed in the claim. Note further the local nature of the definitions. Positions P in the games A[ x ] are all elements of M&δ∞ , ˙ k [g k ]( x , P , γ ) is an element and the map A is an element of M&δ∞ + ω. Each game G of (M&δ∞ + ω)[g∞ ], and the same is certainly true of each set A˙ k [g k ][gk ]( x , P ∗ , γ ∗ ). Furthermore, for any limit ordinal µ > δ∞ , (M&µ)[g∞ ] can figure out the assignments ˙ k [g k ]( x , P , γ ) and $ x , P ∗ , γ ∗ % → A˙ k [g k ]( x , P ∗ , γ ∗ ), restricted to $ x , P , γ % → G γ < µ and γ ∗ < µ. We point out only that this uses the restriction to (M&δk+1 ) in the payoff in Definition 2B.4, see Remark 2B.6. # A 0-sequence is simply a triplet $∅, ∅, γ % for some ordinal γ ∈ M. Thus the end result of our definitions is finite games G0 (∅, ∅, γ ), defined in M for ordinals γ . 2B (2) I wins. Suppose that there exists some γ ∈ M so that I wins the game fixed = G fixed (M, δ∞ , θ, A). ˙ G0 (∅, ∅, γ ). We intend to show that I wins G Fix an imaginary opponent willing to play for II in Gfixed . We describe how to fixed , playing against the imaginary opponent. We shall verify at construct a run of G the end that the run constructed is won by I. Let Apiv be the pivot games map associated to A˙ , δ∞ , and X = M&λ. Let σpiv be the pivot strategies map associated to these objects. (Recall that σpiv is a map which
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2 Games of fixed countable length
provides strategies for II in the pivot games, see Section 1C (3) and Lemma 1C.5.) These maps both belong to M and we are free to move them with elementary embeddings. We divide the construction into ω stages, followed by a “finishing” stage. Stage k fixed , and the finishing stage will concern will concern mega-rounds [ηk + 1, ηk+1 ] of G mega-round θ . At the start of stage k ∈ [0, ω) we shall have the objects of items (A)–(E) below. fixed up to and including mega-round ηk . (A) Moves in G The moves indicated in item (A) produce, among other things: reals $yξ | ξ < ηk %; fixed starts a model Mηk +1 ; and an embedding j1,ηk +1 : M1 → Mηk +1 . (Recall that G with mega-round 1 and M1 = M.) We refer to Mηk +1 as Nk , to j1,ηk +1 as ik , and to $yξ | ξ < ηk % as xk . (B) Some hk = h0 ∗ · · · ∗ hk−1 which is col(ω, i1 (δ0 )) ∗ · · · ∗ col(ω, ik (δk−1 ))generic/Nk . (C) A finite map ϑk : k → Nk &ik (δ∞ ) + 1. xk ], played according to (D) A position Pk of k rounds in the pivot game ik (Apiv )[ the strategy ik (σpiv )[ϑk , xk ]. The position Pk indicated in item (D) includes an iteration tree on Nk of length 2k + 1. k to denote its models, and use π k to We use Uk to denote this tree, use W0k , . . . , W2k ∗,∗ denote its embeddings. Pk further includes a position Pk of k rounds in the auxiliary k ◦ ik )(A)[ xk ]. game (π0,2k k . (E) An ordinal γk in W2k
We point out that Uk only uses extenders with critical points above ik (λ), see Remark 2B.1. In particular i0 (δ0 ), . . . , ik−1 (δk−1 ) are well below the critical points used in Uk . We can thus think of Uk as an iteration tree on Nk [hk ], giving rise to the k [h ]. final model W2k k We shall make sure that: k [hk ]. (i) xk belongs to W2k k [hk ]. We shall Note how this means that $ xk , Pk , γk % is a k-sequence over the model W2k further make sure that: k [hk ] |=“I wins (π k k x , P , γ ).” ˙ (ii) W2k k k k 0,2k ◦ ik )(Gk )[h ](
This last condition serves to keep the construction going. It corresponds to the statement that $ xk , Pk , γk % belongs to a shift of the set Sk mentioned in the brief discussion in Section 2B (1). Remember that intuitively player I can win from Sk to enter a shift of Sk+1 . To perpetuate condition (ii) we simply have to make this intuition precise. For the start of stage 0 set W00 = N0 = M. Condition (i) holds trivially, as x0 = ∅. The existence of some γ0 which satisfies condition (ii) is given by the case assumption, that there is some γ ∈ M so that I wins G0 (∅, ∅, γ ).
2B Limits
61
Let us start stage k of the construction, assuming inductively that the previous stages were taken care of. Fix some ϑ¯ k+1 : k + 1 → Nk &ik (δ∞ ) + 1 which extends ϑk . The precise way we extend ϑk has to do with book-keeping demands and will be explained later on. k [hk ], a winning strategy for I in (π k Appealing to condition (ii) fix σ¯¯ k ∈ W2k 0,2k ◦ ¯ k ˙ k )[h ]( ¯ k denote this game, which follows Diagram 2.2. The ik )(G xk , Pk , γk ). Let G strategy σ¯¯ k opens the game by playing: (a) γ¯¯k+1 < γk ; and k ◦ ik )(A)[ xk ] following the position Pk . (b) a¯¯ k−I , a legal move for I in (π0,2k
We continue the construction through a use of ik (σpiv )[ϑ¯ k+1 , xk ]. This strategy creates an extension of Uk , takes the move a¯¯ k−I of condition (b) which was played in the final even model of Uk , shifts it to the final even model of the extension, and replies to it there. More precisely, ik (σpiv )[ϑ¯ k+1 , xk ] produces a two model extension of Uk , including k k k ¯ k+1 denote the extended tree. models W2k+1 , W2k+2 and an embedding π2k,2k+2 . Let U k k ik (σpiv )[ϑ¯ k+1 , xk ] continues by shifting (π0,2k ◦ ik )(A)[ xk ] to M2k+2 and playing: k ◦ ik )(A)[ xk ] following the shifted position (c) A move a¯ k−II for II in (π0,2k+2 k π2k,2k+2 (Pk −−, a¯¯ k−I ). k k Let γ¯k+1 = π2k,2k+2 (γ¯¯k+1 ). Let a¯ k−I = π2k,2k+2 (a¯¯ k−I ), and let a¯ k = $a¯ k−I , a¯ k−II %. k ¯ Let Pk+1 = π2k,2k+2 (Pk )−−, a¯ k . The triplet γ¯k+1 , a¯ k−I , a¯ k−II represents a run of ¯¯ ), played according to σ¯ = π k k ¯¯ k ). Since σ¯ k is a winning stratπ2k,2k+2 (G k k 2k,2k+2 (σ egy for I we get: k s )[hk ] |= φ [δ s , θ , C s k s (iii) (W2k+2 &δk+1 I k k ˙ k ], where δk = (π0,2k+2 ◦ ik )(δk ), δk+1 = k k (π0,2k+2 ◦ ik )(δk+1 ), and C˙ k = (π0,2k+2 ◦ ik )(A˙ k )[hk ]( xk , P¯k+1 , γ¯k+1 ).
¯ k+1 have critical points above the image Remember that all the extenders used in U of λ = sup{δn | n < ω}. The forcing which gives rise to hk resides well below k k these critical points. The embedding π0,2k+2 : Nk → W2k+2 therefore extends to an k k k k ˙ embedding π0,2k+2 : Nk [h ] → W2k+2 [h ]. Ck is a canonical name for a subset of k k Rθk in W2k+2 [hk ]. As such it resides well below the critical point of π0,2k+2 . Thus C˙ k k k ˙ ˙ belongs to Nk [h ], and π (Ck ) = Ck . Using condition (iii) and pulling back by 0,2k+2
k π0,2k+2 we conclude that:
(iv) (Nk &ik (δk+1 ))[hk ] |= φI [ik (δk ), θk , C˙ k ]. We now appeal to the main inductive assumption in this section, that Theorem 2A.2 holds for θ¯ < θ. Combined with condition (iv) it tells us that player I wins the game
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2 Games of fixed countable length
fixed (N˜ [hk ], ik (δk ), θk , C˙ k ), where N˜ = Nk &ik (δk+1 ). Fix k , a winning strategy for G I in this game. fixed (Nk [hk ], ik (δk ), θk , C˙ k ) are really fixed (N˜ [hk ], ik (δk ), θk , C˙ k ) and G Note that G the same game. The reason is rule (ii) in Section 2A, which forces the iteration trees fixed (Nk [hk ], ik (δk ), θk , C˙ k ) to only use extenders taken from below the played in G image of δk . In particular these trees are then iteration trees on the shorter model fixed (Nk [hk ], ik (δk ), θk , C˙ k ). k is a winning strategy for I in the game G N˜ [hk ]. Thus Remark 2B.9. We could have saved ourselves the roundabout argument in the preceding paragraph, if we used M[g k ] instead of (M&δk+1 )[g k ] in Definition 2B.4. But that would have ruined the local nature of Definition 2B.4. fixed = Nk is equal to Mηk +1 , the starting model for mega-round ηk +1 of the game G k ˙ ˙ Gfixed (M, δ∞ , θ, A). Mega-rounds [1, θk ] in Gfixed (Nk [h ], ik (δk ), θk , Ck ) therefore fixed . correspond precisely to mega-rounds [ηk + 1, ηk + θk = ηk+1 ] in the main game G k (Note that iteration trees on Nk [h ] can be viewed as iteration trees on Nk , see Claim 11 k can thus be viewed as a strategy for I handling mega-rounds in Appendix A.) k play these mega-rounds against the imaginary [ηk + 1, ηk+1 ] in the main game. Let opponent. Together they produce: (d) the reals yξ for ξ ∈ [ηk , ηk+1 ); and (e) iterates of Mηk +1 , leading ultimately to the iterate Mηk+1 +1 and the iteration embedding jηk +1,ηk+1 +1 . fixed If any of the iterates indicated in item (e) is illfounded then I wins the game G by default, regardless of how the game proceeds from here. So let us assume that the iterates are wellfounded. Let zk = $yξ | ηk ≤ ξ < ηk+1 % ∈ Rθk . Let Nk+1 = Mηk+1 +1 and let ik,k+1 : Nk → fixed (Nk [hk ], ik (δk ), θk , C˙ k ) k is a winning strategy for I in G Nk+1 be jηk +1,ηk+1 +1 . As we know that there exists some hk so that: (1) hk is col(ω, ik+1 (δk ))-generic/Nk+1 [hk ]; and (2) zk ∈ ik,k+1 (C˙ k )[hk ]. ¯ k+1 ). Let W k+1 , . . . , W k+1 denote the models of Uk+1 , Let Uk+1 = ik,k+1 (U 0 2k+2 k+1 denote the embeddings. Note that for each l ≤ 2k + 2, W k+1 is simply and let π∗,∗ l ik,k+1 (Wlk ). The same is true of the embeddings. Using the elementarity of ik,k+1 we k+1 k+1 agrees with W0k+1 = Nk+1 up to ik+1 (λ) and crit(π0,2k+2 ) > ik+1 (λ). get that W2k+2 k+1 k Let h = h ∗ hk . Let xk+1 = $yξ | ξ < ηk+1 % = xk −− zk . Using condition (2) we see that certainly xk+1 ∈ Nk+1 [hk+1 ]. Using the agreement between Nk+1 and k+1 k+1 it follows that xk+1 ∈ W2k+2 [hk+1 ]. This secures the inductive condition (i) for W2k+2 k + 1. Let Pk+1 = ik,k+1 (P¯k+1 ) and let γk+1 = ik,k+1 (γ¯k+1 ). Taking condition (2) and folding in the definition of C˙ k from condition (iii) we get:
2B Limits
63
k+1 ◦ ik+1 )(A˙ k )[hk ][hk ]( xk , Pk+1 , γk+1 ). • zk ∈ (π0,2k+2
By Definition 2B.7 this simply says that: k+1 ˙ k+1 )[hk ][hk ]( ◦ ik+1 )(G xk+1 , Pk+1 , γk+1 ). • I wins the game (π0,2k+2
Thus the inductive condition (ii) is secured for k + 1. Let ϑk+1 = ik,k+1 (ϑ¯ k ). Let Pk+1 be the position given by Uk+1 and Pk+1 in the pivot game ik+1 (Apiv )[ xk+1 ]. For the record we note that: (v) ϑk+1 extends ik,k+1 (ϑk ) and Pk+1 extends ik,k+1 (Pk ). k+1 (vi) γk+1 is strictly smaller than (π2k,2k+2 ◦ ik,k+1 )(γk ).
Condition (vi) follows from condition (a) above, which in turn traces back to rule (1) in Definition 2B.4. These final assignments and observations complete the construction for stage k, putting us in a position to start stage k + 1. Running through stages [0, ω) of the fixed . In parconstruction we obtain moves corresponding to mega-rounds [1, θ) of G ticular we obtain the reals $yξ | ξ < θ% which form y, and the models and embeddings $Mξ , jζ,ξ | ζ ≤ ξ < θ%. To complete the proof we must play mega-round θ , and verify that I wins the resulting run. Let Mθ be the direct limit of the system $Mξ , jζ,ξ | ζ ≤ ξ < θ %. This is also the direct limit of the system $Nk , il,k | l ≤ k < ω%. Let ik,∞ : Nk → N∞ = Mθ denote the direct limit embeddings of this system. Let i∞ = i0,∞ : N0 → N∞ . Note that this is the same embedding as j1,θ : M1 → Mθ . If N∞ = Mθ is illfounded then I wins by default, regardless of how this run of fixed continues. So suppose that N∞ is wellfounded. G Let P∞ = k<ω ik,∞ (Pk ). Let ϑ∞ = k<ω ik,∞ (ϑk ). Both assignments make y ], played according to sense by condition (v). P∞ is an infinite run of i∞ (Apiv )[ i∞ (σpiv )[ϑ∞ , y], see item (D) above. Remember that one of the initial assumptions in this section says that M&δ∞ + 1 is countable in V. Using this assumption we can arrange that: (vii) ϑ∞ : ω → N∞ &i∞ (δ∞ ) + 1 is onto. Arranging (vii) is a simple matter of book-keeping, involving the choice of ϑ¯ k+1 extending ϑk in stage k of the construction. (The construction itself places no restrictions on this extension.) Let us only emphasize the use of: (a) the initial assumption that M&δ∞ + 1 is countable; and (b) the fact that all the iteration trees created during the construction are countable—in fact they have length ω by rule (ii) in Section 2A—and use extenders taken from below the image of δ∞ . It follows from these facts that all the relevant embeddings preserve countability (see Appendix A), and N∞ &i∞ (δ∞ ) + 1 is therefore countable in V. Applying Lemma 1C.5, or more precisely Lemma 1C.8, we see that P∞ is a pivot for y over N∞ . This uses (vii), together with the fact that P∞ is by construction played according to i∞ (σpiv )[ϑ∞ , x].
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Let U∞ = k<ω ik,∞ (Uk ). U∞ is the length ω iteration tree played as part of the ∞ denote its embeddings. pivot P∞ . Let Wl∞ , l < ω, denote the models of U∞ . Let π∗,∗ k ∞ Note that Wl is equal to ik,∞ (Wl ) for any k < ω so that l ≤ 2k. The same is true for the embeddings. For each k < ω let γk∞ = ik,∞ (γk ). Using condition (vi) we see that: ∞ < π∞ ∞ (viii) For each k < ω, γk+1 2k,2k+2 (γk ).
It follows that the direct limit along the even branch of U∞ is illfounded. fixed , mega-round θ. We play We are now ready to play the final mega-round of G U∞ , a length ω iteration tree on Mθ , as I’s move. The imaginary opponent, playing for II, picks bθ , a cofinal branch through U∞ . Let Nb = Mθ +1 be the direct limit along bθ , and let i∞,b = jθ,θ +1 be the direct limit embedding. This completes our construction, joint with the imaginary opponent, of a run of fixed . It remains to verify that the run constructed is won by player I. G If by chance bθ is the even branch through U∞ then I wins by default; remember that the even branch of U∞ leads to an illfounded direct limit by condition (viii). We may therefore assume that bθ is odd. Since U∞ is part of an i∞ (A)-pivot for y over N∞ it follows that there exists some h so that: (1) h is col(ω, (i∞,b ◦ i∞ )(δ∞ ))-generic/Nb ; and (2) $ a∞ , y% ∈ (i∞,b ◦ i∞ )(A˙ )[h]. ∞ = $al∞ | l < ω%. In Here al∞ = ik,∞ (a l ) for any sufficiently large k < ω, and a other words a∞ = k<ω ik,∞ (Pk ) is the sequence of auxiliary moves played as part of the pivot P∞ . Now Nb is the same as Mθ+1 , and i∞,b ◦ i∞ is the same as jθ,θ +1 ◦ j1,θ = j : M → ˙ CondiMθ +1 . Remember also that A˙ simply names the product of (M&δ∞ )ω with A. tions (1) and (2) can thus be re-written as: (1 ) h is col(ω, j (δ∞ ))-generic/Mθ+1 ; and ˙ (2 ) y = $yξ | ξ < θ% ∈ j (A)[h]. fixed . The run we constructed is therefore These precisely are the winning conditions in G won by player I, as required. Through the construction above we proved the following claim: Claim 2B.10. Suppose that there exists an ordinal γ ∈ M so that I wins G0 (∅, ∅, γ ). fixed (M, δ∞ , θ, A). fixed = G ˙ # Then I wins the game G As a closing remark let us note that the construction of P above is essentially a matter of taking the construction in case 1 of the sample application Theorem 1E.1 of Section 1E, breaking it up into ω separate stages, and spreading these stages over the models Nk , k < ω. More precisely the passage above from Pk to Pk+1 is similar to round k in the construction in Section 1E, done over Nk and then shifted to Nk+1 . The end use of P∞ is similar to the use of the pivot in Section 1E, done over N∞ .
2B Limits
65
2B (3) II wins. Recall that B˙ is a name for a subset of Rθ in M col(ω,δ∞ ) , that B˙ names ˙ and that B is the mirrored auxiliary game associated the product of (M&δ∞ )ω with B, ˙ to B . In Section 2B(1) we developed a condition which, in Section 2B (2), was shown fixed (M, δ∞ , θ, A). ˙ Here we mirror this development to come up to imply that I wins G fixed (M, δ∞ , θ, B). ˙ with a condition which implies that II wins H x , Q, γ % ∈ Definition 2B.11 (for k < ω). A mirrored k-sequence over M[g k ] is a triplet $ M[g k ] so that: • γ is an ordinal; • x belongs to Rηk ; and • Q is a position of k rounds in B[ x ]. Definition 2B.12 (for k < ω). A mirrored extended k-sequence over M[g k ] is a triplet $ x , Q∗ , γ ∗ % ∈ M[g k ] so that: • γ ∗ is an ordinal; • x belongs to Rηk ; and • Q∗ is a position of k + 1 rounds in B[ x ]. These definitions should be compared with Definitions 2B.2 and 2B.3. The only difference is the change from the auxiliary games map A associated to the name A˙ , to the mirrored auxiliary games map B associated to the name B˙ . x , Q, γ ) and names B˙ k [g k ]( x, P ∗, γ ∗) As in Section 2B(1) we define games Hk ( ∗ by induction on γ and γ . x , Q, γ ) is played according to Diagram 2.3 and the Definition 2B.13. The game Hk ( following rules: (1) (Rule for II) γ ∗ must be an ordinal smaller than γ . (2) (Rule for II and I) bk−II and bk−I must be legal moves in B[ x ] following the position Q. I II
γ∗
bk−I bk−II
Diagram 2.3. The game Hk ( x , Q, γ ).
Once the game is concluded, let bk = $bk−II , bk−I % and Q∗ = Q−−, bk . Player II wins x , Q∗ , γ ∗ )]. just in case that (M&δk+1 )[g k ] |= φII [δk , θk , B˙ k [g k ](
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Definition 2B.13 should be compared with Definition 2B.4. There are the standard differences: instead of working with a k-sequence we work with a mirrored k-sequence; x , Q∗ , γ ∗ ), the payoff now is for player II; the name appearing in the payoff is B˙ k [g k ]( ˙ to be defined below, instead of a name given by A; and the formula used in the payoff condition is φII (which is assumed known for θ¯ < θ ) instead of φI . We point out one more difference. The roles of players I and II are reversed here. This applies of course to the mirrored auxiliary moves bk−II and bk−I , see Section 1D. But it also applies to the move γ ∗ . This move is now played by II. Definition 2B.14. For z ∈ Rθk ∩ M[g k ][gk ] put z ∈ Bk ( x , Q∗ , γ ∗ ) just in case that k ∗ ∗ x −− z, Q , γ ).” M[g ][gk ] |=“II wins Hk+1 ( Definition 2B.14 is the standard mirror image of Definition 2B.7. It determines B˙ k [g k ]( x , Q∗ , γ ∗ ), the canonical name for Bk ( x , Q∗ , γ ∗ ), modulo knowledge of the x −− z, Q∗ , γ ∗ ). As in Section 2B (1), Definitions 2B.14 and 2B.13 are games Hk+1 ( carried out simultaneously, by induction on γ and γ ∗ , inside M. A mirrored 0-sequence is simply a triplet $∅, ∅, γ % for some ordinal γ ∈ M. An argument which mirrors that in Section 2B(2) gives the following claim, which mirrors Claim 2B.10: Claim 2B.15. Suppose that there exists an ordinal γ ∈ M so that II wins H0 (∅, ∅, γ ). fixed (M, δ∞ , θ, B). ˙ # Then II wins H 2B (4) The third case. Suppose that there does not exist a γ ∈ M so that I wins G0 (∅, ∅, γ ), and that there does not exist a γ ∈ M so that II wins H0 (∅, ∅, γ ). We shall show that case (3) of Theorem 2A.2 holds. In other words we shall produce some ˙ ∞ ] nor B[g ˙ ∞ ]. $yξ | ξ < θ% ∈ R θ ∩ M[g∞ ] which belongs to neither A[g Let $γL , γH % be a pair of local indiscernibles of M relative to δ∞ . Recall that this means that γL < γH and: (M&γL + ω) |= ψ[γL , c0 , . . . , ck−1 ] ⇐⇒ (M&γH + ω) |= ψ[γH , c0 , . . . , ck−1 ] for any k < ω, any formula ψ with k + 1 free variables, and any c0 , . . . , ck−1 ∈ (M&δ∞ + ω). Claim 2B.16. Let x be an element of Rηk ∩ M[g k ], and let P be a position of k rounds x , P , γL ) iff I wins Gk ( x , P , γH ). (Note the switch from γL in A[ x ]. Then I wins Gk ( to γH .) Proof. This is immediate from the indiscernibility property of γL and γH , using Claim 2B.8. # The following claim is the mirror image of Claim 2B.16: Claim 2B.17. Let x be an element of Rηk ∩ M[g k ], and let Q be a position of k rounds x , Q, γL ) iff II wins Hk ( x , Q, γH ). # in B[ x ]. Then II wins Hk (
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Let σgen ∈ M[g∞ ] be the generic strategies map associated to A˙ , δ∞ , and X = M&λ. Let τgen ∈ M[g∞ ] be the mirrored generic strategies map associated to B˙ , δ∞ , and X = M&λ. We work in stages k < ω to construct: (A) y = $yξ | ξ < θ% ∈ Rθ ∩ M[g∞ ]; y ]; and (B) a , an infinite run of A[ y ], played according to σgen [ an infinite run of B[ (C) b, y ], played according to τgen [ y ]. The construction takes place inside M[g∞ ]. Note in this respect the fact that σgen and τgen belong to M[g∞ ]. We use xk to denote yηk , At the start of stage k we shall have yηk , a k, and bk. We shall maintain: Pk to denote a k, and Qk to denote bk. xk , Pk , γL ); and (i) I does not win the game Gk ( xk , Qk , γL ). (ii) II does not win the game Hk ( For k = 0 both conditions (i) and (ii) hold because of the case assumption, that I does not win G0 (∅, ∅, γ ) for any γ in M, and similarly with II and H0 . Let us begin stage k of the construction, assuming inductively that the previous stages were taken care of. From condition (i) and Claim 2B.16 if follows that I does not win the game xk , Pk , γH ). This game is finite, hence determined. So II wins. Fix a winning Gk ( xk , Pk , γH ). strategy σk for II. We proceed by using this strategy to construct a run of Gk ( ∗ xk , Pk , γH ). The move is legal since Play γ = γL as a first move, for I, in Gk ( xk , Pk , γH ) continues with moves in round k of A[ xk ] following the γL < γH . Gk ( xk ] and σk together cover these moves, playing ak−I and ak−II reposition Pk . σgen [ spectively. This produces ak , and with it Pk+1 . Note that ak is consistent with σgen [ xk ], as required by condition (B). xk , Pk , γH ), so the run constructed above is won by σk is winning for II in Gk ( player II. It follows that: xk , Pk+1 , γL )]. (iii) (M&δk+1 )[g k ] |= φI [δk , θk , A˙ k [g k ]( Working similarly, but using condition (ii) and Claim 2B.17, we get moves bk−II and bk−I so that bk = $bk−II , bk−I % is consistent with τgen [ xk ] and so that: xk , Qk+1 , γL )]. (iv) (M&δk+1 )[g k ] |= φII [δk , θk , B˙ k [g k ]( By induction Theorem 2A.2 holds for θ¯ < θ. In particular it holds for θk . Applying the theorem and using conditions (iii) and (iv) it follows that there exists some zk ∈ Rθk ∩ k+1 , γL ). xk , Pk+1 , γL ) and B˙ k [g k ][gk ]( xk , Q M[g k ][gk ] which avoids both A˙ k [g k ][gk ]( Fix such a z. Let yηk +ξ = zξ for ξ < θk . This defines xk+1 = yηk+1 . The construction for stage k is now complete. Condition (i) holds for k + 1 because zk
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avoids Ak ( xk , Pk+1 , γL ). Similarly condition (ii) holds for k + 1 because zk avoids Bk ( xk , Qk+1 , γL ). Going through stages [0, ω) we obtain y, a , and b according to conditions (A)–(C). Since everything is done inside M[g∞ ], certainly y ∈ Rθ ∩ M[g∞ ]. Using condi˙ ∞ ]. Using tion (B) and Lemma 1B.2 we see that $ a , y% ∈ A˙ [g∞ ], and hence y ∈ A[g ˙ ∞ ]. So case (3) of Theorem 2A.2 condition (C) and Lemma 1D.2 we see that y ∈ B[g holds, as required. 2B (5) Summary. Our work so far establishes that at least one of the cases in Theorem 2A.2 holds: if there is a γ ∈ M so that I wins G0 (∅, ∅, γ ) then I wins fixed (M, δ∞ , θ, A), ˙ see Section 2B(2); if there is a γ ∈ M so that II wins H0 (∅, ∅, γ ) G ˙ see Section 2B (3); and otherwise there exists y ∈ then II wins Hfixed (M, δ∞ , θ, B), θ ˙ ∞ ] nor B[g ˙ ∞ ], see Section 2B (4). R ∩ M[g∞ ] which belongs to neither A[g Our work also shows that M can distinguish between the three cases. The statements “there exists a γ so that I wins G0 (∅, ∅, γ )” and “there exists a γ so that II wins H0 (∅, ∅, γ )” can be phrased inside M, see Claim 2B.8. Moreover the parameters needed to phrase these statements are definable in M from δ∞ , A˙ and B˙ respectively, and the sequence {ηk }k<ω fixed at the outset in Section 2B. (Note that the auxiliary games map A is definable in M from the name A˙ . The definition is given by Section 1A (2).) Let ˙ be the formula saying that “there exists some strictly increasing sequence φI (δ∞ , θ, A) {ηk }k<ω cofinal in θ so that for some γ player I wins the game G0 (∅, ∅, γ ) defined using fixed (M, δ∞ , θ, A). ˙ then I wins G ˙ Define φII (δ∞ , θ, B) ˙ {ηk }k<ω .” If M |= φI (δ∞ , θ, A) similarly. In conclusion: working with a limit ordinal θ we phrased the two formulae φI (∗, θ, ∗) and φII (∗, θ, ∗) and proved Theorem 2A.2 for θ, assuming that the theorem holds for θ¯ < θ and assuming knowledge of the formulae φI (∗, θ¯ , ∗) and φII (∗, θ¯ , ∗) for θ¯ < θ.
2C Successors Fix a countable θ ≥ 1. We prove Theorem 2A.2 for θ + 1 assuming that it holds for θ . fixed (. . . θ + 1 . . . ) into two parts, the first dealing with The proof involves breaking G mega-rounds [1, θ] and handled using an inductive application to Theorem 2A.2 for θ, and the second dealing with mega-round θ + 1 and handled using an adaptation of the construction in Section 1E. A similar proof shows that Theorem 2A.2 holds for θ = 1, and we shall comment on this later. Fix a ZFC∗ model M with a Woodin cardinal δ . Suppose that −1 + θ + 1 is seen to be countable in M. Suppose that there are −1 + θ + 1 Woodin cardinal of M below δ . Let A˙ ∈ M and B˙ ∈ M be col(ω, δ ) names for subsets of Rθ +1 . Assume that M&δ + 1 is countable in V. Fix g which is col(ω, δ )-generic/M. We will prove Theorem 2A.2 for these objects. Fix δ < δ , a Woodin cardinal of M, least so that there are −1 + θ Woodin cardinals of M below δ. This is possible since there are −1 + θ + 1 Woodin cardinals of M
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below δ . Fix g ∈ M[g ] which is col(ω, δ)-generic/M. We shall use the fact that Theorem 2A.2 holds for θ, applying this theorem with δ, g, and names to be defined shortly. As a matter of notation, we use y for elements of Rθ , and z for elements of R. y−−, z is then an element of Rθ+1 . Working in M, fix a continuous injection ϕ : Rθ +1 → R. The precise nature of this injection is not important, so long as knowledge of y and zk suffices to determine ϕ( y −−, z)k. We think of y−−, z as coded by the real ϕ( y −−, z), and carelessly write y−−, z instead of ϕ( y −−, z) throughout. Let A˙ ∈ M be the canonical name for the set (M&δ )ω × ϕ A˙ [g ] in M[g ]. Similarly let B˙ ∈ M be the canonical name for the set (M&δ )ω × ϕ B˙ [g ]. Let A be the auxiliary games map associated to A˙ , δ , and X = M&δ. Let B be the mirrored auxiliary games map associated to B˙ , δ , and X = M&δ. We carelessly write A[ y −−, z] instead of A[ϕ( y −−, z)] and similarly with B. Note that knowledge of y and zk suffices to determine the rules for the first k rounds of A[ y −−, z] and B[ y −−, z]. Remark 2C.1. The use of the set X = M&δ for the auxiliary games maps will later ensure that the pivots produced in the construction reside above (an image of) δ, see Lemma 1C.7. y ) be the game played according to Diagram 2.4 For each y ∈ Rθ ∩ M[g] let G∗ ( and the rules below: I II
z(0)
a0−I
a1−I a0−II
z(1)
z(2)
...
a1−II
...
Diagram 2.4. The game G∗ ( y ).
I and II alternate playing natural numbers forming together the real z. In addition y −−, z]. The first player to they play moves an−I and an−II in the auxiliary game A[ violate these rules loses. If the rules are successfully followed for ω moves then II wins. y ) played Mirroring the above definition, define for each y ∈ Rθ ∩M[g] a game H ∗ ( according to Diagram 2.5 and the rules below: I II
z(0)
b0−I b0−II
b1−I z(1)
b1−II
z(2)
... ...
Diagram 2.5. The game H ∗ ( y ).
Again I and II collaborate to create the real z. In addition they play auxiliary moves, this time in the mirrored auxiliary game B[ y −−, z]. It is now player I who wins infinite runs. The games G∗ ( y ) and H ∗ ( y ) are defined modulo knowledge of the maps A and B, and the sequence y. A and B exist in M. y exists in M[g]. So the games exist in
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M[g]. Notice that the games are open and closed respectively. So they are determined, in M[g]. Let A˙ ∈ M be the canonical col(ω, δ) name for the set of y ∈ Rθ ∩ M[g] so that I wins G∗ ( y ). Let B˙ ∈ M be the canonical name for the set of y ∈ Rθ ∩ M[g] so that II ∗ y ). wins H ( ˙ We intend to apply Theorem 2A.2 using θ , δ, and the names A˙ and B. fixed (M, δ, θ, A). ˙ Then player I wins Claim 2C.2. Suppose that player I wins G ˙ Gfixed (M, δ , θ + 1, A ). Proof. Fix an imaginary opponent willing to play for II in the game G fixed = ˙ Gfixed (M, δ , θ + 1, A ). We describe how to play against this imaginary opponent, and win. a winning strategy for player I in Using the assumption of the claim, fix , fixed (M, δ, θ, A). against the imaginary opponent. This takes care of mega˙ Pit G rounds [1, θ ] of Gfixed , creating the reals yξ for ξ < θ and the models and embeddings $Mξ , jζ,ξ | ζ ≤ ξ ≤ θ + 1%. If Mθ+1 , or any of the preceding models, is illfounded , regardless of how the run continues. So let us assume that all then player I wins G fixed fixed (M, δ, θ, A) is winning for I in G ˙ models encountered are wellfounded. Since we know that there exists some h so that:
(1) h is col(ω, j1,θ +1 (δ))-generic/Mθ+1 ; and ˙ (2) y ∈ j1,θ +1 (A)[h]. against the imaginary opponent, and It remains to play mega-round θ + 1 of G fixed win. Mega-round θ + 1 consists of moves giving rise to a real z = yθ , an iteration tree Tθ +1 , and a cofinal branch bθ+1 through that tree. Condition (2) and the definition of A˙ together imply that player I, the open player, wins j1,θ +1 (G∗ )( y ). Fix σ ∗ witnessing this. Since winning an open game is absolute we may fix σ ∗ ∈ Mθ+1 [h]. Let Apiv be the pivot games map associated to A˙ , δ , and X = M&δ in M. Let σpiv be the pivot strategies map associated to these objects. Moves given by j1,θ +1 (σpiv ), σ ∗ and shifts of that strategy, and the imaginary y −−, z) opponent combine to produce a real z and a j1,θ+1 (A)-pivot a , U for x = ϕ( over the model Mθ+1 , with the even branch of U illfounded. The construction, which we omit, is similar to that in case 1 in the proof of Theorem 1E.1. j1,θ+1 (σpiv ) takes care of the production of U and of II’s auxiliary moves in shifts of j1,θ+1 (G∗ )( y ) along the even branch of U; the imaginary opponent takes care of II’s part of the real z; and σ ∗ and its shifts along the even branch of U take care of I’s moves in j1,θ+1 (G∗ ), including I’s part of the real z. Remark 2C.3. The construction of z, U, and a involves shifting the strategy σ ∗ along the even branch of U. In Section 1E, σ ∗ was an element of M and could thus be shifted without concern. Here we do not have σ ∗ ∈ Mθ +1 , but only know that σ ∗ is an element
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of Mθ +1 [h] where h is generic for col(ω, j1,θ +1 (δ)). Fortunately the critical points of the extenders used in U are higher than j1,θ +1 (δ). The embeddings of U can thus be extended to act on Mθ+1 [h], and σ ∗ can be shifted using the extended embeddings. The fact that the extenders of U have critical points above j1,θ+1 (δ) traces back to Remark 2C.1, and uses the initial settings of δ > δ. Remark 2C.4. The assumption that M&δ + 1 is countable in V is used implicitly in the construction of z, U, and a . It implies that Mθ +1 &j1,θ+1 (δ ) + 1 is countable in V, and this is needed for the use of j1,θ +1 (σpiv ). The real z constructed above corresponds to rule (i) of Section 2A, in mega-round . Proceeding to rule (ii) we play the tree U constructed above as θ + 1 of the game G fixed the move Tθ +1 . The imaginary opponent responds with a cofinal branch bθ +1 through U. Nθ +2 is set to be the direct limit along bθ+2 , and jθ +1,θ +2 is set to be the direct limit . It remains to verify that the run we constructed embedding. This ends the game G fixed is won by player I. by If b = bθ +1 is the even branch of U, then Mθ +2 is illfounded3 and I wins G fixed default. So assume that bθ+1 is an odd branch. y −−, z) we know that there exists an h Since a , U is a j1,θ +1 (A)-pivot for x = ϕ( so that: (1) h is col(ω, (jb ◦ j1,θ +1 )(δ ))-generic/Mb where Mb is the direct limit along b and jb the direct limit embedding; and (2) $ a , x% ∈ (jb ◦ j1,θ +1 )(A˙ )[h ]. Using the definition of A˙ , and the fact that Mθ +2 = Mb and jθ +1,θ +2 = jb , we get: (1 ) h is col(ω, j1,θ +2 (δ ))-generic/Mθ+2 ; and (2 ) y−−, z ∈ j1,θ +2 (A˙ )[h ]. . This precisely is the winning condition for I in G fixed
#
fixed (M, δ, θ, B). ˙ Then player II wins Claim 2C.5. Suppose that player II wins H fixed (M, δ , θ + 1, B˙ ). H Proof. Mirror the proof of Claim 2C.2.
#
˙ holds true in M&δ , Let φI (δ , θ + 1, A˙ ) be the formula expressing “φI (δ, θ, A) where δ and A˙ are defined as above.” (Note that δ and A˙ are definable in M from θ + 1, δ , and A˙ .) Define φII (δ , θ + 1, B˙ ) similarly. ˙ Using Theorem 2A.2 for θ it If M |= φI [δ , θ + 1, A˙ ] then M&δ |= φI [δ, θ, A]. fixed (M, δ, θ, A), ˙ ˙ follows that I wins Gfixed (M&δ , δ, θ, A). This game is the same as G 3 The fact that the even branch of U is illfounded has to do with the fact that σ ∗ is a winning strategy for the open player. See the construction in case 1 of Theorem 1E.1.
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since both games involve only objects in M at the level of δ. So Claim 2C.2 applies, fixed (M, δ , θ + 1, A˙ ). and it follows that I wins G fixed (M, δ , θ + 1, B˙ ). The next Similarly, if M |= φII [δ , θ + 1, B˙ ] then II wins H claim therefore completes the proof of Theorem 2A.2 for θ + 1: Claim 2C.6. Suppose that M |= φI [δ , θ + 1, A˙ ] and M |= φII [δ , θ + 1, B˙ ]. Then there is a y ∈ Rθ ∩ M[g] and a z ∈ R ∩ M[g ] so that y−−, z ∈ M[g ] belongs to neither A˙ [g ] nor B˙ [g ]. ˙ and φII [δ, θ, B] ˙ fail in M&δ . Theorem 2A.2 Proof. By assumption both φI [δ, θ, A] ˙ ˙ applied in M&δ with θ and the names A and B produces y ∈ Rθ ∩ M[g] which belongs ˙ ˙ to neither A[g] nor B[g]. Using the definition of A˙ and B˙ it follows that II (the closed ∗ y ) and that I (the closed player again) wins H ∗ ( y ). Fix σ ∗ ∈ M[g] and player) wins G ( ∗ τ ∈ M[g] witnessing this. Let σgen ∈ M[g ] be the generic strategies map associated to A˙ , δ , and X = M&δ. Let τgen ∈ M[g ] be the mirrored generic strategies map associated to B˙ , δ , and X = M&δ. Working inside M[g ] combine σ ∗ , τ ∗ , σgen , and τgen to form: (1) z ∈ R; (2) a , a generic infinite run of A[ y −−, z]; and a generic infinite run of B[ (3) b, y −−, z]. The construction, which we omit, is similar to that of case 3 in the proof of Theorem 1E.1. From condition (2) it follows that y−−, z ∈ A˙ [g ]. From condition (3) it follows that y−−, z ∈ B˙ [g ]. # Claims 2C.2, 2C.5, and 2C.6 complete the proof of Theorem 2A.2 for θ +1, assuming that the theorem holds for θ. Remark 2C.7. Adapting the above arguments to the case that θ = 0 one can prove y ) and H ∗ ( y ) thus Theorem 2A.2 for θ + 1 = 1. Rθ = {∅} in the case θ = 0. G∗ ( ∗ ∗ ∗ ∗ degenerate into games G = G (∅) and H = H (∅). Claim 2C.2 should be revised fixed (M, δ , 1, A). ˙ Claim 2C.5 for the case θ = 0 to say that if I wins G∗ then I wins G ˙ is the formula which expresses the statement should be revised similarly. φI (δ , 1, A) ˙ is defined similarly. The proofs of the revised claims for “I wins G∗ .” φII (δ , 1, B) θ = 0 are degenerates of the proofs given above. We leave the exact details to the reader. Remark 2C.7, the work in this section on the successor case, and the work in Section 2B on the limit case complete the proof of Theorem 2A.2. ˙ denote the formula Some words are due on our methodology. Let φIθ (δ, A) θ ˙ φ(δ, θ, A). Define φII similarly. The proof of Theorem 2A.2 for θ used inductive ¯ ¯ knowledge of φIθ and φIIθ , for θ¯ < θ . (This was true for both limit θ and successor θ .) The proof then provided φIθ and φIIθ based on that knowledge. ¯ The passage from $φIθ | θ¯ < θ %—or more precisely from the sequence of Gödel numbers of these formulae—to the formula φIθ was uniform and definable over any
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model of ZFC∗ . This can be seen easily by following the constructions in Sections 2B ˙ which expresses the statement and 2C. We can thus obtain a single formula φI (δ, θ, A) θ ˙ φI (δ, A). Similar reasoning allows us to come up with φII .
2D Limits again We have so far seen how to prove Theorem 2A.2 (Sections 2B and 2C), and how to use Theorem 2A.2 to prove the determinacy of games of countable length ω · (θ + 1) (Theorem 2A.3). Here we handle games of length ω · θ for limit θ < ω1 . We prove: Theorem 2D.1. Let θ be a countable limit ordinal. Suppose that there exists a model M and a cardinal λ in M so that: • M is a class model of ZFC; • M is weakly iterable; • M has θ Woodin cardinals below λ; and • M&λ is countable in V. Then the games Gω·θ (C) are determined for all C ⊂ Rθ in the pointclass <ω2 − 11 . The assumption in Theorem 2D.1 follows from the existence in V of a sharp for θ Woodin cardinals, by the methods at the end of Appendix A. The proof of Theorem 2D.1 takes the rest of this section. We plan to break θ into ω blocks, use Theorem 2A.2 to progress through each of the blocks, and intersperse between the blocks moves in the game of Martin [24] for witnessing membership in a <ω2 − 11 set. Let {δξ }ξ <θ enumerate the Woodin cardinals of M below λ, in increasing order. Without loss of generality we may assume that θ is seen to be countable in M. Fix in M a strictly increasing sequence of ordinals ηk , k < ω, starting with η0 = 0 and cofinal in θ. Let θk = ηk+1 − ηk , or more precisely let θk be the unique ordinal such that ηk+1 = ηk + θk . Let δk be δη k+1 . The following list encapsulates the properties of {δk }k<ω , in M, which we intend to use: • each δk is a Woodin cardinal of M; • there are −1 + θk Woodin cardinals of M below δk and (if k > 0) above δk−1 ; and • for each k, M&δk + 1 is countable in V. These properties allow us to use Theorem 2A.2 with δk , θk , and models M[g k ] where g k is generic for the collapse of δ0 , . . . , δk−1 . (Note that there are still −1 + θk Woodin
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cardinals below δk , left after the collapse.) We shall make such applications of Theorem 2A.2 without further comment. Using Remark 1E.5 every real can be absorbed into a generic extension of an iterate of M, via a forcing which only collapses the first Woodin cardinal of the iterate, and therefore leaves plenty of measurable cardinals in the extension. It follows from this that every real has a sharp. Since M&λ is countable in V, M&λ has a sharp. Replacing M by L(M&λ) if needed let us assume that: • M = L(M&λ). The indiscernibles given by (M&λ) are then Silver indiscernibles for M. Fix C ⊂ Rθ in the pointclass <ω2 − 11 . There is then some α < ω2 and some sequence $Cξ | ξ < α% of 11 sets so that y ∈ C ⇐⇒ the least ξ such that ξ = α or y ∈ Cξ is of the same parity as α. When we say “of the same parity as α” we mean odd if α is odd, and even if α is even. Theorem 2D.1 in the case of α = 0 is trivial. We may therefore assume that α > 0. Increasing α if needed, let us also assume that α is odd. The reader who prefers not to make this assumption should change “odd” to “of parity the same as α” and “even” to “of parity opposite to α” in rules (1)–(3) below. Let m < ω be such that α < ω2 belongs to (ω · m, ω · m + ω]. Let u0 < · · · < um be the first m + 1 uniform Silver indiscernibles. y ] so that Rξ [ y] For each ξ < α there is some Lipschitz continuous map y → Rξ [ is a linear order of ω and y ∈ Cξ iff Rξ [ y ] is wellfounded. Without loss of generality y ], for all ξ and all y. Lipschitz we may assume that 0 is the largest element in Rξ [ continuity above is meant with reference to some coding of elements of Rθ as reals. The precise manner in which this is done is irrelevant, so long as it’s done in M and so y ]k + 1. long as knowledge of yηk suffices to determine Rξ [ Let C[ y ] be the following game, taken from Martin [24]: I and II alternate playing y ] into the ordinals. ordinals, embedding the relations Rξ [ (1) I is responsible for Rξ [ y ] for all even ξ < α, and II is responsible for Rξ [ y ] for all odd ξ < α. y ] must be embedded into ordinals larger than those used for Rξ [ y ]. (This (2) Rξ +1 [ is a rule on I if ξ + 1 is even, and on II if ξ + 1 is odd.) y ] for ξ ∈ [ω · n, ω · n + ω) must be embedded into ordinals in [un−1 , un ). (3) Rξ [ (This is a rule on I if ξ is even, and on II if ξ is odd. In the case n = 0 we mean ordinals below u0 .) We think of a run of C[ y ] as a sequence of ordinals $ρi | i < ω%. I plays ρ2i and II plays ρ2i+1 . ω is divided into α disjoint sets rξ , and {ρi }i<ω rξ is the embedding of y ] into the ordinals. The division is arranged in accordance with rule (1). Rξ [ The division is also arranged so that min(rξ +1 ) > min(rξ ) for all ξ . In other words ordinals corresponding to Rξ +1 [ y ] are only played after the first ordinal corresponding
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y ]. This particular arrangement is needed to make sense of rule (2). Note that to Rξ [ the first ordinal corresponding to Rξ [ y ] is the largest among all ordinals corresponding y ], since 0 is the largest element in Rξ [ y ]. to Rξ [ Observe how rules (2) and (3) imply that Rζ [ y ] is embedded into ordinals below y ] whenever ζ < ξ . those used for Rξ [ We refer the reader to Martin [24] for an exact description of the rules. An auxiliary game of this kind was used by Martin to prove the determinacy of length ω games with <ω2 − 11 payoff. Observe that y → C[ y ] is Lipschitz continuous. We refer to this map as C, and note that knowledge of yηk suffices to determine at least the first k + 1 rounds of C[ y ]. The parameters used in the definition of C are the sequences $Rξ | ξ < α% and $un | n ≤ m%. Both sequences belong to M; the first because C is lightface <ω2 − 11 , and the second because it is finite. The Lipschitz continuous map C therefore belongs to M. We shall use it as we used A and B in Section 2B. y ] is good (over M) if the ordinals played by A position Q = $ρ0 , . . . , ρl−1 % in C[ I, namely ρj for j < l even, are Silver indiscernibles for M; and the ordinals played by II, namely ρi for i < l odd, are definable in M from the ordinals played by I and additional parameters in M&λ ∪ {u0 , . . . , um }. % is called an M-shift of the good position Q if A good position Q = $ρ0 , . . . , ρl−1 for each odd i < l, ρi is definable from M&λ ∪ {u0 , . . . , um } and {ρj | j < l even} in the same way that ρi is definable from M&λ ∪ {u0 , . . . , um } and {ρj | j < l even}. The following facts, and their mirror images listed in Section 2D (2), distill the properties of C[ y ] which we shall need: Fact 2D.2. Suppose that Q is a good position of even length in C[ y ]. (It is I’s turn to play following Q.) Then there is a shift Q of Q, and a Silver indiscernible ρ for M, so that ρ is a legal move for I in C[ y ] following Q . y ]. Suppose Fact 2D.3. Suppose that $Qk | k < ω% is a sequence of good positions in C[ that for each k < ω, Qk+1 extends a shift of Qk . Suppose further that there is no (single) infinite run Q∞ which extends a shift of each Qk , k < ω. Then y ∈ C. The first fact says that it is always possible to ascribe indiscernibles as moves for I; by shifting previous moves it’s possible to make room for the next one. The second fact says that if II can keep up with these indiscernible moves for I, then y does not belong to C. The reader who does not know how to prove the second fact should consult Martin [24]. We comment only that the proof has to do with the specific order imposed by rules (2) and (3). The assumption in Fact 2D.3 that there is no Q∞ which extends a shift of each Qk is needed to make sure that the least ξ so that ξ = α or y ∈ Cξ is not equal to α. 2D (1) II wins. For expository simplicity fix a sequence $gk | k < ω% so that each gk is col(ω, δk )-generic/M[g0 ∗ · · · ∗ gk−1 ]. Let g k denote g0 ∗ · · · ∗ gk−1 . We work in
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the forcing extensions M[g k ] and M[g k ][gk ] though strictly speaking we should work in M and use the forcing languages. We shall follow roughly the outline of definitions in Section 2B, only using C instead of A and B. It is convenient here to start with the case that II wins. So we begin by imitating the definitions in Section 2B(3). Definitions 2B.11 and 2B.12 can be copied verbatim, simply changing B to C. We make only a notational comment: a position Q of k rounds in C[ x ] now consists of ordinals $ρ0 , ρ1 , . . . , ρ2k−1 %. Having defined mirrored k-sequences and mirrored extended k-sequences, let us x , Q∗ , γ ∗ ) and the games Hk ( x , Q, γ ). proceed to the definition of the names B˙ k [g k ]( The definition, as usual, is by induction on γ and γ ∗ . Definition 2B.14, the definition x , Q∗ , γ ∗ ), can be copied verbatim. The definition of Hk ( x , Q, γ ) is modified of Bk ( as follows, to use C[ x ] rather than the auxiliary games map of Section 2B. x , Q, γ ) is played according to Diagram 2.6 and the Definition 2D.4. The game Hk ( following rules: (1) (Rule for II) γ ∗ must be an ordinal smaller than γ . x ] following the (2) (Rule for I and II) ρ2k and ρ2k+1 must be legal moves in C[ position Q. I II
γ∗
ρ2k ρ2k+1
Diagram 2.6. The game Hk ( x , Q, γ ).
Once the game is concluded, let Q∗ = Q−−, ρ2k , ρ2k+1 . Player II wins just in case x , Q∗ , γ ∗ )]. that (M&δk+1 )[g k ] |= φII [δk , θk , B˙ k [g k ]( Recall that the map C is Lipschitz continuous, and knowledge of x ∈ Rηk suffices to determine the first k + 1 rounds of C[ y ] (for any y ∈ Rθ extending x). Rule (2) in Definition 2D.4 therefore makes sense. Claim 2D.5. Suppose that there exists an ordinal γ ∈ M so that II wins H0 (∅, ∅, γ ). Then II wins Gω·θ (C). Proof. Fix an imaginary opponent playing for I in Gω·θ (C). Fix a weak iteration strategy for M. We describe how to play against the imaginary opponent, using , fixed (. . . θk . . .), and indiscernible moves in shifts of C. strategies in various games H As usual the description takes the form of a construction of (among other things) a run y = $yξ | ξ < θ% of Gω·θ (C). We construct by induction on k < ω. At the start of stage k ∈ [0, ω) we have: (A) The reals yξ for ξ < ηk . We use xk to denote $yξ | ξ < ηk %. (B) A -iterate Nk of M = N0 , and an iteration embedding ik : M → Nk .
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(C) Some hk = h0 ∗ · · · ∗ hk−1 which is col(ω, i1 (δ0 )) ∗ · · · ∗ col(ω, ik (δk−1 ))generic/Nk . xk ]. (D) A position Qk of k rounds in ik (C)[ (E) An ordinal γk in Nk . The iteration map ik will be the result of a weak iteration of M, of countable length, using length ω iteration trees, with extenders taken from images of M&δk which is countable in V. It is easy to see that iteration maps of this kind (when their final model is wellfounded) do not move the uniform Silver indiscernibles. Our notation henceforth will use this fact without further comment: we write u0 rather than ik (u0 ), etc. During the construction we shall make sure that: (i) xk belongs to Nk [hk ]. xk , Qk , γk ).” (ii) Nk [hk ] |=“II wins ik (H˙ k )[hk ]( xk ], where good is meant over Nk . (iii) Qk is a good position in ik (C)[ (iv) γk is definable in Nk from Qk and additional parameters in Nk &ik (λ) ∪ {u0 , . . . , um }. We informally use tk (Qk ) to refer to the “definition” of γk given by condition (iv). If Q is a shift of Qk we write tk (Q ) for the result of interpreting tk using Q instead of Qk . The objects needed for the start of stage 0 are for the most part given trivially. The assumption of Claim 2D.5 states that there is some γ ∈ M so that II wins H0 (∅, ∅, γ ). Let γ0 be the least such. H0 is definable in M from {u0 , . . . , um } and a real parameter. (Note that these objects suffice to define C.) The least γ so that II wins H0 (∅, ∅, γ ) is thus definable in N0 = M using parameters allowed in condition (iv). We note before we begin that the sequences $H˙ k | k < ω% and $B˙ k | k < ω% are definable in M from {u0 , . . . , um } and a real parameter. (The real parameter codes, among other things, the sequence {θk }k<ω .) Let us start stage k of the construction. Let Qk be the shift of Qk given by Fact 2D.2, and let ρ be the Silver indiscernible given by that fact. Let γk = tk (Qk ). Since Qk is a shift of Qk , there is an elementary embedding τ : Nk [hk ] → Nk [hk ] which fixes the uniform indiscernibles and sends Qk to Qk . By necessity τ sends γk to γk . H˙ k , being definable from uniform indiscernibles and a real, is not moved by τ . Applying τ to xk , Qk , γk ). Let σk ∈ Nk [hk ] condition (ii) it follows that II wins the game ik (H˙ k )[hk ]( k witness this. We may assume that σk is definable in Nk [h ] from {u0 , . . . , um }, Qk , and parameters from (Nk &ik (λ))[hk ]. (Note that xk belongs to (Nk &ik (λ))[hk ], and that γk xk , Qk , γk ) is definable from the parameters listed. It follows that the game ik (H˙ k )[hk ]( is definable from the parameters listed.) Let γk+1 be the first move played by σk . Play the Silver indiscernible ρ as the next move, ρ2k , for I. Let ρ2k+1 be the response played by σk . ¯ k+1 = Q −−, ρ2k , ρ2k+1 . Since σk is a winning strategy for player II we get: Let Q k
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¯ k+1 , γ¯k+1 )], where δ s = ik (δk ) (v) (Nk &ik (δk+1 ))[hk ] |= φII [δks , θk , B˙ ks [hk ]( xk , Q k and B˙ ks = ik (B˙ k ). By our choice of a definable σk , both γ¯k+1 and ρ2k+1 are definable in Nk from parameters in Nk &ik (λ) ∪ {u0 , . . . , um }, Qk , and ρ. (Initially we get definability in Nk [hk ] using xk as a parameter. But we can pass to Nk and use forcing conditions, which belong to Nk &ik (λ), as parameters.) Thus: ¯ k+1 is a good position in ik (C)[ xk ]. (vi) Q ¯ k+1 . (vii) γ¯k+1 is definable in Nk using parameters in Nk &ik (λ) ∪ {u0 , . . . , um } and Q For the record let us also note that: ¯ k+1 2k). (viii) γ¯k+1 < tk (Q ¯ k+1 2k = Q .) This follows from rule (1) in Definition 2D.4. (Note that Q k Applying Theorem 2A.2 to condition (v) we see that player II wins the game fixed (N˜ k [hk ], δ s , θk , B˙ s [hk ]( ¯ k+1 , γ¯k+1 )), where N˜ k = (Nk &ik (δk+1 )). Notice H xk , Q k k fixed (Nk [hk ], . . . . . . ), since N˜ k is a rank that this game is precisely the same as H initial segment of Nk taken above all relevant objects. So player II wins the game fixed (Nk [hk ], δ s , θk , B˙ s [hk ]( k witnessing this. ¯ k+1 , γ¯k+1 )). Fix H xk , Q k k k , the imaginary opponent, and combine to produce a complete run of fixed (Nk [hk ], δ s , θk , B˙ s [hk ]( ¯ k+1 , γ¯k+1 )), including reals $yξ | ξ ∈ [ηk , ηk+1 )% xk , Q H k k and a weak iteration of Nk of length −1+θk +1. Let Nk+1 be the final model of this weak iteration and let ik,k+1 : Nk → Nk+1 be the iteration embedding. Let ik+1 = ik,k+1 ◦ ik . fixed (Nk [hk ], . . . . . . ) involve iteration trees Remark 2D.6. Strictly speaking moves in H on Nk [hk ]. But we can view them as trees on Nk (see Appendix A). The use of the iteration strategy guarantees that Nk+1 is wellfounded. Let zk = k is winning for II, there exists some hk so that: $yξ | ξ ∈ [ηk , ηk+1 )%. Since (1) hk is col(ω, ik,k+1 (δks ))-generic/Nk+1 [hk ]; and ¯ k+1 ), ik,k+1 (γ¯k+1 )). xk , ik,k+1 (Q (2) zk ∈ ik,k+1 (B˙ ks )[hk ][hk ]( ¯ k+1 ), and let γk+1 = ik,k+1 (γ¯k+1 ). Let xk+1 = xk −− zk , let Qk+1 = ik,k+1 (Q Conditions (i)–(iv) for k + 1 are now easy to verify. We only point out that condition (ii) for k + 1 follows from condition (2) and that condition (iv) for k + 1 follows from condition (vii) above. This completes the construction in stage k, putting us in a position to start stage k + 1. Inductively the construction is now complete. For future record let us note that: (ix) Qk+1 extends a shift of ik,k+1 (Qk ). (x) tk+1 (Qk+1 ) < ik,k+1 (tk )(Qk+1 2k), where both tk+1 and ik,k+1 (tk ) here are interpreted inside Nk+1 .
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I II
γ∗
79
ρ2k ρ2k+1
Diagram 2.7. The game Gk ( x , P , γ ).
The last condition follows from condition (viii), and ultimately traces back to rule (1) in Definition 2D.4. It remains to verify that $yξ | ξ < θ% is won by player II in Gω·θ (C). Let N∞ be the direct limit of the models Nk . Let ik,∞ : Nk → N∞ be the direct limit embeddings. Let i∞ denote i0,∞ . The use of the iteration strategy during the xk ] construction guarantees that N∞ is wellfounded. Let Q∗k = ik,∞ (Qk ). Since i∞ (C)[ consists precisely of the first k +1 rounds of i∞ (C)[ y ], Q∗k is a position in i∞ (C)[ y ]. By condition (iii) each Q∗k is good, where this is meant over N∞ . Moreover by condition (ix), Q∗k+1 extends a shift of Q∗k for each k. The following claim will allow us to apply Fact 2D.3: Claim 2D.7. There is no Q∗∞ which extends a shift of each Q∗k . Proof. Suppose otherwise. Let γk∗ = ik,∞ (tk )(Q∗∞ 2k), where the definition ik,∞ (tk ) ∗ < γk∗ for each k. But this is interpreted in N∞ . Using condition (x) we get γk+1 # contradicts the fact that N∞ is wellfounded. Applying Fact 2D.3 we finally conclude that y ∈ C. Playing for II against some imaginary opponent we produced a run y = $yξ | ξ < θ% of Gω·θ (C). We then went on to verify that y ∈ C so that this run is won by II. This completes the proof of Claim 2D.5. # 2D (2) I wins. We work here to mirror the argument of Section 2D (1). The process of mirroring is similar to that in Section 2B, except that we keep using the same map C for the mirrored argument. Let us start with the basic definitions. The definitions of k-sequences and extended k-sequences are verbatim copies of Definitions 2B.2 and 2B.3 in Section 2B (1), only changing A to C. Definition 2B.7 can also be copied verbatim. Definition 2B.4 should be modified to the following: x , P , γ ) is played according to Diagram 2.7 and the Definition 2D.8. The game Gk ( following rules: (1) (Rule for II) γ ∗ must be an ordinal smaller than γ . x ] following the (2) (Rule for I and II) ρ2k and ρ2k+1 must be legal moves in C[ position P . Once the game is concluded let P ∗ = P −−, ρ2k , ρ2k+1 . Player I wins just in case that (M&δk+1 )[g k ] |= φI [δk , θk , A˙ k [g k ]( x , P ∗ , γ ∗ )].
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Note that γ ∗ is played by II, not by I. This is the one essential difference between the definition here and Definition 2B.4. Let $γL , γH % be the least pair of local indiscernibles of M relative to um . Note that γL is definable in M from um . Claim 2D.9. Suppose that I wins G0 (∅, ∅, γL ). Then I wins Gω·θ (C). Proof sketch. For the purpose of this argument, a position P in C[ y ] is called good if the ordinals played by II are indiscernibles for M, and the ordinals played by I are definable from the ordinals played by II. Note how this mirrors the earlier definition of a good position Q. Mirror the definition of a shift similarly. The following facts are adapted from Martin [24]: Fact 2D.10. Suppose that P is a good position of odd length in C[ y ]. (It is II’s turn to play following P .) Then there is a shift P of P , and a Silver indiscernible ρ for M, so that ρ is a legal move for II in C[ y ] following P . y ]. Fact 2D.11. Suppose that $Pk | k < ω% is a sequence of good positions in C[ Suppose that for each k < ω, Pk+1 extends a shift of Pk . Then y belongs to C. Fact 2D.10 states that it is always possible to ascribe indiscernibles as moves for II. Fact 2D.11 states that if I can keep up with these indiscernible moves, then y ∈ C. These facts mirror the earlier facts, 2D.2 and 2D.3. But notice that the condition of Fact 2D.3 that $Qk | k < ω% cannot be combined to form an infinite run Q∞ is not mirrored into the current Fact 2D.11. That condition was needed earlier to make sure that the least ξ so that ξ = α ∨ y ∈ Cξ is not equal to α. Here there is no need for such an assumption; if the least ξ so that ξ = α ∨ y ∈ Cξ is equal to α, then y belongs to C. The proof of the current Claim 2D.9 mirrors the earlier proof of Claim 2D.5, using Facts 2D.10 and 2D.11 instead of Facts 2D.2 and 2D.3. We omit the details and only make the following observations: The strategy σk (obtained during the proof) is now a strategy for I. Since γ ∗ is still played by II it is our responsibility during the construction to come up with γ¯k+1 ; it is not played for us by σk . We play γ¯k+1 ourselves using the usual trick of switching between local indiscernibles, starting with γL . Conditions (viii) and (x) must therefore be relinquished; as we switch between local indiscernibles we certainly do not obtain an infinite decreasing sequence. As a result we cannot mirror Claim 2D.7. But we don’t need to. Fact 2D.11—unlike the earlier Fact 2D.3—can be applied knowing only that for each ∗ extends a shift of P ∗ . # k < ω, Pk+1 k 2D (3) Determinacy. Remember that our goal is to prove the determinacy of Gω·θ (C). Given Claims 2D.5 and 2D.9 it is enough to prove: Claim 2D.12. Either there exists some γ so that II wins H0 (∅, ∅, γ ), or else I wins G0 (∅, ∅, γL ).
2D Limits again
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Proof. Assume for contradiction that I does not win G0 (∅, ∅, γL ), and that for all γ , II does not win H0 (∅, ∅, γ ). In particular II does not win H0 (∅, ∅, γL ). Recall that for each k < ω, gk is col(ω, δk )-generic/M[g k ] where g k = g0 ∗ · · · ∗ gk−1 . We work by induction on k < ω to construct: (A) xk ∈ Rηk ∩ M[g k ]; xk ]; and (B) Pk , a position of k rounds in C[ (C) an ordinal γk . We shall make sure that xk+1 extends xk ; that Pk+1 extends Pk ; and most importantly that γk+1 is smaller than γk . This last clause will yield the desired contradiction. As we construct we shall maintain: xk , Pk , γk ); and (i) I does not win Gk ( xk , Pk , γk ). (ii) II does not win Hk ( We start with x0 = ∅, P0 = ∅, and γ0 = γL . Conditions (i) and (ii) hold by the initial assumption for contradiction. Suppose xk , Pk , and γk have been constructed. Suppose that conditions (i) and (ii) xk , Qk , γk ) is a finite game, condition (i) implies that it is won by hold at k. Since Gk ( II. Fix a strategy τk witnessing this. Using condition (ii) similarly, fix a strategy σk , xk , Pk , γk ). winning for I in Hk ( xk , Pk , γk ) and Let σk and τk play against each other. (Note that the games Hk ( xk , Pk , γk ) have identical moves, see Diagrams 2.6 and 2.7.) This produces γk+1 , Gk ( ρ2k , and ρ2k+1 . The rules of Gk ( xk , Pk , γk ) are such that: (iii) γk+1 < γk . xk , Pk , γk ), we get: Let Pk+1 = Pk −−, ρ2k , ρ2k+1 . Since τk is winning for II in Gk ( xk , Pk+1 , γk+1 )]. • (M&δk+1 )[g k ] |= φI [δk , θk , A˙ k [g k ]( xk , Pk , γk ), we get: Since σk is winning for I in Hk ( xk , Pk+1 , γk+1 )]. • (M&δk+1 )[g k ] |= φII [δk , θk , B˙ k [g k ]( We may therefore apply Theorem 2A.2 and obtain zk ∈ Rθk ∩ M[g k ][gk ] which belongs xk , Pk+1 , γk+1 ) nor B˙ k [g k ][gk ]( xk , Pk+1 , γk+1 ). Using the defto neither A˙ k [g k ][gk ]( initions of these two sets (see Definitions 2B.7 and 2B.14) it follows that I does not xk −− zk , Pk+1 , γk+1 ) and that II does not win Hk+1 ( xk −− zk , Pk+1 , γk+1 ). win Gk+1 ( Setting xk+1 = xk −− zk completes the construction at stage k, securing conditions (i) and (ii) for k + 1. Inductively the construction is now complete. Condition (iii) implies that {γk }k<ω is an infinite descending sequence of ordinals, a contradiction. # Claims 2D.5, 2D.9, and 2D.12 together imply that Gω·θ (C) is determined. This completes the proof of Theorem 2D.1.
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2E Universally Baire sets By a tree on a set Y we mean a subset of Y <ω , closed under initial segments. An infinite branch through T is a sequence y ∈ Y ω so that yn ∈ T for all n < ω. [T ] denotes the set of infinite branches through T . We make the natural identification between U ω ×W ω and (U ×W )ω : Given u ∈ U ω and w ∈ W ω we use $u, w% to denote both the actual pair $u, w% ∈ U ω × W ω , and the sequence $$u(i), w(i)% | i < ω% ∈ (U × W )ω . We make a similar identification for products of more than two spaces. Let T be a tree on X × U . Define p[T ] ⊂ X ω , the projection of T to X ω , by setting x ∈ p[T ] iff there exists some u ∈ U ω so that $x, u% ∈ [T ]. Notice that p[T ] is not absolute between models of set theory. The following exercises touch on this issue of absoluteness. Exercise 2E.1. Let M and M[g] be (wellfounded) models of ZFC∗ . Suppose that M ⊂ M[g]. Let S and S ∗ be trees in M, on ω × U and ω × U ∗ respectively. Suppose that M |=“p[S] ∩ p[S ∗ ] = ∅.” Prove that p[S] ∩ p[S ∗ ] = ∅ also in M[g]. Hint. Look at the tree of attempts to construct $x, u, u∗ % so that $x, u% ∈ [S] and # $x, u∗ % ∈ [S ∗ ], and use the absoluteness of being wellfounded. Exercise 2E.2. Give an example of a generic extension V[g] of V, and trees T and T ∗ in V, on ω × ω1 and ω × ω respectively, so that “p[T ] ∪ p[T ∗ ] = ωω ” holds in V but not in V[g]. Let X be countable, so that Xω is a Polish space. A set C ⊂ Xω is λ-universally Baire if all its continuous preimages, to topological spaces with regular open bases of cardinality ≤ λ, have the property of Baire. C is ∞-universally Baire if it is λuniversally Baire for all cardinals λ. Feng–Magidor–Woodin [5] provides the following convenient characterization of universally Baire sets: Definition 2E.3. A pair of trees T and T ∗ on X × U and X × U ∗ respectively is exhaustive for a poset P if the statement “p[T ] ∪ p[T ∗ ] = X ω ” is forced to hold in all generic extensions of V by P. Theorem 2E.4 (Feng–Magidor–Woodin [5]). Let X be countable, let C ⊂ Xω , and let λ be an infinite cardinal. C is λ-universally Baire iff there are trees T and T ∗ so that: (1) p[T ] = C and p[T ∗ ] = X ω − C; and (2) the pair $T , T ∗ % is exhaustive for all posets of size ≤ λ.
#
We work with this characterization, rather than the definition. Our aim is first to prove determinacy for games with λ-universally Baire payoff for sufficiently large λ, and second to propagate the property of being universally Baire along the hierarchy of definability. We work below with substructures of initial segments of the universe. By “rank initial segment of V” we always mean a rank initial segment which satisfies enough of ZFC∗ for the relevant application.
2E Universally Baire sets
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Exercise 2E.5. Let C ⊂ ωω be λ-universally Baire, and let T , T ∗ witness this. Let Vθ be a rank initial segment of V, large enough that T , T ∗ , and λ belong to Vθ . Let N be a countable model which embeds elementarily into Vθ , say by σ . Suppose T , T ∗ , and λ belong to the range of σ . Let T¯ , T¯ ∗ , and λ¯ be such that σ (T¯ ) = T , σ (T¯ ∗ ) = T ∗ , and ¯ = λ. σ (λ) ¯ Let h ∈ V be col(ω, λ)-generic over N . Let x belong to ωω ∩ N[h]. Prove that N[h] |=“x ∈ p[T¯ ]” if and only if x ∈ C. The next exercise obtains determinacy for universally Baire sets. It reduces the large cardinal assumption in Feng–Magidor–Woodin [5, Theorem 5.4], where determinacy is obtained using two Woodin cardinals. The exercise involves a modification of the proof of Theorem 1E.1. Combined with Fact 2E.7 below it basically leads back to the determinacy proved in Section 1E. Exercise 2E.6. Let λ be a Woodin cardinal (in V). Let C ⊂ ωω be λ-universally Baire. Prove that Gω (C) is determined. Hint. Let T and T ∗ witness that C is λ-universally Baire. Let Vθ be a rank initial segment of V, large enough that T , T ∗ , and λ belong to Vθ . Let H be a countable elementary substructure of Vθ , with T , T ∗ , λ ∈ H . Let M be the transitive collapse of H and let π : M → H be the anti-collapse embedding. Let S, S ∗ , and δ be such that π(S) = T , π(S ∗ ) = T ∗ , and π(δ) = λ. Let g be col(ω, δ)-generic over M. Let A˙ ∈ M be the canonical name for the set (M&δ)ω × p[S] as computed in M[g]. Let B˙ ∈ M be the canonical name for (M&δ)ω × p[S ∗ ]. Imitate the proof of Theorem 1E.1 using these names. In case 3 use the elementarity of π to see that $S, S ∗ % is exhaustive for col(ω, δ) over M. In case 1 pick the branch b using Theorem 12 in Appendix A, so that you get an embedding σ : Mb → Vθ with σ ◦ jb = π . (A branch b for which such an embedding σ exists is called realizable.) # Then replace the use of 12 absoluteness with a use of Exercise 2E.5. We pass now to the matter of propagating the property of being universally Baire. The starting point is the following fact: Fact 2E.7 (Feng–Magidor–Woodin [5, Theorem 3.4]). Let X be a countable set, let λ be an infinite cardinal, and let C ⊂ X ω be 12 . Suppose that (Vλ+1 ) exists. Then C is λ-universally Baire. Given C ⊂ X ω × ωω and x ∈ X ω define Cx to be the set {y ∈ ωω | $x, y% ∈ C}. Define the length ω game quantifier ω , acting on subsets of Xω × ωω , by setting x ∈ ω C iff player I has a winning strategy in the game Gω (Cx ). Suppose now that C ⊂ Xω × ωω is λ-universally Baire, and let T and T ∗ witness this. For x ∈ Xω let Tx and Tx∗ be the trees {$t, u% | $x lh(t), t, u% ∈ T } and {$t, u∗ % | $x lh(t), t, u∗ % ∈ T ∗ } respectively. It is easy to check that Tx and Tx∗ witness that Cx is λ-universally Baire. In particular the pair $Tx , Tx∗ % is exhaustive for col(ω, λ). This last fact is true not only for x ∈ V, but also for x in small generic extensions of V.
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The following exercise makes this statement (or rather the equivalent statement over a countable substructure) precise. Exercise 2E.8. Let Vθ be a rank initial segment of V, large enough that T , T ∗ , and λ belong to Vθ . Let H be a countable elementary substructure of Vθ , with T , T ∗ , λ ∈ H . Let P be the transitive collapse of H and let π : P → H be the anti-collapse embedding. Let S, S ∗ , and δ be such that π(S) = T , π(S ∗ ) = T ∗ , and π(δ) = λ. Suppose for simplicity that X ∈ H and π(X) = X. (This is certainly true if X = ω.) Let M = P [h] be a generic extension of P by a poset of size strictly less than δ. Let x belong to X ω ∩ M. Prove that the pair $Sx , Sx∗ % is exhaustive for col(ω, δ) over M. We now have everything set up for propagating the property of being universally Baire. The next exercise phrases the propagation precisely. Exercise 2E.9. Let λ be a Woodin cardinal (in V). Let X be a countable set. Let C ⊂ X ω × ωω be λ-universally Baire. Prove that ω C is κ-universally Baire for all κ < λ. Hint. Since X is countable you may assume that X = ω. Let T and T ∗ witness that C is λ-universally Baire. Let Vθ be a rank initial segment of V, large enough that T , T ∗ , and λ belong to Vθ . Let Z be the set of tuples $P , π, S, S ∗ , δ, h% so that P is a countable transitive model, π is an elementary embedding of P into Vθ , π(S) = T , π(S ∗ ) = T ∗ , π(δ) = λ, and h is generic over P for a poset of size less than δ. Let $P , π, S, S ∗ , δ, h% belong to Z. Let M = P [h]. Notice that δ is a Woodin cardinal in M, since M is a generic extension of P by a poset of size less than δ. Given x ∈ ωω ∩ M let A˙ x be the canonical name, in the forcing col(ω, δ) over M, for the set (M&δ)ω × p[Sx ]. Let G∗ be defined as in the proof of Theorem 1E.1, using the name A˙ x over M. Let φ(x, S, δ) be the statement that player I wins this game G∗ . (This is a statement made over M, using x, S, and δ as parameters.) Show that if there is a tuple $P , π, S, S ∗ , δ, h% ∈ Z so that P [h] |= φ[x, S, δ], then x belongs to ω C. You will essentially imitate your work on case 1 in Exercise 2E.6. Show that if there is a tuple $P , π, S, S ∗ , δ, h% ∈ Z so that P [h] |= ¬φ[x, S, δ], then x belongs to ωω − ω C. You will essentially imitate your work on cases 2 and 3 in Exercise 2E.6. You can use Exercise 2E.8 to exclude case 3. The tuples in Z can be coded by countable sequences. Using this fact, define trees R and R ∗ so that: (1) x ∈ p[R] iff there is $P , π, S, S ∗ , δ, h% ∈ Z so that P [h] |= φ[x, S, δ]; and (2) x ∈ p[R ∗ ] iff there is $P , π, S, S ∗ , δ, h% ∈ Z so that P [h] |= ¬φ[x, S, δ]. Your work above shows that p[R] ⊂ ω C and p[R ∗ ] ⊂ ωω − ω C. It remains to show that the pair $R, R ∗ % is exhaustive over V for all posets of size strictly less than λ. Here you will at last use the way Z allows not just countable substructures of Vθ , but also their small generic extensions. Your precise definition of the trees R and R ∗ is also important here. The argument works with the natural definition of these trees, but there are plenty of unnatural definitions with which the argument would fail. #
2E Universally Baire sets
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It is easy to check that {ω C | C ∈ 1n } is precisely the pointclass 1n+1 . The following is therefore immediate from Fact 2E.7 and repeated applications of Exercise 2E.9: Corollary 2E.10. Let n > 0. Let κ be an infinite cardinal. Suppose that there are n Woodin cardinals λ1 < λ2 · · · < λn above κ, and suppose that (Vλn +1 ) exists. Then all 1n+2 sets are κ-universally Baire. # Let us now pass to games of arbitrary fixed countable length. Exercise 2E.11. Let α ≥ 1 be a countable ordinal. Let λ be a Woodin cardinal and suppose that there are −1+α Woodin cardinals below λ. (Altogether then V is assumed to have −1 + α + 1 Woodin cardinals.) Let C ⊂ Rα be λ-universally Baire. Prove that Gω·α (C) is determined. Hint. Let T and T ∗ witness that C is λ-universally Baire. Let Vθ be a rank initial segment of V large enough that T , T ∗ , λ ∈ Vθ . Let H be a countable elementary substructure of Vθ , with T , T ∗ , λ ∈ H . Let M be the transitive collapse of H and let π : M → H be the anti-collapse embedding. Let S, S ∗ , and δ∞ be such that π(S) = T , π(S ∗ ) = T ∗ , and π(δ∞ ) = λ. Let g be col(ω, δ∞ )-generic over M. Let A˙ ∈ M be the canonical name for p[S] as computed in M[g]. Let B˙ ∈ M be the canonical name for p[S ∗ ]. Imitate the proof of Theorem 2A.3 using these names. As in Exercise 2E.6 you should use Theorem 12 in Appendix A to choose branches through the iteration trees that come up during the construction, so that all your branches are realizable. In case 1 you will then reach an iteration embedding j1,α+1 : M → Mα+1 , and an embedding σ : Mα+1 → Vθ so that σ ◦ j1,α+1 = π. As in Exercise 2E.6, use Exercise 2E.5 to argue that the run you constructed belongs to C. # Let α ≥ 1 be countable. Given C ⊂ X ω × Rα and x ∈ X ω define Cx to be the set { y ∈ Rα | $x, y% ∈ C}. Define the length ω · α game quantifier ω·α , acting on subsets of X ω × Rα , by setting x ∈ ω·α C iff player I has a winning strategy in the game Gω·α (Cx ). Exercise 2E.12. Let α ≥ 1 be a countable ordinal. Let λ be a Woodin cardinal. Let κ < λ, and suppose that there are −1+α Woodin cardinals between κ and λ. (Altogether then V is assumed to have −1 + α + 1 Woodin cardinals above κ.) Let X be countable. Let C ⊂ Xω × Rα be λ-universally Baire. Prove that ω·α C is κ-universally Baire. Hint. Imitate your solution to Exercise 2E.9, using the formula φI of Theorem 2A.2 and appealing to your determinacy proof in the last exercise. # Remark 2E.13. From the last exercise it follows that if there is a class of Woodin cardinals then the pointclass of ∞-universally Baire sets is closed under applications of ω·α for each countable α. This was proved previously by Woodin, using stationary tower forcing. The stationary tower proofs do not require iterability, and this becomes an advantage when one deals with stronger large cardinal axioms, beyond the level where
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iterability is known. For example Woodin showed, using stationary tower forcing, that if there is a class of measurable Woodin cardinals then the pointclass of ∞-universally Baire sets is closed under applications of the quantifier associated to open games of length ω1 . It should be noted that determinacy for these games is not yet known from large cardinals in V. Assuming full iterability, through realizable branches, for countable substructures of rank initial segments of V, this determinacy would follow from measurable Woodin cardinals by the results in Neeman [33]. But the necessary iterability is open. Our determinacy proofs can be applied not only to games where the payoff sets are universally Baire, but also to games where the payoff sets are homogeneously Suslin (see Martin–Steel [18, §2] for the definition). We shall give a concrete example of this in the context of games of continuously coded length, in Exercise 3E.6. Let us here only say that the argument in Section 2D can be adapted to deal with homogeneously Suslin payoff instead of <ω2 − 11 payoff, producing the following result: Exercise 2E.14. Let α be a countable limit ordinal. Let λ be an infinite cardinal. Suppose there are α Woodin cardinals below λ. Let C ⊂ Rα be λ-homogeneously Suslin. Then Gω·α (C) is determined. # Hint. Adapt the proof of Theorem 2D.1 to replace Martin’s games C[ y ] with games on the tree witnessing that C is homogeneously Suslin. For more details see the hint to Exercise 3E.6. # For limit ordinals α = ω · β with β additively closed, the determinacy in Exercise 2E.14 is due to Woodin, obtained as a consequence of his work on equiconsistency results connecting ω · β Woodin cardinals under choice with measures on [Pω1 (R)]β under AD.
Chapter 3
Games of continuously coded length
Given a set C ⊂ R<ω1 and a partial function ν : R → ω, the game Gcont (ν, C) is played as follows: In mega-round α the two players collaborate as usual to produce a real yα . If ν(yα ) is undefined, the game ends. Player I wins iff $yξ | ξ ≤ α% ∈ C. If ν(yα ) is defined, set nα to be its value. If nα ∈ {nξ | ξ < α} then again the game ends. Again I wins iff $yξ | ξ ≤ α% ∈ C. Otherwise the game continues. I II
.........
yα (0)
yα (2) yα (1)
......... yα (3)
...
Diagram 3.1. The game Gcont (ν, C).
Let $yξ | ξ ≤ α% be a position in Gcont (ν, C). By necessity ξ → nξ , ξ < α, is injective; else the game would have ended before reaching mega-round α. α is therefore countable. For limit α the injection ξ → nξ is produced continuously as the game approaches α. Gcont (ν, C) is said to have continuously coded length in light of this fact. The notion traces back to Steel [41]. We develop here methods for proving the determinacy of games of continuously coded length. For example we show that: Theorem (3D.1). Suppose that there exists an iterable class model M with cardinals κ < δ so that (a) δ is a Woodin cardinal of M; (b) κ is strong to δ + 1 in M (see Section 3B); and (c) M&δ + 1 is countable in V. Then the games Gcont (ν, C) are determined for all ν in the class 02 and all C in the pointclass <ω2 − 11 . Other results of similar flavor are listed later in the chapter. The methods developed here, apart from yielding the determinacy of games of continuously coded length, also serve as a foundation for some of the work in the next chapters.
3A Codes We begin with a way to code positions in Gcont by reals. The coding is continuous at limits, in a manner made precise in Claim 3A.5. Fix a partial map ν : R → ω. Let $yξ | ξ < α% be a countable sequence of reals. Let nξ = ν(yξ ) for ξ < α. $yξ | ξ < α% is said ro be a ν-position (a position for short) if the nξ -s are distinct. Claim 3A.1. Suppose that $yξ | ξ < λ% is a position of limit length λ. Then nξ → ∞ as ξ → λ.
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Proof. Immediate using the fact that the nξ -s are distinct.
#
y ) be the partial linear order on ω defined Given a position y = $yξ | ξ < α% let o( y ) = {nξ | ξ < α} ⊂ ω. d( y ) is the domain by o( y ) = {$nζ , nξ % | ζ < ξ < α}. Let d( of the order o( y ). It’s clear that the order type of o( y ) is α. Moreover for any ξ < α y ) is precisely ξ . the order type of nξ in o( Let R be the Polish space consisting of R and an extra isolated point “↑” which we think of as standing for “undefined.” Define z( y ) ∈ (R )ω by: yξ if n = nξ ; z( y )n = ↑ if n ∈ d( y ). Fix some recursive map ϕ, injecting (R )ω × {partial linear orders on ω} into R. We y , to denote ϕ(z( y ), o( y )). We refer to yξ | ξ < α as the use yξ | ξ < α, or real coding $yξ | ξ < α%. We say that x is a code if it is a real coding some position y ), o( y )%. The position y can y = $yξ | ξ < α%. Note that y can be recovered from $z( thus be recovered from the real coding it. The precise specifications of ϕ are not important, so long as ϕ satisfies the following property: Property 3A.2. For any position y and any natural number l, y l depends only on y )l−1 l%. o( y )(l × l) and on $z( y )0 l, . . . , z( Using Property 3A.2 one can easily prove the following claim: Claim 3A.3. Let $yξ | ξ < α% be a position of length α. Let $yξ | ξ < α + 1% be a position of length α + 1 which extends $yξ | ξ < α%. Then yξ | ξ < α and yξ | ξ < α + 1 agree to nα . # In fact one can prove: Claim 3A.4. For any position $yξ | ξ < λ% and any α < λ, if nξ ≥ n for all ξ ∈ [α, λ), # then yξ | ξ < α and yξ | ξ < λ agree to n. Combining this with Claim 3A.1 immediately yields: Claim 3A.5. Let $yξ | ξ < λ% be a position of limit length λ. For each α ≤ λ let xα = yξ | ξ < α. Then xα → xλ as α → λ. # A ν-run (a run for short) is a sequence y = $yξ | ξ < α + 1% so that yα is a position; yα is a real; and either ν(yα ) is not defined, or else ν(yα ) ∈ {ν(yξ ) | ξ < α}. A ν-run is thus a terminal stage in the game of continuously coded length associated to ν. Fix C ⊂ R<ω1 . Fix a pointclass . We say that C is in the codes if there exists some C ∗ ⊂ R × R in so that for any run y = $yξ | ξ < α + 1%, y ∈ C ⇐⇒ $ y α, yα % ∈ C ∗ . We say that C belongs to the pointclass if it is in the codes.
3B First determinacy result, part I
89
3B First determinacy result, part I Work with a fixed ν : R → ω, and assume that ν is continuous. Fix some C ⊂ R<ω1 which is <ω2 − 11 in the codes. We aim to prove that Gcont (ν, C) is determined. We shall use the following large cardinal assumption: there exists a model M with cardinals κ < δ so that: • M is a class model of ZFC∗ ; • M is mildly iterable (see Appendix A); • δ is a Woodin cardinal of M; • κ is strong to δ + 1 in M. In other words, for every Z ∈ M&δ + 1, there exists some extender E in M so that crit(E) = κ and Z ∈ Ult(M, E); • M&δ + 1 is countable in V. This assumption is well above a measurable limit of Woodin cardinals (κ is easily seen to be a measurable limit of Woodin cardinals in M), but well below a Woodin limit of Woodin cardinals. By passing to L(M&δ + 1) we may add the following assumption: • M = L(M&δ + 1). We shall use this extra condition in the proof. 3B (1) Names. Using the assumption that ν is continuous, fix some partial map ν˜ : ω<ω → ω so that for any y ∈ ωω , ν(y) = n iff ν˜ (yi) = n for all sufficiently large i < ω. ν˜ is coded by a real. We may assume that this real belongs to M; if not, use Remark 1E.5 or Woodin [45] with the first Woodin cardinal of M to absorb this real into a small (relative to the image of κ) generic extension of an iterate of M and replace M by this generic extension. Fix C ∗ ⊂ R × R in the pointclass <ω2 − 11 so that for any position y = $yξ | ξ < y α. We α + 1% of successor length α + 1, y ∈ C ⇐⇒ $x, yα % ∈ C ∗ where x = ∗ may assume that C is lightface in a real which belongs to M; if not then again replace M by a small generic extension of an iterate which absorbs the necessary real. For $x, yα % ∈ R × R let C[x, yα ] be the Martin auxiliary game associated to $x, yα %, and C ∗ ; see Section 2D and Martin [24]. The map C = (x, yα → C[x, yα ]) is Lipschitz continuous and belongs to M. We shall refer to this map in our definition; it will be our aid in deciding membership in C. Using the assumption that κ is strong to δ + 1 in M, fix a sequence of extenders in M, all with critical point κ, all with M&δ ⊂ Ult(M, E) and E = $Eξ | ξ < lh(E)% M so that iE (κ) > δ, and such that for every Z ∈ M&δ + 1 there exists some ξ < lh(E) Z ∈ Ult(M, Eξ ). For expository simplicity fix g which is col(ω, δ)-generic/M. Observe that g is also generic over Ult(M, E) for any extender E in M.
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3 Games of continuously coded length
Claim 3B.1. Suppose x is a real which belongs to M[g]. Then there exists some so that x belongs to Ult(M, Eξ )[g]. ξ < lh(E) Proof. This follows immediately from the properties of E using some Z ∈ M&δ + 1 which codes a name for the real x. # Definition 3B.2 (for a real x ∈ M[g]). ord(x) (the order of x) is the least ξ so that x belongs to Ult(M, Eξ )[g]. For a code x = yξ | ξ < α ∈ M[g], a sequence a ∈ (M&δ)ω ∩ M[g], and an ordinal γ , we intend to define a game G∗ ( a , x, γ ) played inside M[g]. G∗ ( a , x, γ ) will consist of moves to produce a real yα in the usual way, and auxiliary moves through which player I tries to witness that either: (1) $yξ | ξ < α +1% is terminal in Gcont (ν, C) a∗, x ∗, γ ∗) and won by I; or (2) $yξ | ξ < α + 1% is non-terminal and I can win π(G∗ )( where x ∗ = yξ | ξ < α + 1, π(G∗ ) is a shift of G∗ by some embedding π , a ∗ is a shift by π of some initial segment of the auxiliary moves made in G∗ ( a , x, γ ), and γ ∗ is smaller than the shift of γ . Condition (2), when made precise, is self perpetuating and will enable a construction that continues until securing condition (1). It will be a while before we get to the actual definition of G∗ . We start by developing some notation around it. ˙ ] is the canonical name for the set of $ Definition 3B.3. A[γ a , x% ∈ M[g] so that: ω a , x, γ ) in M[g]. a ∈ (M&δ) ; x ∈ ωω is a code; and player I wins G∗ ( ˙ ], δ, and X = M&κ. Let A[γ ] denote the auxiliary games map associated to A[γ Remark 3B.4. The use of the set X = M&κ instead of X = ω for the auxiliary games map will later ensure that the pivots we construct reside above κ. a , x, γ ) will make reference to the auxiliary games maps A[γ ∗ ] The definition of G∗ ( ∗ for γ < γ , and in fact to the association Aγ = (γ ∗ → A[γ ∗ ]). The definition is thus an induction on γ . Fix an ordinal γ . Work under the assumption that Aγ is known. We shall end ˙ ] and by extension to A[γ ]. We start with a definition of G∗ (. . . , γ ), giving rise to A[γ with the definition of an intermediary game. Let M[g] ¯ be a small generic extension of M, where small means of size less than κ. ¯ Let P ∈ (M&δ)<ω . Let s ∈ ω<ω . Let x = yξ | ξ < α be a code belonging to M[g]. For simplicity assume that lh(s) ≥ lh(P ). Suppose that ν˜ (s) is defined. (Later on we shall work in situations where lh(P ) = ν˜ (s), but for the time being let us not make this assumption.) We define the game G∗main (x, s, P , γ ) associated to these objects. ¯ according to Diagram 3.2 Set n = lh(P ). G∗main (x, s, P , γ ) is played inside M[g] and rules (1)–(3) below. We use m to index rounds in G∗main (x, s, P , γ ). The game starts with m = lh(P ). ∗ , a∗ ∗ The objects played in round m are γm∗ , am−I m−II , and yα (m). We use Pn to denote P , ∗ ∗ ∗ ∗ and inductively define Pm+1 = Pm −−, am−I , am−II . We let yα n = sn, and inductively define yα m + 1 = yα m−−, yα (m). We let x ∗ denote $yξ | ξ < α%−−, yα . Note that x ∗ depends continuously on yα .
3B First determinacy result, part I
I II
γn∗
yα (n)
∗ an−I
∗ an−II
∗ γn+1
yα (n + 1)
∗ an+1−I
∗ an+1−II
91
······
Diagram 3.2. The game G∗main (x, s, P , γ ).
(1) (Rule for I) γm∗ is an ordinal chosen so that: (a) Pm∗ is a legal position in A[γm∗ , x ∗ ]; (b) γm∗ < γ if m = n; and ∗ if m > n. (c) γm∗ < γm−1
(2) (Rule for I and II) yα (m) is a natural number, played by I if m is even and by II if m is odd. For m < lh(s) we require yα (m) = s(m). ∗ ∗ and am−II are legal moves in the game A[γm∗ , x ∗ ], fol(3) (Rule for I and II) am−I ∗ lowing the position Pm .
The first player to violate any of the rules of G∗main (x, s, P , γ ) loses. Note that, because of rule (1c), there are no infinite runs of G∗main (x, s, P , γ ); one of the players must violate a rule at some point. Observe how rule (2) and the assignment yα n = sn together force yα to extend s. (The assumption for simplicity that lh(s) ≥ n is used in the indexing of moves, allowing us to start with yα (n) in round n.) The fact that yα extends s implies that ν(yα ) = ν˜ (s). o( y ∗ ) is therefore determined from the start, and does not depend on the moves yα (m). Using Property 3A.2 it follows that x ∗ m depends only on (yξ for ξ < α and) yα m. So the rules for the first m + 1 rounds in A[γm∗ , x ∗ ] are known once yα (m) is played. Rule (3) therefore makes sense. The rules for the first m rounds of A[γm∗ , x ∗ ] are known before yα (m) is played, and so rule (1a) too makes sense. Note finally that γm∗ < γ by rules (1b) and (1c). Knowledge of Aγ is thus sufficient for the definition of G∗main (x, s, P , γ ). Let φmain (A)[x, s, P , γ ] express the statement “player I wins G∗main (x, s, P , γ ).” (The only part of A relevant to the truth value of φmain (A)[. . . , γ ] is Aγ .) Since winning a clopen game is absolute, player I wins G∗main (x, s, P , γ ) iff M[g] ¯ |= φmain (A)[x, s, P , γ ]. Remark 3B.5. A position of length m in G∗main (x, s, P , γ ) consists of the tuples yα m ∗ . The game G∗ and Pm∗ , and the ordinals γn∗ , . . . , γm−1 main (x, s, P , γ ) from that position ∗ ∗ ). (s and y m are comonward is simply the game Gmain (x, max{s, yα m}, Pm∗ , γm−1 α patible sequences. By their maximum we mean the longer of the two.) In particular it ∗ % is a winning position for I in G∗ follows that $yα m, Pm∗ , γm∗ , . . . , γm−1 main (x, s, P , γ ) ∗ ), and similarly for II. iff I wins G∗main (x, max{s, yα m}, Pm∗ , γm−1
92
3 Games of continuously coded length
Let us now define the game G∗ ( a , x, γ ). We work with some code x = yξ | ξ < α belonging to M[g], and some a ∈ (M&δ)ω , again belonging to M[g]. a , x, γ ) is played inside M[g] according to Diagram 3.3. Players I and II G∗ ( collaborate as usual to produce a real yα = $yα (i) | i < ω%. In addition they play auxiliary moves in the game C[x, yα ]. They continue this way until, if ever, reaching an i < ω so that: (a) ν˜ (yα i) is defined; (b) ν˜ (yα i) ∈ {nξ | ξ < α}; and (c) i ≥ ν˜ (yα i). The first player to violate any of the rules of C[x, yα ] loses. Infinite runs where an i < ω satisfying conditions (a)–(c) was not reached are won by player I. If an i < ω satisfying conditions (a)–(c) is reached, the game ends. We set s = yα i and P = a ˜ν (s). We let N = Ult(M, Eord(x) ) and let π be the ultrapower map. (See Definition 3B.2 for the definition of ord(x).) I wins iff N[g] |= φmain (π(A))[x, s, π(P ), π(γ )]. I II
yα (0), ρ0
yα (2), ρ2 yα (1), ρ1
... yα (3), ρ3
...
Diagram 3.3. G∗ ( a , x, γ ), so long as ν˜ (yα i) is not defined or belongs to {nξ | ξ < α}.
An i satisfying conditions (a)–(c) corresponds to a stage where it’s clear that $yξ | a , x, γ ) consists of moves ξ < α+1% will not be terminal in Gcont (ν, C). Intuitively G∗ ( to produce yα , and auxiliary moves verifying membership in C for as long as it seems that $yξ | ξ < α + 1% may be terminal. If ever it becomes clear that $yξ | ξ < α + 1% is not going to be terminal then the game ends, and instead the players pass to a shift of G∗main through the reference to φmain in the payoff. Strictly speaking condition (c) in the definition of G∗ is not necessary. It has to do with the assumption for simplicity that lh(s) ≥ lh(P ) in the definition of G∗main (. . . ), and amounts to nothing more than some delay in the ending of G∗ ( a , x, γ ). We could drop this condition at the price of complicating Diagram 3.2. Note that N[g] is a small generic extension of N, where small is meant with respect to π(κ). This follows from the fact that the extenders in E send κ above δ and has to do with the large cardinal assumption of this section. Moreover x belongs to this small generic extension of N by Definition 3B.2 (see also Claim 3B.1). The use of φmain in the payoff of G∗ therefore makes sense. Note finally that the definitions here are made by induction on γ . The definition of A[γ ] requires knowledge of G∗ (. . . , γ ), and the definition of G∗ (. . . , γ ) in turn require knowledge of the map γ ∗ → A[γ ∗ ] for γ ∗ < γ .
3B First determinacy result, part I
93
˙ ] for 3B (2) The basic step. The work in Section 3B (1) defines, in M, a name A[γ M ˙ ]. ˙ each ordinal γ . Let A be the map which assigns to each γ < jump (δ) the name A[γ M ˙ The restriction to γ < jump (δ) is in line with Section 1F. Its point is to make A a set in M, rather than a class over M. There is nothing of importance about the specific use of jumpM (δ), except that it is larger than the γ -s we shall care about later on. ˙ δ, and X = M&κ. Let Let Amix be the mixed pivot games map associated to A, σmix be the mixed pivot strategies map associated to the same objects. See Section 1F for the definition of these maps. We work below with a code x = yξ | ξ < β and a mixed pivot P for x, and show how I can win to construct yβ and a mixed pivot P∗ for x ∗ = yξ | ξ < β + 1, so that P∗ extends Pν(yβ ). The construction will be used later for the successor mega-round in a construction of a run of Gcont (ν, C). Let us be more precise. Fix some small (with respect to κ) generic extension M[g] ¯ ¯ Fix some : ω → M&δ + 1, of M. Fix some code x = yξ | ξ < β in M[g]. not necessarily a surjection. Fix an infinite run P of Amix [x], played according to σmix [, x]. P consists of the objects T , a , f , and γ . We constantly use the terminology of Section 1F when working with these objects. Fix some s ∈ ω<ω . Suppose that ν˜ (s) is defined. Suppose that lh(s) ≥ ν˜ (s). Let P = a ˜ν (s). Let n denote ν˜ (s) = lh(P ). Let b be an infinite odd branch of T . Let Mb be the direct limit along b and let jb : M → Mb be the direct limit map. Recall that γb = γ (P, b) is the shift to Mb of the ordinal of γ which corresponds to the root of b, see Definition 1F.6. Suppose that: ¯ |= φmain (jb (A))[x, s, P , γb ]. (1) Mb [g] Remark 3B.6. The extenders used in T have critical points above κ, by Lemma 1F.12 and through the use of X = M&κ noted in Remark 3B.4. The embeddings of T ¯ therefore extend to act on M[g], ¯ and x ∈ M[g] ¯ also belongs to Mb [g]. Fix an imaginary opponent, willing to play for II in the standard length ω game on natural numbers, starting from s. Fix some ∗ : ω → M&δ + 1, which agrees with to n. We shall construct: (A) A real yβ extending s. yβ (i) for odd i ≥ lh(s) will be played by the imaginary opponent. (B) An infinite run P∗ of Amix [x ∗ ], played according to σmix [∗ , x ∗ ], which agrees with P on the first n rounds. x ∗ here equals $yξ | ξ < β%−−, yβ . If ∗ is chosen onto M&δ + 1, then P∗ is mixed pivot for x ∗ by Lemma 1F.11. Observe that ν(yβ ) = n, since yβ extends s. If follows using Claim 3A.3 that x ∗ n = xn. This implies that Amix [x] and Amix [x ∗ ] have the same rules for the first n rounds. Similarly σmix [, x] and σmix [∗ , x ∗ ] agree on the first n rounds. Condition (B) therefore makes sense.
94
3 Games of continuously coded length
Let n¯ < ω be largest so that f (n) ¯ belongs to b. n¯ is equal to the root of b in the terminology of Section 1F. Let k be the first element of the odd branch b which is larger than f (n) ¯ and larger than e(n). Let γ equal jf (n),k ¯ (γn¯ ). γ is the shift to Mk of the ordinal of γ corresponding to the root of b. Since γb is the shift of that ordinal to Mb , γb = jk,b (γ ). By Lemma 1F.13, P = a n belongs to Mk , and is not moved by jk,b . Pulling assumption (1) down using jk,b therefore gives: (2) Mk [g] ¯ |= φmain (j0,k (A))[x, s, P , γ ]. (Note that jk,b extends to act on Mk [g], ¯ and therefore on x, see Remark 3B.6.) Using ¯ a winning strategy for player I in j0,k (G∗main )(x, s, P , γ ). this condition fix σ ∗ ∈ Mk [g], Let us begin round n of P∗ . (P∗ n is set equal to Pn.) Playing for I we play f ∗ (n) = k. This forces us to play an extension T ∗ k + 1 of the iteration tree T ∗ e(n) + 1 = T e(n) + 1. We play T ∗ k + 1 = T k + 1. Thus Mf∗ ∗ (n) = Mk . This move is presented in Diagram 3.4. The top line in the diagram presents the old configuration of P. The bottom line shows how P∗ departs from P in round n. We leave it to the reader to check that the conditions of rule (1) in Section 1F (1) are satisfied by our move for P∗ . For rule (1d) one uses the fact that T satisfies rule (1d) and conditions (i)–(iii) in Section 1F(1). II
$
P
Me(n)
_ _I _
P∗
Me(n)
_ _ _ _ _ _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ _ Mk
Mf (n)
Mf (n)+1
Mf (n)+2
_ _I _ Mf (n+1)
Mk
(Mf∗ ∗ (n) )
(Me∗∗ (n) )
Diagram 3.4. First move in round n of P∗ .
The rest of the construction follows the lines of Section 1E. We proceed in rounds indexed by m, starting with m = n. ∗ ∗ , p ∗ , u∗ , E ∗ ∗ The moves γm∗ , lm m m f ∗ (m) , Ef ∗ (m)+1 , and wm are played by a combined ∗ ∗ effort involving σmix [ , x ] and appropriate shifts of the strategy σ ∗ fixed using condition (2) above. We refer the reader to Diagram 3.5. The natural number yβ (m) is determined by s if m < lh(s), played by the imaginary opponent if m ≥ lh(s) is odd, and played by the shift of σ ∗ if m is even. (Diagram 3.5 is drawn on the assumption that n is even.) Note that once yβ (m) is played, enough of x ∗ is known to determine the behavior of σmix [∗ , x ∗ ] in round m. We set f ∗ (m) equal to e∗ (m) in all rounds m > n. This means that I does not play an interval of T ∗ in those rounds; we simply set T ∗ f ∗ (m) + 1 equal to T ∗ e∗ (m) + 1. Remark 3B.7. The construction requires shifting σ ∗ along the branch k, k+2, k+4, . . . ¯ a small generic extension of T ∗ . This shifting is possible since σ ∗ belongs to Mk [g], of Mk relative to κ, and since all the extenders used in T ∗ have critical points above κ
3B First determinacy result, part I
_ _ _ _ _ _
∗ Mk+1
Mk
(Mf∗ ∗ (n) )
∗ Mk+2
∗ Mk+3
(Mf∗ ∗ (n+1) )
∗ Mk+5
∗ Mk+4
(Mf∗ ∗ (n+2) )
95
∗ Mk+6
(Mf∗ ∗ (n+3) )
_ σ∗
γn∗
σ∗
yβ (n)
σ∗
σmix
[∗ ,x ∗ ]
ln∗
/o /o /o /o /o /o /
pn∗ u∗n
_
wn∗
∗ jk,k+2 (σ ∗ )
∗ γn+1
Oppnt or s(n+1)
yβ (n+1)
∗ (σ ∗ ) jk,k+2
∗ ln+1
u∗n+1
σmix
[∗ ,x ∗ ]
/o /o /o /o /o /o /
∗ pn+1
_
∗ wn+1
∗ jk,k+4 (σ ∗ )
∗ γn+2
∗ jk,k+4 (σ ∗ )
yβ (n+2)
∗ (σ ∗ ) jk,k+4
∗ ln+2
/o /o /o /o /o /
∗ pn+2
u∗n+2
σmix [∗ ,x ∗ ]
_
∗ wn+2
Diagram 3.5. The construction of P∗ .
by Lemma 1F.12 and Remark 3B.4. The fact that σ ∗ belongs to Mk [g] ¯ follows form the assumption that x belongs to M[g]. ¯ Another use of this assumption was made in Remark 3B.6 and in passing from assumption (1) to condition (2).
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3 Games of continuously coded length
We leave it to the reader to check that σmix , shifts of σ ∗ , and the imaginary opponent together cover all the necessary moves for the construction of P∗ , and only note the following properties of the construction: (i) γn∗ < γ . (Recall that γ = jf (n),k ¯ (γn¯ ).) ∗ ∗ ) for m > n. (ii) γm∗ < jk+2(m−n)−2,k+2(m−n) (γm−1
Both properties follow from the rules of j0,k (G∗main )(x, s, P , γ ) and its shifts to the ∗ . Property (i) follows from rule (1b) in Section 3B (1), and property models Mk+2(m−n) (ii) follows from rule (1c). Lemma 3B.8. If P is useful (see Definition 1F.8), then so is P∗ . ¯ T ∗ f ∗ (m). We wish to show Proof. Fix m > 0. Let m ¯ < m be largest so that f ∗ (m) that: ∗ ). (∗) γm∗ < jf∗ ∗ (m),f ∗ (m) (γm ¯ ¯
If m < n then condition (∗) follows from the assumption that P is useful, since P∗ agrees with P for the first n rounds. If m > n then condition (∗) follows immediately from property (ii) above. f ∗ (m) is equal to k + 2(m − n) in this case, and m ¯ is simply equal to m − 1. ¯ < m is largest so that So suppose that m = n. f ∗ (m) is equal to k in this case. m ¯ T ∗ k. Since T ∗ k + 1 was set equal to T k + 1, and since f ∗ n was set equal f ∗ (m) to f n, it follows that f (m) ¯ T k. Since k belongs to the odd branch b it follows that f (m) ¯ ∈ b. Recall that n¯ < ω is largest so that f (n) ¯ ∈ b. Combining this with the conclusion of the previous paragraph we see that f (m) ¯ ∈ [0, f (n)] ¯ T . Using the fact that P is useful it follows from this that: (iii) γn¯ ≤ jf (m),f ¯ ). ¯ (n) ¯ (γm Equality holds if m ¯ = n. ¯ If m ¯ < n¯ then the inequality of condition (iii) follows directly from the conditions placed on γ by Definition 1F.8. Shifting condition (iii) to Mk via jf (n),k ¯ , and composing the result with the inequality of condition (i), we obtain: (iv) γn∗ < jf (m),k ¯ ). ¯ (γm and jf (m),k are the same since T ∗ k + 1 = T k + 1. f ∗ (m) ¯ and f (m) ¯ are the jf∗ (m),k ¯ ¯ ∗ same since f n = f n. γm∗¯ and γm¯ are the same since γ ∗ n = γ n. Folding these facts into condition (iv) we immediately obtain condition (∗) in the case m = n. # 3B (3) I wins. Work in M. Define the game G∗ini to be played according to Diagram 3.6 and rules (1) and (2) below. ∗ ∗ ∗ for $a ∗ , a ∗ ∗ We write am m−I m−II % and Pm for $a0 , . . . , am−1 %.
3B First determinacy result, part I
I II
γ0∗
∗ a0−I
∗ a0−II
γ1∗
∗ a1−I
∗ a1−II
97
... ...
Diagram 3.6. The game G∗ini .
(1) (Rule for I) γm∗ is an ordinal chosen so that: (a) Pm∗ is a legal position in A[γm∗ , ∅]; (b) γm∗ < jumpM (δ) if m = 0; and ∗ if m > 0. (c) γm∗ < γm−1 ∗ ∗ (2) (Rule for I and II) am−I and am−II are legal moves in the game A[γm∗ , ∅], ∗ following the position Pm .
The first player to violate any of the rules loses. Because of rule (1c) there are no infinite runs of G∗ini . G∗ini is similar to the games G∗main (. . . , γ ) defined in Section 3B (1). There are a couple of differences. There is no yβ played here and we work with the code x ∗ = ∅ for the empty position. There is no ordinal γ given at the start; instead γ0 must be smaller than jumpM (δ). There is also no Pn∗ = P given at the start; the game begins with round 0. Observe that G∗ini , being a clopen game, is determined. The winning player has a winning strategy in M. Lemma 3B.9. Suppose that I wins G∗ini . Then I wins the long game Gcont (ν, C). Proof. Fix an imaginary opponent, willing to play for II in Gcont (ν, C). Fix a mild iteration strategy for M. Working with and the imaginary opponent we construct (among other things) a run of Gcont (ν, C). We shall verify at the end that the run constructed is won by player I. We work in stages, starting with stage 0. At the start of stage α we have the objects indicated in items (A)–(E). (A) Reals yξ for ξ so that ξ + 1 < α. Note the restriction to ξ + 1 < α in item (A). In stage α, for α a successor, we construct the real yα−1 . We do not construct any reals in stage 0, and we do not construct any reals in limit stages. Let nξ denote ν(yξ ). We shall end the construction as soon as we construct some yα so that ν(yα ) is not defined, or defined and an element of {nξ | ξ < α}. At the start of each stage α we therefore have: (1) nξ , ξ + 1 < α, are all distinct.
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3 Games of continuously coded length
We use xξ to denote yζ | ζ < ξ . When ending the construction—when constructing some yα so that ν(yα ) is not defined or defined and an element of {nξ | ξ < α}—we shall make sure that $xα , yα % ∈ C ∗ . We shall thus see that our run of Gcont (ν, C) is won by player I, as required. (B) A -iteration of M = M0 giving rise to models $Mξ | ξ < α% and embeddings $jζ,ξ | ζ ≤ ξ < α% between them. (C) Maps ξ : ω → Mξ &j0,ξ (δ) + 1 for ξ < α. (D) Infinite runs Pξ of j0,ξ (Amix )[xξ ], played according to j0,ξ (σmix )[ξ , xξ ], for ξ < α. Using Claim 3A.4 we know that: (2) (For n < ω and ζ < α.) Suppose nξ ≥ n for all ξ ∈ [ζ, α). Then xα agrees with xζ to n. Precisely, xα n = xζ n. By means to be described shortly we shall obtain the following additional agreement condition: (3) (For n < ω and ζ < α.) Suppose nξ ≥ n for all ξ ∈ [ζ, α). Then α agrees with the image of ζ to n. Precisely, α n = jζ,α (ζ n). As we construct in stage α we shall make sure further that: (i) Pα is useful. (ii) (For n < ω and ζ < α.) Suppose nξ ≥ n for all ξ ∈ [ζ, α). Then Pα agrees with the image of Pζ on the first n rounds. Precisely, Pα n = jζ,α (Pζ n). Note that conditions (2) and (3) are needed to make sense of condition (ii). More precisely, condition (2) is needed to ensure that the first n rounds of j0,α (Amix )[xα ] and the first n rounds of the image to α of j0,ζ (Amix )[xζ ] have the same rules, and both conditions are needed to make sure that j0,α (σmix )[α , xα ] and the image to α of j0,ζ (σmix )[ζ , xζ ] are the same for the first n rounds. Finally we would like to know that: (4) α is onto Mα &j0,α (δ) + 1. This is a matter of bookkeeping, but the bookkeeping is not entirely straightforward. For limit α condition (3) completely determines α , leaving us no freedom in trying to obtain condition (4). We must thus plan ahead, before reaching α, to make sure that condition (4) holds at α. To best describe the bookkeeping involved in the construction of the maps α : ω → Mα &j0,α (δ)+1 it is convenient to change from maps with domain ω to maps with domain ω ∪ (ω × ω). Fix some bijection e : ω → ω ∪ (ω × ω). The precise nature of e is not important, so long as:
3B First determinacy result, part I
99
(a) For each n < ω, e n ⊂ ω ∪ (n × ω). Fix some surjection φ0 : ω → M&δ + 1. This is possible using the initial large cardinal assumption, specifically the condition saying that M&δ + 1 is countable in V. At the start of stage α we have, in addition to the objects described in items (A)–(D) above: (E) Maps φξ +1 : {nξ } × ω → Mξ +1 for ξ + 1 < α. We do not construct any φλ for limit λ. φα for successor α will be constructed during stage α. We shall pick φα so that: (iii) (For successor α.) φα is onto Mα &j0,α (δ) + 1. By induction then we shall know that φξ +1 : {nξ } × ω → Mξ +1 &j0,ξ +1 (δ) + 1 is onto for each ξ + 1 ≤ α. Let us now describe how to construct α in stage α. Using the surjections φ0 , {φξ }ξ +1<α , and (if α is a successor) φα , define a partial map ψα : ω ∪ (ω × ω) → Mα &j0,α (δ) + 1 by: ψα = (j0,α ◦ φ0 ) ∪ (jξ +1,α ◦ φξ +1 ). The domain of ψα is equal to ω ∪
ξ +1≤α ξ +1≤α ({nξ } × ω).
Moreover:
(b) The range or ψα contains j0,α (M0 &δ + 1). (c) For each ξ + 1 ≤ α, the range of ψα contains jξ +1,α (Mξ +1 &j0,ξ +1 (δ) + 1). Condition (b) follows from the fact that φ0 is onto M&δ + 1. Condition (c) follows from the fact that φξ +1 is onto Mξ +1 &j0,ξ +1 (δ) + 1. Define α : ω → Mα &j0,α (δ) + 1 by: ψα (e(i)) if e(i) ∈ dom(ψα ); and α (i) = (3.1) ∅ otherwise. This definition takes care of both conditions (3) and (4) above. Condition (3) follows immediately from the definition of α , the definition of ψα , and condition (a). For successor α condition (4) follows from condition (c) with ξ + 1 = α. For α = 0 condition (4) follows from condition (b). For limit α condition (4) follows from condition (c) so long as we make sure that: (iv) (For limit α.) Every element of Mα is of the form jξ +1,α (w) for some ξ + 1 < α and some w ∈ Mξ +1 . It is time now to start the construction. We divide into the cases α = 0; α is a limit; and α is a successor. For α = 0 we must simply construct P0 . For limit α we must construct Mα and Pα . For successor α we must construct Mα , yα−1 , φα , and Pα . In each case we must verify conditions (i)–(iv).
100
3 Games of continuously coded length
Initial stage. We start with α = 0. Using the assumption of Lemma 3B.9 fix a winning strategy σ ∗ ∈ M for player I in G∗ini . Recall that 0 , defined above using φ0 , is a map from ω onto M0 &δ + 1. Let P0 be the run of Amix [∅] generated as follows: • σmix [0 , ∅] supplies moves for player II. • f0 (m) is set equal to e0 (m) in each round m. This means that the iteration tree created has a single even branch consisting of {f0 (m) = 2m | m < ω}. • σ ∗ and its shifts along the even branch supply moves for player I. These three conditions cover all moves in Amix [∅], giving rise to an infinite run P0 . The ordinals γm∗ supplied by σ ∗ and its shifts must satisfy rule (1c) of G∗ini and its shifts (else σ ∗ would lose). It follows that P0 is useful. Condition (i) therefore holds for α = 0. Conditions (ii)–(iv) are vacuous. # (Initial stage) Successor stage. Let α = β + 1 be a successor ordinal. By conditions (D), (4), and (i), Pβ is a useful mixed pivot for xβ where mixed pivot is interpreted over Mβ and using j0,β (A). Let bβ be the cofinal branch through Tβ (the iteration tree given by Pβ ) chosen by the iteration strategy . Let Qβ be the direct limit along bβ . Let kβ : Mβ → Qβ be the direct limit embedding. Since Pβ is useful (see Definition 1F.8), bβ must be an odd branch. Since Pβ is a mixed pivot for xβ there must exist some hβ so that: (S1) hβ is col(ω, (kβ ◦ j0,β )(δ))-generic/Qβ ; and (S2) $ aβ , xβ % ∈ (kβ ◦ j0,β )(A)[γβ ], where γβ = γ (Pβ , bβ ). aβ , xβ , γβ ). Combining condition (S2) Let G∗β denote the game (kβ ◦ j0,β )(G∗ )( with Definition 3B.3 we see that I wins G∗β in Qβ [hβ ]. Fix a strategy σβ∗ ∈ Qβ [hβ ] which witnesses this. G∗β is played according to Diagram 3.3. Let us begin to form a “run” of this game. We use σβ∗ to obtain moves for I. The imaginary opponent supplies natural number moves for II, playing yβ (i) for odd i. We ascribe indiscernibles for Qβ as ordinal moves for II. More precisely: Let λ = (kβ ◦ j0,β )(δ) + 1. Mβ &j0,β (δ) + 1 is countable in V—this is witnessed by β —and from this it follows that Qβ &λ is countable in V. The large cardinal assumption at the start of the section is more than enough to guarantee that every real has a sharp. (This is seen as usual using the genericity iterations, Remark 1E.5 or Woodin [45], with the first Woodin cardinal of M.) Qβ &λ therefore has a sharp. The initial assumption on M includes the condition M = L(M&δ + 1). Thus Qβ = L(Qβ &λ). The Silver indiscernibles for Qβ &λ are therefore indiscernibles for Qβ . Working by induction on i < ω we define yβ i and sequences Pi of ordinals, of length i. We construct so that yβ i and Pi together form a position in G∗β according to σβ∗ ; so that II’s move in Pi are Silver indiscernibles; and so that II’s moves in yβ i are the ones given by the imaginary opponent. The argument is standard, see for example Martin [24].
3B First determinacy result, part I
101
We continue this way until, if ever, reaching an i which satisfies conditions (a)–(c) of Section 3B (1). If no such i is reached we end up producing a real yβ so that ν(yβ ) is not defined, or defined and an element of {nξ | ξ < β}. We also produce a sequence of positions Pi of increasing lengths in (kβ ◦ j0,β )(C)[xβ , yβ ] consisting of indiscernibles for II and moves given by σβ∗ for I. The argument of Martin [24] shows under such circumstances that $xβ , yβ % belongs to C ∗ . This is the end of the construction. We have reached a position $yξ | ξ < β + 1% where Gcont (ν, C) ends, and ends with a win for player I. So suppose an i satisfying conditions (a)–(c) of Section 3B (1) is reached. G∗β ends at the first such i. Let sβ denote the position yβ i obtained in G∗β . Since σβ∗ was picked a winning strategy for I, G∗β ends with a win for I. Looking at the winning condition indicated in Section 3B(1) we see that: (S3) Nβ [hβ ] |= φmain ((πβ ◦ kβ ◦ j0,β )(A))[xβ , sβ , πβ (Pβ ), πβ (γβ )]. Recall that γβ = γ (Pβ , bβ ). Nβ here is equal to Ult(Qβ , Fβ ), where Fβ is the πβ is the ultrapower embedding. (kβ ◦ j0,β )(ord)(xβ )-th extender of (kβ ◦ j0,β )(E). Pβ is equal to aβ ˜ν (sβ ). Note how all this corresponds to the winning condition of G∗ (. . . ) defined in Section 3B(1). The current configuration of models (Mβ , Qβ , Nβ ) and the configuration of embeddings between them are presented in the lower line of Diagram 3.7. To proceed we shall have to look at these models in a different light. jˆβ
jβ,β+1
j jjjj j j j jTT / Mβ T TTTT kβ Qβ TT Tβ
jjj jjjj j j T Mβ TTTT TTTTkβ
j jjjj j j j / Qβ Mβ+1 TjTTTT TTTTkβ O Tβ D τβ copy with jβ,β+1 πβ / Qβ / Nβ Fβ
$
Fβ
Tβ
Diagram 3.7. Fβ applied to Qβ (lower line); and Fβ applied to Mβ (upper line) followed by copying.
Let κβ = j0,β (κ). By Lemma 1F.12 and Remark 3B.4, Pβ resides above κβ . It follows first that Mβ and Qβ agree beyond κβ , and secondly that crit(kβ ) > κβ . Now Fβ has critical point kβ (κβ ), equal to κβ since kβ does not move this ordinal. So crit(Fβ ) < crit(kβ ). Fβ can therefore be applied to Mβ .
102
3 Games of continuously coded length
Let Mβ+1 = Ult(Mβ , Fβ ). Let jβ,β+1 be the ultrapower embedding. The situation is presented in the upper left part of Diagram 3.7. We have so far defined Mα = Mβ+1 . Let us already here note that regardless of how we continue to construct the real yβ = yα−1 , we shall have: (S4) ν(yβ ) = ν˜ (sβ ). This is because ν˜ (sβ ) is defined and yβ , which we shall finish constructing shortly, is going to extend sβ = yβ i. We thus know already here the value of nβ ; it is equal to ν˜ (sβ ). From the fact that Mβ &j0,β (δ) + 1 is countable in V and the fact that the iteration leading form Mβ to Mβ+1 is countable it follows that Mβ+1 &j0,β+1 (δ) + 1 is countable in V. Fix a surjection φβ+1 : {˜ν (sβ )} × ω → Mβ+1 &j0,β+1 (δ) + 1. Let β+1 , the surjection of item (C), be defined using φβ+1 via the bookkeeping algorithm leading to equation (3.1) on page 99. Note that ψβ+1 and jβ,β+1 ◦ ψβ agree on ω ∪ (˜ν (sβ ) × ω); the only difference between these two maps is the addition of φβ+1 , whose domain is {˜ν (sβ )} × ω. It follows using condition (a) on page 99 that: (S5) β+1 agrees with jβ,β+1 ◦ β to ν˜ (sβ ).
Let β denote jβ,β+1 ◦ β . Observe that β is then equal to m<ω jβ,β+1 (β m). Let Pβ equal m<ω jβ,β+1 (Pβ m). Pβ is thus a run of j0,β+1 (Amix )[xβ ], played according to j0,β+1 (σmix )[β , xβ ]. Tβ , the iteration tree given by Pβ , is simply the copy of Tβ from Mβ to Mβ+1 via jβ,β+1 . The copying is indicated in dotted lines in Diagram 3.7. Let Qβ be the direct limit of the models of Tβ along bβ , and let kβ be the direct limit embedding. Let jˆβ : Qβ → Q be the copy embedding. β
It is easy to see that γ (Pβ , bβ ) = jˆβ (γ (Pβ , bβ )). It is also true, though not as easy, that a m = jˆβ ( aβ m) for each m < ω. To prove this remember that aβ m is not β
moved by a tail-end of the embedding of Tβ along bβ , and similarly with aβ and Tβ . Lemma 3B.11, to be proved in Section 3B(5), states under the current circumstances that there exists an elementary embedding τβ : Nβ → Qβ , with crit(τβ ) > πβ (κβ ) and such that τβ ◦ πβ = jˆβ . This embedding is presented in dashed line in Diagram 3.7. Since crit(τβ ) > πβ (κ), τβ extends to act on Nβ [hβ ], a small generic extension of Nβ relative to πβ (κβ ). Applying the extended embedding τβ : Nβ [hβ ] → Qβ [hβ ] to condition (S3) and using the equality τβ ◦ πβ = jˆβ we get: Qβ [hβ ] |= φmain ((jˆβ ◦ kβ ◦ j0,β )(A))[xβ , sβ , jˆβ (Pβ ), jˆβ (γβ )]. By standard copying arguments, jˆβ ◦ kβ is equal to kβ ◦ jβ,β+1 . We can thus re-write the last equation as: Qβ [hβ ] |= φmain ((kβ ◦ j0,β+1 )(A))[xβ , sβ , jˆβ (Pβ ), jˆβ (γβ )].
3B First determinacy result, part I
103
Now Pβ is equal to aβ ˜ν (sβ ). jˆβ (Pβ ) is therefore equal to aβ ˜ν (sβ ). γβ is equal to γ (Pβ , bβ ). jˆβ (γβ ) is therefore equal to γ (Pβ , bβ ). Making these substitutions in the equation we get: (S6) Qβ [hβ ] |= φmain ((kβ ◦ j0,β+1 )(A))[xβ , sβ , Pβ , γβ ], where Pβ = aβ ˜ν (sβ ) and γβ = γ (Pβ , bβ ). This precisely is the assumption (1) of Section 3B (2), only shifted to Mβ+1 . (Remember that Qβ is the direct limit of models of Tβ along bβ , and kb : Mβ+1 → Qβ is the direct limit embedding.) We can now run the construction of Section 3B (2) with Pβ standing for P, β standing for , xβ standing for x, sβ standing for s, bβ standing for b, and using β+1 for ∗ (note that β+1 agrees with β to ν˜ (sβ ) by condition (S5)). Remark 3B.10. The construction in Section 3B (2) is made under the assumption that x belongs to a small—with respect to κ—generic extension of M. (The need for this assumption is explained in Remark 3B.7.) In appealing to the construction of Section 3B (2) we are therefore making use of the fact that xβ belongs to a small— with respect to j0,β+1 (κ)—generic extension of Mβ+1 . This fact traces back to the large cardinal assumption on M, which allowed picking the extender Fβ in such a way that xβ was absorbed into Ult(Qβ , Fβ )[hβ ]. From x ∈ Ult(Qβ , Fβ )[hβ ] a quick agreement calculation shows that xβ belongs to Ult(Mβ , Fβ )[hβ ]. hβ is generic for col(ω, (kβ ◦ j0,β )(δ)). This is a small forcing relative to j0,β+1 (κ) since jβ,β+1 , the ultrapower embedding by Fβ , sends its critical point above (kβ ◦ j0,β )(δ). Running the construction of Section 3B(2) we obtain the real yβ , extending sβ , and a run P∗ of j0,β+1 (Amix )[xβ+1 ], played according to j0,β+1 (σmix )[β+1 , xβ+1 ]. Let Pβ+1 be this run. By construction, see Section 3B (2), Pβ+1 agrees with Pβ on the first ν˜ (sβ ) rounds. In other words: (S7) Pβ+1 nβ = jβ,β+1 (Pβ nβ ). By Lemma 3B.8: (S8) Pβ+1 is useful. The construction for stage α = β + 1 is now complete. Condition (i) holds for α by condition (S8) above. Condition (ii) for α follows from the same condition for β, and condition (S7). Condition (iii) was secured by the choice of φβ+1 . Condition (iv) is vacuous. # (Successor stage) Limit stage. Work now with a limit ordinal α. We have the picture of models presented in Diagram 3.8. Let Mα be the direct limit of the models $Mξ | ξ < α% under the embeddings $jζ,ξ | ζ ≤ ξ < α%. Every element of Mα has pre-images in Mξ for all sufficiently large ξ < α. Condition (iv) therefore holds trivially. Condition (iii) is vacuous here. So we need simply construct Pα and verify conditions (i) and (ii). Using Claim 3A.1 fix an increasing sequence {βn }n<ω so that:
104
3 Games of continuously coded length jξ,ξ +1
Mξ
kkk Sk k Sk SSSS k / ξ
Qξ
jξ +1,ξ +2 Fξ
#
Mξ +1
Tξ
$ kk SkSkSkS / Qξ +1 Fξ +1 Mξ +2 k SSkξ +1 Tξ +1
Diagram 3.8. The situation at limit stages.
• Each βn is smaller than α and sup{βn | n < ω} = α. • nξ ≥ n for each ξ ∈ [βn , α). For n ≤ m < ω it follows from the last item and from condition (ii) at βm that: (L1) jβn ,βm (Pβn n) is equal to Pβm n. For each n < ω let P∗n = jβn ,α (Pβn n). By condition (L1) then P∗n is an initial segment of P∗m for n < m. Define: P∗n . Pα = n<ω
P∗n
is a position of length n in j0,α (Amix )[xβn ]. By condition (2), xβn n = xα n. P∗n is therefore a position in j0,α (Amix )[xα ]. A similar argument using condition (3) shows that P∗n is according to j0,α (σmix )[α , xα ]. So Pα is a run of j0,α (Amix )[xα ] played according to j0,α (σmix )[α , xα ], as required. For each n < ω, Pβn n is useful. This follows from condition (i) at βn . Using the elementarity of jβn ,α we see that P∗n is useful. By Claim 1F.9, Pα is useful. This takes care of condition (i) at α. Condition (ii) at α follows directly from the definition of Pα , # (Limit stage) and the same condition at the βn -s. The initial, successor, and limit stage constructions together complete the proof of Lemma 3B.9. # (Lemma 3B.9) 3B (4) Discussion. Morally the limit case is always the most important part in a determinacy proof (in the context of long games). Each determinacy proof involves perpetuating a carefully engineered condition through a transfinite construction. It is easy enough to engineer the condition so that it propels itself through successor stages. But propelling through limits requires special care. The condition must involve advance planning in anticipation of future limit stages. Consider for example the situation of Section 2B. We had there a model with −1 + θ + 1 Woodin cardinals, δξ for ξ ∈ [1, θ ]. We had a name A˙ for a subset of Rθ in M col(ω,δθ ) , and an associated auxiliary games map A. θ was a limit ordinal, and we had a fixed sequence of ordinals $ηk | k < ω% converging to θ . For illustrative purposes let G∗ ( y ), where y ∈ Rθ , denote the game in which players I and II alternate moves in A[ y ] and player I in addition plays a descending sequence of ordinals. The game is presented in Diagram 3.9.
3B First determinacy result, part I
I II
γ0
a0−I
γ1
a1−II
a0−II
105
...... a1−II
Diagram 3.9. The illustrative game G∗ ( y ).
Note how round k of G∗ ( y ), following the position consisting of $a0 , . . . , ak−1 % and x , P , γ ) of Diagram 2.2 with $γ0 , . . . , γk−1 % say, is exactly the same as the game Gk ( x = yηk , P = $a0 , . . . , ak−1 % and γ = γk−1 . The determinacy proof in Section 2B y ) into its constituent rounds, and plugging round k was thus a matter of breaking G∗ ( of the game in stage ηk following the production of yηk . As the construction in Section 2B approached θ, moves in these rounds accumulated to produce the kind of condition needed in stage θ. The planning for the limit stage θ was thus achieved y ) over stages cofinal in θ . through spreading G∗ ( Consider now the case of continuously coded length. Let y be a position of limit length α in Gcont (ν, C), and let γ be an ordinal. For illustrative purposes consider the y , γ ), played along the format of Diagram 3.9, but with the following rules game G∗ ( (where A[. . . ] is the map defined in Section 3B (1)): (1) (Rule for I) γm is an ordinal chosen so that: y ]; (a) $a0 , . . . , am−1 % is a legal position in A[γm , (b) γm < γ if m = 0; and (c) γm < γm−1 if m > 0. (2) (Rule for I and II) am−I and am−II are legal moves for I and II respectively in the y ], following the position $a0 , . . . , am−1 %. game A[γm , Note the similarities between this game and the game G∗main (. . . ) displayed in Diagram 3.2. Here too the determinacy proof can be viewed as breaking G∗ ( y, γ ) into its constituent rounds, and distributing these rounds over earlier stages. A careful look at the limit stage construction in Section 3B (3) shows precisely how this happens. y , γ ) is plugged into stage η = ξ + 1 for ξ < α largest so that nξ ≤ m. Round m of G∗ ( Let ηα,m denote this stage. A couple of points which distinguish the “breaking and spreading” done here from that done in Section 2B are worth highlighting. First, note that the manner of spreading here is not fixed at the outset. The sequence $ηα,m | m < ω% leading to α depends not only on α but also on yα. In other words the sequence depends on the actual moves played in the game Gcont (ν, C) up to α. Next, note that a given η may serve more than one limit α. In fact it may have to serve infinitely many; there may be some m and infinitely many distinct limit ordinals αi so that ηαi ,m = η for all i. This is the main point which distinguishes the method of approaching limits here from the method used in Chapter 2. It creates a substantial difficulty. A tail-end of the moves in the auxiliary game at stage η must be reserved to propel the inductive
106
3 Games of continuously coded length
condition to stage η + 1. Thus only a finite part of this auxiliary game can be devoted to advance planning for future limits. Yet this finite part must somehow plan for infinitely many limits. Our solution is to engineer the proof in such a way that for α > α, the auxiliary game at stage α is the image under some elementary embedding of the auxiliary game at stage α. The moves planned in stage η and then used in stage α can thus be shifted via this elementary embedding and used again in stage α . In fact the moves in round m of stage η are shifted and used in all limit stages α with ηα,m = η. Planning for one limit now becomes the same as planning for all. The key to this solution is lifting moves in the current auxiliary game so that they can be used again in the next one. The game G∗ (. . . ) designed in Section 3B (1) does precisely this lifting, through its use of an ultrapower embedding by Eord(x) for the reference to φmain . 3B (5) Internal ultrapowers vs. copying. During the proof of Lemma 3B.9 we appealed to a lemma which allowed us to switch between an internal ultrapower embedding and a copy map. Here we phrase and prove this lemma. We work with a model M, and a cardinal κ in M. Let T be a length ω iteration tree on M, using only extenders with critical points above κ. Let b be a cofinal branch through T , let Q denote the direct limit model along b, and let k : M → Q be the direct limit embedding. Fix an extender F ∈ Q with critical point equal to k(κ) = κ. Note then that F can be applied to M. Let M = Ult(M, F ), and let j : M → M be the ultrapower embedding. Let T be the result of copying T to a tree on M . Let Q be the direct limit of the models of T along b, and let k : M → Q be the direct limit embedding. Let jˆ : Q → Q be the copy map. The situation is similar to that presented in the upper line of Diagram 3.7. Let N = Ult(Q, F ), and let π : Q → N be the ultrapower embedding. The situation is similar to that presented in the lower line of Diagram 3.7. We wish to prove: Lemma 3B.11. There is an elementary embedding τ : N → Q with crit(τ ) > π(κ) and such that τ ◦ π = jˆ. Proof. We use the following simple claim throughout the proof: Claim 3B.12. Let X ⊂ κ <ω be given. Then jˆ(X) is equal to π(X). Proof. First calculate that: jˆ(X) = (jˆ ◦ k)(X) = (k ◦ j )(X) = j (X). The first equality uses the fact that crit(k) > κ (all extenders in T have critical points above κ by assumption). The second equality uses the commutativity of Diagram 3.10. The third equality uses the fact that crit(k ) > j (κ).
3B First determinacy result, part I
107
j jjjj j j j jTT / MO T TTTT k QO TT F ∈Q
T
jˆ
j
jj jjjj j j j / jTTT MT TTTT k Q TT T
Diagram 3.10. F ∈ Q applied to M, and copying.
Now note that j (X) is equal to π(X); both j and π are ultrapower embeddings # by F , and so agree on subsets of crit(F )<ω = κ <ω . Every element of N is of the form π(f )(a) for some a ∈ π(κ <ω ) in the support of F and some function f : κ lh(a) → Q in Q. π(κ lh(a) ) = jˆ(κ lh(a) ) by Claim 3B.12, so a belongs to the domain of jˆ(f ). We can therefore consider the object jˆ(f )(a). Definition 3B.13. For w = π(f )(a) ∈ N , where a belongs to the support of F and f : κ lh(a) → Q is a function in Q, define τ (w) = jˆ(f )(a). Definition 3B.13 is somewhat premature, but the next claim shows that it makes sense. Claim 3B.14. Suppose a ∈ π(κ)<ω belongs to the support of F . Suppose f, g ∈ Q are both functions from κ lh(a) into Q. Suppose that π(f )(a) = π(g)(a). Then jˆ(f )(a) = jˆ(g)(a). Proof. Let X = {t ∈ κ lh(a) | f (t) = g(t)}. By assumption a ∈ π(X). By Claim 3B.12, jˆ(X) = π(X). It follows that a ∈ jˆ(X), so jˆ(f )(a) = jˆ(g)(a) as required. # τ is now well defined. It remains to show that it satisfies the requirements of Lemma 3B.11. Claim 3B.15. τ : N → Q is elementary. Proof. Fix w ∈ N , say w = π(f )(a). Fix a formula ϕ. Suppose N |= ϕ[w]. We wish to show that Q |= ϕ[τ (w)]. Let X = {t ∈ κ lh(a) | Q |= ϕ[f (t)]}. Since N |= ϕ[π(f )(a)], a ∈ π(X). Using # Claim 3B.12 it follows that a ∈ jˆ(X), so Q |= ϕ[jˆ(f )(a)] as required. Claim 3B.16. τ ◦ π = jˆ.
108
3 Games of continuously coded length
Proof. Let w ∈ Q be given. Let f : κ 0 → Q be the function whose value at ∅ is w. π(w) is equal to π(f )(∅). τ (π(f )(∅)) by definition is jˆ(f )(∅), which is equal to jˆ(w). Thus (τ ◦ π)(w) = jˆ(w) as required. # Note that τ (π(κ)) = jˆ(κ) = π(κ) by commutativity and Claim 3B.12. To complete the proof we must verify that τ π(κ) is the identity. We shall then know that crit(τ ) > π(κ). Fix some w ∈ π(κ). Then w = π(f )(a) for some a in the support of F and some f : κ lh(a) → Q with range(f ) ⊂ κ. Now τ (w) = jˆ(f )(a) by definition, and jˆ(f ) = π(f ) by Claim 3B.12. (f can be regarded as a subset of κ <ω ; this uses the fact that range(f ) ⊂ κ.) It follows that τ (w) = π(f )(a) = w, as required. # (Lemma 3B.11) We end by noting that, with a slight extra assumption involving the extenders used in T , Lemma 3B.11 actually improves to the statement that Q is equal to N and jˆ is equal to π. Lemma 3B.17. Suppose that the extenders used in T are all κ closed. Then Q equals N and jˆ equals π . Proof. It is enough to show that the embedding τ given by Lemma 3B.11 is onto. We would then have an ∈-preserving map from one transitive structure onto another. The structures would have to be equal and the map, τ , would have to be the identity. So fix some u ∈ Q . We wish to show that u belongs to the range of τ . : M → Q denote the Let $Mi | i < ω% denote the models of T . For i ∈ b let ki,b i b embedding given by T . Let Mi and ki,b denote the corresponding objects for T . Fix a (w) = u. node l ∈ b large enough that u has a pre-image in Ml . Fix w ∈ Ml so that kl,b Now Ml , being a finite iterate of M , is actually contained in M which is equal to Ult(M, F ). So there is some a ∈ j (κ)<ω in the support of F and some function g : κ lh(a) → M in M so that w = j (g)(a). Since w belongs to Ml , g can be picked with range(g) ⊂ Ml . Ml is closed under sequences of length κ in M; this uses the assumption that the extenders of T are κ closed. It follows that g belongs to Ml . Let f = kl,b (g). This is a function which belongs to Q. By standard copying arguments, specifically the commutativity of Diagram 3.11, (j (g)). Using this identity calculate that: jˆ(f ) = kl,b (j (g))(a) jˆ(f )(a) = kl,b = kl,b (j (g)(a)) = kl,b (w)
= u. ) > j (κ) and a ∈ The equality leading to the second line uses the fact that crit(kl,b j (κ)<ω .
3C First determinacy result, part II
MO
k0,l
kl,b
k0,l
/ Ml
/ Q O jˆ
j Ml
j
M
/ M Ol
109
kl,b
/Q
Diagram 3.11.
jˆ(f )(a) belongs to the range of τ by Definition 3B.13. The calculation above shows that u = jˆ(f )(a). So u ∈ range(τ ) as required. # The closure assumed in Lemma 3B.17 is easy to arrange. But the only advantage of the conclusion of Lemma 3B.17 over the conclusion of Lemma 3B.11 is aesthetical; it simplifies slightly the presentation of Diagram 3.7, eliminating the need for τβ . It is hard to decide which is better: putting in the extra work needed to secure the assumption of Lemma 3B.17; or settling for the weaker conclusion of Lemma 3B.11. We chose the latter.
3C First determinacy result, part II Let us continue with the proof that Gcont (ν, C) is determined. Our plan is to mirror the ∗ ; prove that if II wins H ∗ definitions of the previous section to end with a game Hini ini then II wins Gcont (ν, C); and most importantly rule out the possibility that II wins G∗ini ∗ . and I wins Hini 3C (1) II wins. Mirroring the definitions in Section 3B(1) we work to define games x, γ ) associated to a b ∈ (M&δ)ω ∩ M[g], a code x = yξ | ξ < α ∈ M[g], H ∗ (b, and an ordinal γ . The games are clopen and played inside M[g]. Simultaneously we set: ˙ ] is the canonical name for the set of $b, x% ∈ M[g] so that: Definition 3C.1. B[γ ω ω ∗ b ∈ (M&δ) ; x ∈ ω is a code; and player II wins H (b, x, γ ). ˙ ], δ, We use B[γ ] to denote the mirrored auxiliary games map associated to B[γ and X = M&κ. Fix for a moment a small, relative to κ, generic extension M[g] ¯ of M. Fix an ¯ Let Q ∈ (M&δ)<ω . ordinal γ . Let x = yξ | ξ < α be a code belonging to M[g]. Let s ∈ ω<ω . Assume that lh(s) ≥ lh(Q). Suppose that ν˜ (s) is defined. We define
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∗ (x, s, Q, γ ). The definition mirrors that of under these assumptions the game Hmain G∗main (x, s, P , γ ) in Section 3B(1). ∗ (x, s, Q, γ ) is played inside M[g] Set n = lh(Q). Hmain ¯ according to Diagram 3.12 and rules (1)–(3) below.
I II
γn∗
yα (n)
∗ bn−II
∗ bn−I
∗ γn+1
yα (n + 1)
∗ bn+1−II
∗ bn+1−I
······
∗ (x, s, P , γ ). Diagram 3.12. The game Hmain ∗ (x, s, Q, γ ). The game starts with m = lh(Q). We use m to index rounds in Hmain ∗ ∗ , and y (m). We use Q∗ to denote , bm−I The objects played in round m are γm∗ , bm−II α n ∗ ∗ ∗ . We let y n = sn, and in∗ Q, and inductively define Qm+1 = Qm −−, bm−II , bm−I α ductively define yα m + 1 = yα m−−, yα (m). We let x ∗ denote $yξ | ξ < α%−−, yα .
(1) (Rule for II) γm∗ is on ordinal chosen so that: (a) Q∗m is a legal position in B[γm∗ , x ∗ ]; (b) γm∗ < γ if m = n; and ∗ (c) γm∗ < γm−1 if m > n.
(2) (Rule for I and II) yα (m) is a natural number, played by I if m is even and by II if m is odd. For m < lh(s) we require yα (m) = s(m). ∗ ∗ and bm−I are legal moves in the game B[γm∗ , x ∗ ], (3) (Rule for II and I) bm−II ∗ following the position Qm . ∗ (x, s, P , γ ) loses. Note that The first player to violate any of the rules of Hmain sooner or later one of the players must violate a rule; there are no infinite runs of the game, because of rule (1c). ∗ (x, s, Q, γ ).” Only Let ψmain (B)[x, s, Q, γ ] express the statement “II wins Hmain Bγ is relevant to the truth value of ψmain (B)[. . . , γ ]. x, γ ). Work with a code x = yξ | ξ < α ∈ M[g], some We can now define H ∗ (b, ω x, γ ) is played according to Diagram 3.3. b ∈ (M&δ) ∩M[g], and an ordinal γ . H ∗ (b, Players I and II collaborate as usual to produce a real yα . In addition they play auxiliary moves in the game C[x, yα ]. They continue this way until, if ever, reaching an i < ω which satisfies condition (a)–(c) in Section 3B(1). The first player to violate any of the rules of C[x, yα ] loses. Infinite runs, where an i < ω which satisfies conditions (a)–(c) has not been reached, are won by player I, just as they were in Section 3B (1). If an i < ω which satisfies conditions (a)–(c) is reached, the game ends. We set s = yα i ν (s). As in Section 3B(1) we let N = Ult(M, Eord(x) ) and let π be the and Q = b˜ ultrapower embedding. II wins iff N |= ψmain (π(B))[x, s, π(Q), π(γ )].
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The only difference between H ∗ (. . . ) and G∗ (. . . ) is in the payoff condition when an i < ω which satisfies conditions (a)–(c) is reached. In the case of G∗ (. . . ) we used φmain to determine the payoff for I. Here we use ψmain to determine the payoff for II. H ∗ (. . . , γ ) is defined modulo knowledge of the map γ ∗ → B[γ ∗ ] for γ ∗ < γ . The definition of B[γ ] requires knowledge of H (. . . , γ ). As usual both definition are made by simultaneous induction on γ . ∗ , mirroring the game G∗ of Section 3B (3). H ∗ is played Define finally a game Hini ini ini according to Diagram 3.13 and rules (1) and (2) below.
I II
γ0∗
∗ b0−II
∗ b0−I
γ1∗
∗ b1−II
∗ b1−I
... ...
∗ . Diagram 3.13. The game Hini ∗ ∗ ∗ ∗ for $b∗ ∗ We write bm m−II , bm−I % and Qm for $b0 , . . . , bm−1 %.
(1) (Rule for II) γm∗ is an ordinal chosen so that: (a) Q∗m is a legal position in B[γm∗ , ∅]; (b) γm∗ < jumpM (δ) if m = 0; and ∗ if m > 0. (c) γm∗ < γm−1 ∗ ∗ (2) (Rule for II and I) bm−II and bm−I are legal moves in the game B[γm∗ , ∅], ∗ following the position Qm .
The first player to violate any of the rules loses. The game is clopen; because of rule (1c) there are no infinite runs. The following lemma is a direct mirror image of Lemma 3B.9: ∗ . Then II wins Lemma 3C.2. Suppose that there exists an ordinal γ so that II wins Hini # the long game Gcont (ν, C).
Proof. Simply mirror the construction of Section 3B (3). For the most part the adaptation is trivial. Let us only comment on the situation at the end of the construction. The construction ends in the successor stage β + 1 if, while playing Hβ∗ , an i < ω which satisfies conditions (a)–(c) of Section 3B (1) is not reached. The situation, adapted from the successor stage in Section 3B (3), is as follows: ∗ Hβ denotes the game (kβ ◦ j0,β )(H ∗ )(bβ , xβ , γβ ), played according to Diagram 3.3. Infinite runs where an i < ω which satisfies conditions (a)–(c) has not been reached are won by player I, see the definition of H ∗ (. . . ) above. We have a strategy τβ ∈ Qβ [hβ ] which is winning for II in Hβ∗ , see the successor stage in Section 3B (3), particularly conditions (S1), (S2) and the paragraph following them. Working by induction on i < ω we define yβ i and sequences Qi of ordinals, of length i. We construct so that yβ i and Qi together form a position in Hβ∗ according
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to τβ∗ ; so that I’s ordinal moves in Qi are Silver indiscernibles; and so that I’s natural number moves in yβ i are the ones given by the imaginary opponent. Suppose we never reach an i < ω which satisfies conditions (a)–(c) of Section 3B (1). Gcont (ν, C) then ends with mega-round β. We must check that the end position is won by player II, in other words we must check that $xβ , yβ % ∈ C ∗ . The sequences Qi are positions in Martin’s game (kβ ◦ j0,β )(C)[xβ , yβ ]. Note that the positions Qi cannot be shifted and combined to form an infinite run of (kβ ◦ j0,β )(C)[xβ , yβ ]. This is because infinite runs of Hβ∗ are won by player I, yet τβ∗ is a winning strategy for player II. It follows from this that $xβ , yβ % does not belong to C ∗ , as required. The argument is standard; let us only refer the reader to Fact 2D.3. # 3C (2) Determinacy. To complete the proof that Gcont (ν, C) is determined, it is now sufficient to verify that the hypotheses of Lemmas 3B.9 and 3C.2 cannot both fail. Suppose they do. In other words suppose that I does not win G∗ini and II does not win ∗ . We shall derive a contradiction. We work inside M[g] throughout. In particular, Hini when we say a small (relative to κ) generic extension of M we mean a small extension M[g] ¯ with g¯ ∈ M[g]. Let γL equal cardM (M&δ + ω + 1). Let γH be the successor in M of γL . This is precisely jumpM (δ). Claim 3C.3. Let x be a code belonging to a small generic extension of M. Let P ∈ (M&δ)<ω and s ∈ ω<ω be given, with ν˜ (s) defined and lh(s) ≥ lh(P ). Then I wins G∗main (x, s, P , γL ) iff I wins G∗main (x, s, P , γH ). A similar claim holds ∗ . for Hmain Proof. We prove the claim for G∗main . Notice that the only dependence of G∗main on γ is through rule (1b) on page 91. Increasing γ in this rule simply adds more possible moves for I. It’s clear therefore that G∗main (. . . , γ ) becomes easier for I as γ increases. Working in M[g] let Uγ be the set of triples $x, s, P % so that x is a code belonging to a small generic extension of M, P ∈ (M&δ)<ω , s ∈ ω<ω with ν˜ (s) defined and lh(s) ≥ lh(P ), and player I wins G∗main (x, s, P , γ ). From the discussion in the previous paragraph it follows that Uγ ⊂ Uγ wherever γ < γ . In other words $Uγ | γ ∈ ON% is ⊂-increasing. Using a Skolem hull and absoluteness argument one can check that $Uγ | γ ∈ ON% stabilizes before the successor of cardM (M&δ + ω) in M. The claim follows. # ˙ L ], δ, and X = M&κ. Let σgen [γL ] be the generic strategies map associated to A[γ ˙ L ], δ, and M&κ. Let τgen [γL ] be the mirrored generic strategies map associated to B[γ Definition 3C.4. A foothold is a quadruplet x, s, P , Q so that: (1) x is a code, for a position yξ | ξ < α say, which belongs to a small, relative to κ, generic extension of M; (2) s ∈ ω<ω , ν˜ (s) is defined, ν˜ (s) ∈ {ν(yξ ) | ξ < α}, and lh(s) ≥ ν˜ (s);
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(3) P is a position of ν˜ (s) rounds in A[γL , x], played according σgen [γL , x]; (4) player II wins G∗main (x, s, P , γL ); (5) Q is a position of ν˜ (s) rounds in B[γL , x], played according τgen [γL , x]; ∗ (x, s, Q, γ ). (6) player I wins Hmain L
We think of P and Q as sequences of length ν˜ (s). Note that lh(P ) = ν˜ (s) ≤ lh(s), so the reference to G∗main (x, s, P , γL ) in condition (4) makes sense. The reference to ∗ (x, s, Q, γ ) in condition (6) makes sense similarly. Hmain L Condition (4) in Definition 3C.4 is an indication that $yξ | ξ < α% $s% should, at least in some weak sense, be a winning position for II in Gcont (ν, C). Condition (6) on the other hand is some indication that $yξ | ξ < α% $s% should be a winning position for I. Our expectation is that these two conditions combined should lead to a contradiction. We should be able to reach this contradiction by progressing from one foothold to the next. A progression of this kind could never reach an end position in Gcont (ν, C), since that would be a position which is won by both players. On the other hand we shall see that the progression cannot go on forever. Definition 3C.5. Let x , s , P , Q and x, s, P , Q be two footholds, with x equal to yξ | ξ < α , and x equal to yξ | ξ < α say. x , s , P , Q extends x, s, P , Q if: (1) α > α; (2) yξ = yξ for all ξ < α; (3) yα extends s (hence ν(yα ) = ν˜ (s)); (4) ν(yξ ) ≥ ν˜ (s) for all ξ > α; (5) P and P agree. Precisely: if ν˜ (s) ≤ ν˜ (s ) then P is an initial segment of P , and if ν˜ (s ) ≤ ν˜ (s) then P is an initial segment of P ; and (6) Q and Q agree. Conditions (2), (3), and (4) together imply that x ˜ν (s) is equal to x˜ν (s). It follows that A[γL , x] and σgen [γL , x] agree with A[γL , x ] and σgen [γL , x ] on the first ν˜ (s) rounds. Condition (5) therefore makes sense, and similarly with condition (6). Claim 3C.6. Suppose that x, s, P , Q is a foothold. Say x = yξ | ξ < α . Then there are yα , a ∗ , and b∗ in M[g] so that: (1) yα is a real which extends s; (2) a ∗ is an infinite run of A[γL , x ∗ ], played according to σgen [γL , x ∗ ], where x ∗ = $yξ | ξ < α%−−, yα ; (3) a ∗ extends P ;
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(4) b∗ is an infinite run of B[γL , x ∗ ], played according to τgen [γL , x ∗ ]; and (5) b∗ extends Q. Proof. We work in M[g] to construct yα , a ∗ , and b∗ maintaining the following conditions for each m ≥ ν˜ (s): (i) yα m is compatible with s; (ii) a ∗ m is a position in A[γL , x ∗ ] played according to σgen [γL , x ∗ ]; (iii) player II wins G∗main (x, max{s, yα m}, a ∗ m, γL ); (iv) b∗ m is a position in B[γL , x ∗ ] played according to τgen [γL , x ∗ ]; and ∗ (x, max{s, y m}, b ∗ m, γL ). (v) player I wins Hmain α
We start with m = ν˜ (s), setting yα m = sm; a ∗ m = P ; and b∗ m = Q. Conditions (ii)–(v) follow directly from conditions (3)–(6) in Definition 3C.4, since x and x ∗ agree to ν˜ (s). Suppose now that yα m, a ∗ m, and b∗ m are known, and that conditions (i)–(v) hold for m. Using condition (iii) and Claim 3C.3 fix a winning strategy σm∗ for II in G∗main (x, max{s, yα m}, a ∗ m, γH ). Note the switch from γL to γH here. Working ∗ (x, max{s, y m}, b ∗ m, γH ). similarly with (v) fix a winning strategy τm∗ for I in Hmain α We construct moves in round m of G∗main (x, max{s, yα m}, a ∗ m, γH ), and in round ∗ (x, max{s, y m}, b ∗ m, γH ): m of Hmain α • We play γm∗ = γL in both games. This is a legal move since γL < γH . • τm∗ (if m is even) or σm∗ (if m is odd) plays yα (m). ∗ ∗ and bm−II . • σgen [γL , x ∗ ] and τgen [γL , x ∗ ] respectively play am−I ∗ ∗ . and bm−I • σm∗ and τm∗ respectively play am−II
This completes the construction. The position given by a ∗ m + 1, yα m + 1, and γL must be a winning position for II in G∗main (x, max{s, yα m}, a ∗ m, γH ), since σm∗ is a winning strategy for II in that game. Condition (iii) for m + 1 follows from this using Remark 3B.5. A similar argument gives condition (v). The other conditions are immediate. # Claim 3C.7. Suppose x ∗ , a ∗ , b∗ ∈ M[g] are such that: (1) x ∗ is a code, x ∗ = yξ | ξ < λ say; (2) a ∗ is an infinite run of A[γL , x ∗ ], played according to σgen [γL , x ∗ ]; (3) b∗ is an infinite run of B[γL , x ∗ ], played according to τgen [γL , x ∗ ].
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Then there exists some s ∗ ∈ ω<ω so that: (1) ν˜ (s ∗ ) is defined, ν˜ (s ∗ ) ∈ {ν(yξ ) | ξ < λ}, and lh(s ∗ ) ≥ ν˜ (s ∗ ); (2) player II wins π(G∗main )(x ∗ , s ∗ , π(P ∗ ), π(γL )) in N[g], where P ∗ denotes a ∗ ˜ν (s ∗ ); ∗ )(x ∗ , s ∗ , π(Q∗ ), π(γ )) in N[g], where Q∗ denotes (3) player I wins π(Hmain L b∗ ˜ν (s ∗ ).
In conditions (2) and (3), N denotes Ult(M, Eord(x ∗ ) ) and π : M → N denotes the ultrapower embedding. Note then that x ∗ belongs to N[g]. Proof. Using Lemma 1B.2 and condition (2) in the hypothesis of the claim we see ˙ L ][g]. By Definition 3B.3, and since both a ∗ and x ∗ belong to that $ a ∗ , x ∗ % ∈ A[γ a ∗ , x ∗ , γL ) in M[g]. Since G∗ ( a ∗ , x ∗ , γL ) is M[g], this means that I does not win G∗ ( ∗ a closed game, it must be that II wins. Fix then σ ∈ M[g], a winning strategy for II in G∗ ( a ∗ , x ∗ , γL ). Working similarly with condition (3) of the hypothesis of the claim ∗ fix τ ∈ M[g], a winning strategy for I in H ∗ (b∗ , x ∗ , γL ). a ∗ , x ∗ , γL ) and H ∗ (b∗ , x ∗ , γL ) are played according to the same rules exactly. G∗ ( The only difference between the games is in the payoff when reaching an i < ω which satisfies conditions (a)–(c) in Section 3B(1). We can thus pit σ ∗ (playing for II in Diagram 3.3) against τ ∗ (playing for I). They must at some point reach an i < ω which satisfies conditions (a)–(c) in Section 3B(1); for otherwise they play the exact same game, with the exact same payoff, yet they both win. Now both G∗ ( a ∗ , x ∗ , γL ) and H ∗ (b∗ , x ∗ , γL ) end at the first i which satisfies conditions (a)–(c). Let s ∗ = yα i be the position reached when the games end. Let P ∗ = a ∗ ˜ν (s ∗ ) and let Q∗ = b∗ ˜ν (s ∗ ). Since σ ∗ is a winning strategy for II in G∗ ( a ∗ , x ∗ , γL ), we know that N[g] |= ∗ ∗ ∗ φmain (π(A))[x , s , π(P ), π(γL )]. In other words, player I does not win the game π(G∗main )(x ∗ , s ∗ , π(P ∗ ), π(γL )) in N [g]. This game is clopen, hence determined in N [g]. So it must be won by II. A similar argument using the fact that τ ∗ is winning for I in H ∗ (b∗ , x ∗ , γL ) shows ∗ )(x ∗ , s ∗ , π(Q∗ ), π(γ )) in N[g]. # that player I wins π(Hmain L Claim 3C.7 can be thought of as producing s ∗ so that x ∗ , s ∗ , π(P ∗ ), π(Q∗ ) is a foothold in the sense of N. Let us now reflect this to obtain some x so that x , s ∗ , P ∗ , Q∗ is a foothold in the sense of M. We have to replace x ∗ , which only belongs to M[g], by some approximation x which belongs to a small generic extension of M. The next claim does this. Conditions (2)–(6) specify the ways in which x “approximates” x ∗ . Claim 3C.8. Continue with the terminology of Claim 3C.7. Let s ∗ be given by the conclusion of that claim. Suppose that α < λ (strictly) is such that $yξ | ξ < α% belongs (not only to M[g] but also) to a small generic extension of M. Let j < ω be given. Then there is code x , for a sequence $yξ | ξ < λ % say, which satisfies:
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(1) x , s ∗ , P ∗ , Q∗ is a foothold (in particular x belongs to a small generic extension of M); (2) λ > α strictly; (3) yξ = yξ for all ξ < α; (4) yα j is equal to yα j ; (5) x j = xj ; and (6) {˜ν (yξ ) | ξ < λ } ∩ j is equal to {˜ν (yξ ) | ξ < λ} ∩ j . Proof. Increasing j if necessary we may assume that j ≥ ν˜ (s ∗ ). Let r denote xj . Let I = {˜ν (yξ ) | ξ < λ} ∩ j . Let t = yα j . Note that I , r, and t are finite sequences, and so belong to M. Let θ[x , P , Q, γL , $yξ | ξ < α%] be the formula stating that: (i) x is a code for a position, $yξ | ξ < λ % say; (ii) the sequence coded by x extends $yξ | ξ < α% strictly; (iii) yα j = t; (iv) x j = r; (v) {yξ | ξ < λ } ∩ j = I ; (vi) {ν(yξ ) | ξ < λ } does not have ν˜ (s ∗ ) as an element; (vii) player II has a winning strategy in G∗main (x , s ∗ , P ∗ , γL ); and ∗ (x , s ∗ , Q∗ , γ ). (viii) player I has a winning strategy in Hmain L
By assumption $yξ | ξ < α% belongs to a small generic extension of M. Fix g¯ witnessing this. Note that π : M → N , the ultrapower embedding by Eord(x ∗ ) , extends to an embedding of M[g] ¯ into N[g]. ¯ $yξ | ξ < α% is not moved by this embedding. Now N [g] ¯ is a model of the statement (∗) “There exists a cardinal µ < π(κ) so that after forcing with col(ω, µ) over N one obtains some x so that θ[x , π(P ∗ ), π(Q∗ ), π(γL ), $yξ | ξ < α%] holds.” To see this take µ = δ and x = x ∗ . The formula θ holds with this assignment: Conditions (i)–(v) hold directly because x = x ∗ . Condition (vi) holds because of condition (1) in the conclusion of Claim 3C.7. Conditions (vii) and (viii) hold because of conditions (2) and (3) in the conclusion of Claim 3C.7. Using the elementarity of the embedding π : M[g] ¯ → N[g] ¯ we may pull the statement (∗) back to M[g]. ¯ We obtain some µ < κ, some g¯ which is col(ω, µ)-generic/M, and some x ∈ M[g¯ ] so that θ[x , P ∗ , Q∗ , γL , $yξ | ξ < α%] holds in M[g¯ ]. It is easy to see that x , s ∗ , P ∗ , Q∗ is a foothold. Conditions (1), (2), (4), and (6) of Definition 3C.4 hold because of the definition of θ and, in the case of condition (2),
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known facts about s ∗ . Conditions (3) and (5) hold because of known facts about P ∗ and Q∗ , more precisely because of conditions (2) and (3) in the hypothesis of Claim 3C.7. Note here the importance of clause (iv) in the definition of θ, which implies that x ∗ and x agree to j ≥ ν˜ (s ∗ ). # Corollary 3C.9. Suppose x, s, P , Q is a foothold. Then there exists a foothold x , s ∗ , P ∗ , Q∗ which extends x, s, P , Q. Proof. First apply Claim 3C.6. The claim produces a ∗ , b∗ , and yα which gives rise to x ∗ = $yξ | ξ < α%−−, yα . Now apply Claim 3C.7 with λ = α + 1. Follow this with an application of Claim 3C.8, using j = lh(s). (Note then that j ≥ ν˜ (s).) It is easy to see that the resulting foothold extends x, s, P , Q. # Equipped with the previous claims we can begin our quest for a contradiction. We know to begin with that both the hypothesis of Lemma 3B.9 and the hypothesis of ∗ . We shall Lemma 3C.2 fail, in other words I does not win G∗ini and II does not win Hini use this knowledge to obtain a foothold. We shall then use the previous claims to extend this foothold to the point of contradiction. ∗ ). There exists Claim 3C.10 (assuming that I does not win G∗ini and II does not win Hini a foothold.
Proof. Begin by constructing a ∗ , b∗ ∈ M[g] so that: (i) a ∗ is an infinite run of A[γL , ∅], played according to σgen [γL , ∅]; and (ii) b∗ is an infinite run of B[γL , ∅], played according to τgen [γL , ∅]. The construction is similar to the construction of yα , a ∗ , and b∗ in the proof of ∗ instead of G∗ ∗ Claim 3C.6, only using G∗ini and Hini main and Hmain . We leave the details to the reader, and only note that the construction is based on the condition that II wins G∗ini ∗ . This condition follows from the initial assumption for contradiction and I wins Hini since both games are clopen and hence determined. Next apply Claim 3C.7 with x ∗ = ∅. Let s ∗ ∈ ω<ω be given by that claim. Let ∗ P = a ∗ ˜ν (s ∗ ) and let Q∗ = b∗ ˜ν (s ∗ ). Let N = Ult(M, Eord(∅) ) and let π be the ultrapower embedding. The conclusion of Claim 3C.7 is such that: (iii) player II wins π(G∗main )(x ∗ , s ∗ , π(P ∗ ), π(γL )); and ∗ )(x ∗ , s ∗ , π(Q∗ ), π(γ )). (iv) player I wins π(Hmain L
Since x ∗ = ∅ belongs to N, both games above in fact belong to N. Conditions (iii) and (iv) in fact hold in N. Pulling these conditions down to M using the elementarity of π we get: (v) player I wins G∗main (x ∗ , s ∗ , P ∗ , γL ); and ∗ (x ∗ , s ∗ , Q∗ , γ ). (vi) player II wins Hmain L
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It is now easy to verify that x ∗ = ∅, s ∗ , P ∗ , Q∗ is a foothold.
#
An n-foothold is a foothold x, s, P , Q so that ν˜ (s) = n. Let n0 be least so that there exists an n0 -foothold. Note the implicit use of Claim 3C.10 here. Let x0 , s0 , P0 , Q0 be some n0 -foothold. Working inductively on k < ω let nk+1 be least so that there exists an nk+1 -foothold which extends xk , sk , Pk , Qk . Note the implicit use of Corollary 3C.9 here. Let xk+1 , sk+1 , Pk+1 , Qk+1 be some nk+1 -foothold which extends xk , sk , Pk , Qk . Claim 3C.11. Suppose k ≥ l. Then nk ≥ nl and xk+1 , sk+1 , Pk+1 , Qk+1 extends xl , sl , Pl , Ql . Proof. We work by induction on k. Fix l ≤ k < ω. We first show that nk ≥ nl . If l = 0 this follows immediately from the minimality condition in the definition of n0 . So suppose l > 0. By induction xk , sk , Pk , Qk extends xl−1 , sl−1 , Pl−1 , Ql−1 . By the minimality condition in the definition of nl it follows immediately that nk ≥ nl . Next let us show that xk+1 , sk+1 , Pk+1 , Qk+1 extends xl , sl , Pl , Ql . If l = k this follows immediately from the definition of xk+1 , sk+1 , Pk+1 , Qk+1 . So suppose l < k. We know by induction that: (i) xk , sk , Pk , Qk extends xl , sl , Pl , Ql . We know by definition that: (ii) xk+1 , sk+1 , Pk+1 , Qk+1 extends xk , sk , Pk , Qk . We know from the previous paragraph that: (iii) nk ≥ nl . Using conditions (i)–(iii) it is easy to verify that xk+1 , xk+1 , Pk+1 , Qk+1 extends xl , sl , Pl , Ql . We refer the reader directly to Definition 3C.5, and only comment that condition (iii) is used in the verification. # The sequence coded by xk+1 extends the sequence coded by xk . Let $yξ | ξ < λ% be the union of the sequences coded by xk , k < ω. Intuitively $yξ | ξ < λ% is a position given by a “maximal” union of footholds, maximal in the sense that {nk | k < ω} is made as dense as possible. We plan to get a contradiction by taking the construction a step further, and extending this “maximal” sequence. This will ultimately produce a natural number n∗ which could have been used in {nk | k < ω}, but wasn’t. Let x ∗ = yξ | ξ < λ. Let αk be the length of the sequence coded by xk . We have xk = yξ | ξ < αk . Note that ν(yξ ) ≥ nk for all ξ ≥ αk . This follows from Claim 3C.11 and conditions (3) and (4) in Definition 3C.5. Using Claim 3A.4 it follows that xk and x ∗ agree to nk . Pk is a position of nk rounds in A[γL , xk ], played according to σgen [γL , xk ]. Using the agreement between x ∗ and xk it follows that Pk is also a position of nk rounds in A[γL , x ∗ ], played according to σgen [γL , x ∗ ]. By similar reasoning, Qk is a position of nk rounds in B[γL , x ∗ ], played according to τgen [γL , x ∗ ]. Claim 3C.12. Suppose l < k. Then nl < nk strictly.
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Proof. We know already that nl ≤ nk . But nk = nl is impossible: By condition (2) in Definition 3C.4, nk ∈ {ν(yξ ) | ξ < αk }. In particular nk = ν(yαl ). But ν(yαl ) = ν˜ (sl ) # by condition (3) in Definition 3C.5. So nk = ν˜ (sl ). Using Claims 3C.11 and 3C.12 we see that Pk strictly extends Pl for l < k < ω, and similarly with Q. Let a ∗ = k<ω Pk and let b∗ = k<ω Qk . Then: (a) a ∗ is an infinite run of A[γL , x ∗ ], played according to σgen [γL , x ∗ ]; and (b) b∗ is an infinite run of B[γL , x ∗ ], played according to τgen [γL , x ∗ ]. Remark 3C.13. Each xk belongs to a small generic extension of M. This follows from condition (1) in Definition 3C.4. But we cannot expect the whole sequence $xk | k < ω% to belong to a small generic extension of M. The reason is the implicit use of g in the definition of $nk , xk , sk , Pk , Qk | k < ω%. The maps σgen and τgen are defined relative to g, and so g is implicitly used already in the very definition of a foothold. We therefore cannot expect $yξ | ξ < λ% and x ∗ to belong to a small generic extension of M. They need only belong to M[g]. Apply Claim 3C.7 to x ∗ , a ∗ , and b∗ . (Note that all three belong to M[g], since the entire construction takes place inside M[g].) Let s ∗ ∈ ω<ω be given by that claim. Let n∗ = ν˜ (s ∗ ). Note that n∗ ∈ {ν(yξ ) | ξ < λ} by Claim 3C.7. In particular n∗ ∈ {nk | k < ω}. This is the first whiff of a contradiction. n∗ should have been put into {nk | k < ω} by the minimality conditions in the construction of $nk , xk , sk , Pk , Qk | k < ω%. Let us make the contradiction precise. Let P ∗ = a ∗ n∗ and let Q∗ = b∗ n∗ . Let k < ω be large enough that n∗ < nk+1 . Such a k exists by Claim 3C.12. Apply Claim 3C.8 with α = αk and j = lh(sk ). The result is some code x so that x , s ∗ , P ∗ , Q∗ is a foothold, and so that x , s ∗ , P ∗ , Q∗ extends xk , sk , Pk , Qk . But n∗ = ν˜ (s ∗ ) is smaller than nk+1 . This contradicts the minimality condition in the definition of nk+1 . The contradiction was reached under the assumption that both the hypothesis of Lemma 3B.9 and the hypothesis of Lemma 3C.2 fail (see Claim 3C.10). We can thus conclude that one of these hypotheses holds true. Using the corresponding lemma it follows that Gcont (ν, C) is determined.
3D A slight improvement In the previous sections we proved the determinacy of Gcont (ν, C) for C in the pointclass <ω2 − 11 and a continuous ν. Here we improve the result slightly. We use the same large cardinal assumption to handle ν in the class 02 . Precisely we prove: Theorem 3D.1. Suppose that there exists a mildly iterable class model M with cardinals κ < δ so that (a) δ is a Woodin cardinal of M; (b) κ is strong to δ + 1 in M; and (c) M&δ + 1 is countable in V. Then the games Gcont (ν, C) are determined for all ν in the class 02 and all C in the pointclass <ω2 − 11 .
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It is not clear if this result is optimal. The conclusion could conceivably be strengthened to handle ν in classes above 02 . At the moment it is only known that it cannot be strengthened beyond 06 , see Neeman [35]. Theorem 3D.1 is the beginning of a hierarchy. For countable α ≥ 1 we say that “κ is strong to δ + α in M” if for every Z ∈ M&δ + α there exists some extender E ∈ M with crit(E) = κ and Z ∈ Ult(M, E). The following result generalizes Theorem 3D.1: Theorem 3D.2 (for countable α ≥ 1). Suppose that there exists a mildly iterable class model M with cardinals κ < δ so that (a) δ is a Woodin cardinal of M; (b) κ is strong to δ + α in M; and (c) M&δ + α is countable in V. Then the games Gcont (ν, C) are determined for all ν in the class 01+α and all C in the pointclass <ω2 − 11 . In both theorems the assumptions can be obtained from large cardinals in V using the methods described at the end of Appendix A. 3D (1) 02 functions. Let us prove Theorem 3D.1. Fix M, κ, and δ which satisfy the large cardinal assumptions listed in Section 3B. Fix a sequence E = $Eξ | ξ < lh(E)% as in Section 3B (1). We intend to imitate the determinacy proof of Sections 3B and 3C, but working now with ν : R → ω in the class 02 . Fix such a function ν. As usual we may assume that a real coding ν belongs to M. Work in M[g] where g is col(ω, δ)-generic/M. For each n < ω the pre-image of {n} under ν is 02 . Fix a list of closed sets {Ci | i < ω} large enough to generate all n. By this we mean that for each n < ω there is some In ⊂ ω so that ν −1 {n} for ν −1 {n} = i∈In Ci . We shall convert ν into a “continuous” function by unraveling the closed sets. We refer the reader to Martin [25] for the exact definitions of covers which unravel given sets. Here we shall quote only facts we need, and not even all of these. Let K0 be the tree ω<ω . Construct $Ki+1 , fi+1 , ϕi+1 % so that for each i < ω, $Ki+1 , fi+1 , ϕi+1 % is an i-covering of Ki which unravels fi−1 ◦ · · · ◦ f1−1 (Ci ). This can be done in such a way that: Fact 3D.3. Nodes of length at least 2i in the tree Ki are sequences of the form $d0 , . . . , dl−1 % where d0 , . . . , d2i−1 are real numbers and d2i , . . . , dl−1 are natural numbers. For a node s ∈ Ki of length 2i let Ki (s) denote the tree of sequences t ∈ Ki which extend s, or equal an initial segment of s. Let fi,0 denote f1 ◦ · · · ◦ fi . We alternate between thinking of fi,0 as a Lipschitz continuous map from [Ki ] into ωω , and as a map from Ki into ω<ω . From Fact 3D.3 we directly get the following claim: Claim 3D.4 (for nodes s ∈ Ki of length 2i). s, Ki (s), and fi,0 Ki (s) are all coded by real numbers. # From Claim 3D.4 and the strength assumptions on the sequence of extenders E (see Section 3B (1)) we get the following corollary:
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so Corollary 3D.5. For s ∈ Ki a node of length 2i, there exists some ξ < lh(E) that s, Ki (s), and fi,0 Ki (s) belong to Ult(M, Eξ )[g]. We use ord(s) to denote the least such ξ . # For each i < ω and each j < i, fi,0 −1 (Cj ) is clopen (see Martin [25]). In fact for w ∈ [Ki ], knowledge of w2(j + 1) suffices to determine whether w belongs to fi,0 −1 (Cj ). This, the fact that Kj +1 and Ki agree on nodes of length 2(j + 1), and the fact that fi,j +1 is the identity on such nodes, allow us to pick sets Sj for j < ω so that: Fact 3D.6. Sj is a set of nodes of length 2(j + 1) in Kj +1 . For every j < i < ω and every w ∈ [Ki ], w ∈ fi,0 −1 (Cj ) iff w2(j + 1) ∈ Sj . Let K∞ be the inverse limit of the trees Ki . Let f∞,i be the inverse limit embeddings. We alternate between thinking of f∞,i as a Lipschitz continuous map from [K∞ ] into [Ki ], and as a map from K∞ into Ki . We have the standard commutativity: f∞,0 = fi,0 ◦ f∞,i . Fact 3D.7. K∞ and Ki have the same nodes of length 2i. f∞,i is the identity on nodes of length 2i. Define a partial map ν˜ : K∞ → ω by setting: Definition 3D.8. ν˜ (s) = n iff there is some j so that s2(j + 1) ∈ Sj and j ∈ In . (Recall that In ⊂ ω are such that for each n, ν −1 {n} = j ∈In Cj .) The following claims can be deduced easily from the previous facts: Claim 3D.9. For each w ∈ [K∞ ], (ν ◦ f∞,0 )(w) = n just in case that ν˜ (wi) = n for all sufficiently large i. Proof. Note that (ν ◦ f∞,0 )(w) = n iff w belongs to f∞,0 −1 (Cj ) for some j ∈ In . By Fact 3D.6, w belongs to f∞,0 −1 (Cj ) iff w2(j + 1) ∈ Sj . # Claim 3D.10. Suppose that w ∈ [Ki ], and that ν˜ (w2i) is defined and equal to n. Then # (ν ◦ fi,0 )(w) = n. We can now imitate the determinacy proof of Sections 3B and 3C, only replacing the function ν˜ defined above for the one used in those sections. We briefly sketch the argument. Fix a payoff set C ⊂ ω<ω1 which is <ω2 − 11 in the codes. We aim to show that Gcont (ν, C) is determined by imitating the proof in Sections 3B and 3C. We begin by adapting the definition of the games G∗main (. . . ) of Section 3B (1). Let M[g] ¯ be a small generic extension of M. Let x = yξ | ξ < α be a code belonging ¯ to M[g]. ¯ Let P ∈ (M&δ)<ω . Let n = lh(P ). Let s be some finite sequence in M[g]. For simplicity assume that lh(s) ≥ n. Let K ∈ M[g] ¯ be a tree which contains s as a node. Let f ∈ M[g] ¯ be a Lipschitz embedding from [K] into ωω . We also think of f as a map from K into ω<ω . Suppose that:
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(∗) For every w ∈ [K] which extends s, (ν ◦ f )(w) is equal to n. Let sˆ denote the triplet $s, K, f %. We define a game G∗main (x, sˆ , P , γ ). The game is played inside M[g] ¯ according to Diagram 3.14 and rules (1)–(3) below. I II
γn∗
wα (n)
∗ an−I
∗ an−II
∗ γn+1
wα (n + 1)
∗ an+1−I
∗ an+1−II
······
Diagram 3.14. The game G∗main (x, sˆ , P , γ ).
We use m to index rounds in G∗main (x, sˆ , P , γ ). The game starts with m = lh(P ). ∗ , a∗ ∗ The objects played in round m are γm∗ , am−I m−II , and wα (m). We use Pn to de∗ ∗ ∗ ∗ note P , and inductively define Pm+1 = Pm −−, am−I , am−II . We let wα n = sn, and inductively define wα m + 1 = wα m−−, wα (m). Rule (2) below guarantees that wα belongs to [K] and extends s. We let yα = f (wα ). We let x ∗ denote $yξ | ξ < α%−−, yα . (1) (Rule for I) γm∗ is an ordinal chosen so that: (a) Pm∗ is a legal position in A[γm∗ , x ∗ ]; (b) γm∗ < γ if m = n; and ∗ (c) γm∗ < γm−1 if m > n.
(2) (Rule for I and II) wα (m), played by I if m is even and by II if m is odd, is subject to the restriction that wα m−−, wα (m) is a node in K. For m < lh(s) we require further that wα (m) = s(m). ∗ ∗ (3) (Rule for I and II) am−I and am−II are legal moves in the game A[γm∗ , x ∗ ], fol∗ lowing the position Pm .
The first player to violate any of the rules of G∗main (x, s, P , γ ) loses. Because of rule (1c) there are no infinite runs of this game. The reader should compare the definition here with the definition given in Section 3B (1). The natural number moves yα (m) of Section 3B (1) are replaced here by moves on the tree K. Moves on K are translated to natural numbers via the Lipschitz map f . Note how the assumption (∗) above replaces the assumption in Section 3B (1) that “˜ν (s) is defined.” The current assumption has a similar effect. It guarantees that ν(yα ) is known from the start—here it must in fact equal n—and does not depend on the moves wα (m). This implies that x ∗ depends in a Lipschitz manner on yα , which in turn depends in a Lipschitz manner on wα . To sum, knowledge of wα m suffices to determine x ∗ m. Rules (1a) and (3) therefore make sense.
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a , x, γ ). We Continuing the parallel with Section 3B (1) let us define the games G∗ ( ω work with a ∈ (M&δ) ∩ M[g], a code x = yξ | ξ < α in M[g], and an ordinal γ . a , x, γ ) is played inside M[g] according to Diagram 3.15. G∗ ( I II
wα (0), ρ0
wα (2), ρ2 wα (1), ρ1
... wα (3), ρ3
...
Diagram 3.15. G∗ ( a , x, γ ), so long at ν˜ (wα i) is not defined or belongs to {nξ | ξ < α}.
Players I and II collaborate to produce an infinite branch wα ∈ [K∞ ]. K∞ here is the covering tree constructed above. Player I plays wα (m) for even m < ω, and player II plays wα (m) for odd m < ω. We set yα = f∞,0 (wα ). In addition to playing wα , I and II play auxiliary moves in the game C[x, yα ]. (Note that f∞,0 is Lipschitz, so knowledge of wα m suffices to determine yα m. Knowledge of yα m in turn suffices to determine the rules of the first m rounds of C[xα , yα ].) They continue this way until, if ever, reaching an i < ω so that: (a) ν˜ (wα 2i) is defined; (b) ν˜ (wα 2i) ∈ {nξ | ξ < α}; and (c) 2i ≥ ν˜ (wα 2i). The map ν˜ : K∞ → ω used here is the one given by Definition 3D.8. The first player to violate any of the rules of C[x, yα ] loses. Infinite runs where an i < ω satisfying conditions (a)–(c) was not reached are won by player I. If an i < ω satisfying conditions (a)–(c) is reached, the game ends. We set s = wα 2i, n = ν˜ (s), and P = a n. We let N = Ult(M, Eord(s) ) and let π be the ultrapower map. (See Corollary 3D.5 for the definition of ord(s).) We let K = Ki (s) and let f = fi,0 Ki (s). Set sˆ = $s, K, f %. I wins just in case that N[g] |=“Player I has a winning strategy in π(G∗main )(x, sˆ , π(P ), π(γ )).” Remark 3D.11. Both Corollary 3D.5 and Claim 3D.10 are needed to make sense of the use of π(G∗main )(x, sˆ , π(P ), π(γ )) in the payoff condition. Corollary 3D.5 implies that sˆ belongs to N [g]. Claim 3D.10 implies that the condition (∗) preceding the definition of G∗main (. . . ) holds for sˆ . The reader should compare the game G∗ (x, a , γ ) described here with the game described in Section 3B(1). Here instead of directly playing the real yα , I and II play on the tree K∞ . This allows use of the function ν˜ of Definition 3D.8. By passing to the cover K∞ we have essentially converted the 02 function ν to the continuous function given by ν˜ . This is a good place to end our brief sketch of the proof of Theorem 3D.1. The rest of the proof is a matter of following the argument of Sections 3B and 3C, using the revised games described above, and using the techniques of Martin [25] to convert strategies on K∞ or Ki to strategies on natural numbers.
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Exercise 3D.12. Complete the proof of Theorem 3D.1. 3D (2) 01+α functions. The proof of Theorem 3D.2 is similar to the proof of Theorem 3D.1. But with Theorem 3D.2 one has to unravel sets which reside higher up in the Borel hierarchy. One therefore has to allow for covers which reside higher up in the von-Neumann hierarchy; Fact 3D.3 no longer holds when the sets being unraveled are more complicated than closed. Fact 3D.3 led us to Claim 3D.4, which in turn combined with our large cardinal strength assumption to give Corollary 3D.5. Corollary 3D.5 was essential to the definitions in Section 3D(1), see specifically Remark 3D.11. To maintain the validity of Corollary 3D.5 when dealing with covers of greater size, we simply have to increase our large cardinal assumption. We have to make sure that for each of the trees Ki in the covers used, and for each node s ∈ Ki of length 2i, there is some extender E ∈ M so that: (1) crit(E) = κ; (2) E is at least δ-strong; and (3) s, Ki (s), and fi,0 Ki (s) all belong to Ult(M, E)[g]. So long as our large cardinal assumption is strong enough to secure these conditions, we can run the proof described in Section 3D(1) essentially unchanged. Condition (3) is the one which relates the necessary large cardinal strength to the complexity of ν. Readers familiar with Martin [25] can easily compute the size of the trees Ki needed to handle a function ν in a given Borel class, and thereby connect the complexity of ν with a large cardinal assumption on M. We leave this computation to those readers and only note that the end result is Theorem 3D.2.
3E Variation In the spirit of the previous chapter we develop here a mechanism which reduces games of continuously coded length to iteration games. This mechanism will allow us to compose games of continuously length with games of the kind handled in Chapter 2, and to prove determinacy for games with universally Baire payoff. Let ν : R → ω be given. By position we mean a ν-position and by run we mean a ν-run, see Section 3A. Let M be a ZFC∗ model with a Woodin cardinal δ∞ . Let A˙ be a name for a subset of cont = G cont (M, δ∞ , ν, A) ˙ in which players I R × R in M col(ω,δ∞ ) . We define a game G and II collaborate to construct: • a run $yξ | ξ ≤ α%;
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• a sequence of models of the form presented in Diagram 3.16, starting with M0 = M; and • embeddings jζ,ξ : Mζ → Mξ for ζ ≤ ξ ≤ α, kξ : Mξ → Qξ for ξ ≤ α, and k∞ : Qα → M∞ , of the form presented in Diagram 3.16.
j0,1
M0
j1,2
kkk kk SS k SSSS k / Q0 0
F0
#
kkk kk SS k SSSS k / Q1 1
M1
T0
F1
#
M2
T1
jξ,ξ +1
Mξ
kkk Sk k Sk SSSS k / ξ
Qξ
Fξ
#
Mξ +1
Mα
Tξ
kk kk / Qα kSkSkSkS / kSkSkSkS SS kα SS k∞ Tα
M∞
T∞
cont . Diagram 3.16. The iterates in G
For limit λ we let Mλ be the direct limit of the models Mξ for ξ < λ. M0 is set equal to M. Mega-round λ, when λ is a limit or 0, is played as follows: (L1) Player I plays a length ω iteration tree Tλ on Mλ . (L2) Player II plays a cofinal branch bλ through Tλ . This completes mega-round λ. We let Qλ be the direct limit along bλ , and let kλ be the corresponding embedding. At the start of a successor mega-round β + 1 we know already the iterate Qβ . Mega-round β + 1 proceeds as follows: (S1) Players I and II collaborate in the usual fashion to produce the real yβ . If the real yβ produced following rule (S1) satisfies the condition (∗) ν(yβ ) is defined and ν(yβ ) ∈ {ν(yξ ) | ξ < β}, then mega-round β + 1 continues according to rules (S2)–(S4). (S2) Player I plays an extender Fβ ∈ Qβ with critical point smaller than crit(kβ ). We set Mβ+1 = Ult(Mβ , Fβ ) and let jβ,β+1 be the ultrapower embedding. (S3) Player I plays a length ω iteration tree Tβ+1 on Mβ+1 . (S4) Player II plays a cofinal branch bβ+1 through Tβ+1 .
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3 Games of continuously coded length
This completes mega-round β + 1. We let Qβ+1 be the direct limit along bβ+1 , and let kβ+1 be the direct limit embedding. We are then in a position to begin mega-round β + 2. If the real yβ produced following rule (S1) does not satisfy the condition (∗) above, then mega-round β + 1 continues with rules (F1) and (F2) instead of rules (S2)–(S4). Note that mega-round β + 1 finishes the game in this case. (F1) Player I plays a length ω iteration tree T∞ on Qβ . (F2) Player II plays a cofinal branch b∞ through T∞ . We let M∞ be the direct limit along b∞ , and let k∞ be the direct limit embedding. We let j0,∞ : M0 → M∞ be equal to k∞ ◦ kα ◦ j0,α . By the failure of condition (∗) we know that $yξ | ξ ≤ β% is a run—a terminal stage in the game of continuously coded cont (M, δ∞ , ν, A) ˙ ends at this point. If M∞ or any of the length associated to ν. G models produced along the way is illfounded, then player I wins. Otherwise player I wins just in case that there exists some h so that: (1) h is col(ω, j0,∞ (δ∞ ))-generic/M∞ ; and ˙ where xβ = yξ | ξ < β. (2) $xβ , yβ % ∈ j0,∞ (A)[h], cont (M, δ∞ , ν, A). ˙ As usual The above rules complete the description of the game G ˙ we define a mirrored version of this game. Let B be a name for a subset of R × R in cont (M, δ∞ , ν, B) ˙ is played according to the rules listed above except M col(ω,δ∞ ) . H that it is now player II who plays the extenders in rule (S2) and the trees in rules (L1), (S3) and (F1); while I is the player responsible for branches in rules (L2), (S4) and (F2). It is now player II who wins if any of the models produced is illfounded. If all the models produced are wellfounded, then player II wins just in case that there exists ˙ an h which satisfies condition (1) above and condition (2) with A˙ replaced by B. cont (M, δ∞ , ν, A) ˙ should be viewed as a reduction mechanism A strategy for I in G which converts an iteration strategy for M into a strategy for player I in the game of cont (M, δ∞ , ν, B) ˙ continuously coded length associated to ν. A strategy for II in H should similarly be viewed as a reduction mechanism which converts an iteration strategy for M into a strategy for player II in the game of continuously coded length. The reader should compare this to Section 2A. The next theorem should be compared to Theorem 2A.2. Theorem 3E.1. Suppose that there are κ < δ < δ∞ so that (a) δ and δ∞ are Woodin cardinals of M; and (b) κ is strong to δ + 1 in M. Suppose that ν belongs to the class 02 and that (a code for) ν belongs to M. Suppose that M&δ∞ + 1 is countable in V. Fix g∞ ∈ V which is col(ω, δ∞ )-generic/M. At least one of the following three cases holds: cont (M, δ∞ , ν, A); ˙ (1) player I has a winning strategy in G cont (M, δ∞ , ν, B); ˙ or (2) player II has a winning strategy in H
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(3) in M[g∞ ] there is a run $yξ | ξ ≤ α% so that $yξ | ξ < α, yα % belongs to ˙ ˙ neither A[g] nor B[g]. Moreover M can distinguish this. To be precise, there are formulae φI and φII so that: ˙ then case (1) holds; if M |= φII [δ∞ , ν, B] ˙ then case (2) holds; if M |= φI [δ∞ , ν, A] and otherwise case (3) holds. We sketch a proof of Theorem 3E.1 in Section 3E (1). Let us here notice a few corollaries. Corollary 3E.2. Suppose that there exists a mildly iterable class model M with cardinals κ < δ0 < δ1 so that (a) δ0 and δ1 are Woodin cardinals of M; (b) κ is strong to δ0 + 1 in M; and (c) M&δ1 + 1 is countable in V. Then the games Gcont (ν, C) are determined for all ν in the class 02 and all C in the pointclass 12 . Proof. Fix ν and C. Let C ∗ witness that C is 12 in the codes. Passing to a generic extension of an iterate of M if needed, we may assume that a code for ν exists in M and that C ∗ is 21 (z) for some real z which exists in M. Let A˙ ∈ M be a col(ω, δ1 ) name for the set of pairs $x, y% so that x is a code for a ν-position, y is a real, and $x, y% satisfies the 21 (z) statement which defines C ∗ . Let B˙ name the set of pairs which do not satisfy that 21 (z) statement. be a winning Apply Theorem 3E.1 with δ∞ = δ1 . Suppose case (1) holds. Let ˙ strategy for player I in Gcont (M, δ1 , ν, A). Let be a mild iteration strategy for M. and combine in the natural way to produce a strategy for player I in Gcont (ν, C). It is easy to check that this is a winning strategy for player I. This uses 21 absolute cont , and the fact that -iterates of M are always ness, condition (2) in the payoff of G wellfounded. If case (2) of Theorem 3E.1 holds, a similar argument shows that player II has a winning strategy in Gcont (ν, C). Case (3) of Theorem 3E.1 is impossible, since A˙ and B˙ name complementary sets. # Corollary 3E.3. Suppose that κ < λ0 < λ1 are such that (a) λ0 and λ1 are Woodin cardinals in V; and (b) κ is strong to λ0 + 1, again in V. Let ν be 02 and let C be λ1 -universally Baire in the codes. Then Gcont (ν, C) is determined. Proof sketch. This is simply a matter of applying the methods of Section 2E, more specifically Exercise 2E.11, to the current context. The argument involves referring to Theorem 3E.1 where the solution to Exercise 2E.11 referred to Theorem 2A.2, and using Claim 14 in Appendix A in addition to Theorem 12. # Corollary 3E.4. Let cont be the game quantifier associated to the class of games Gcont . Suppose that κ < λ0 < λ1 satisfy the assumptions in Corollary 3E.3. Let ν be 02 and ¯ Baire for let C be λ1 -universally Baire in the codes. Then cont (ν, C) is κ-universally each κ¯ < κ. Proof sketch. Refer to Exercises 2E.9 and 2E.12, and apply the arguments there to the current context. #
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Let θ be a countable ordinal. Let C ∗ be a subset of R1+θ . Gcont−1+θ (ν, C ∗ ) is played as follows: Players I and II first follow the game of continuously coded length associated to ν, and then continue playing −1 + θ additional reals. More precisely players I and II collaborate as usual to produce a sequence of reals $yξ | ξ < α + θ% so that (a) for each ξ < α, ν(yξ ) is defined and is not an element of {ν(yζ ) | ζ < ξ }; and (b) ν(yα ) is not defined, or else it is an element of {ν(yξ ) | ξ < α}. The game is won by player I iff yξ | ξ < α, −−$yα+ξ | ξ < θ% belongs to C ∗ . Combining Theorem 3E.1 with the results connected to Theorem 2A.2 one easily sees that: Corollary 3E.5 (for a countable ordinal θ ≥ 1). Suppose that there exists a mildly iterable class model M, an increasing sequence $δξ | ξ < θ%, and κ < δ0 so that (a) each δξ is a Woodin cardinals of M; (b) κ is strong to δ0 + 1 in M; and (c) M& sup{δξ + 1 | ξ < θ} is countable in V. Then the games Gcont−1+θ (ν, C ∗ ) are # determined for all ν in the class 02 and all C ∗ in the pointclass <ω2 − 11 . Similar combinations are possible also with the current results, namely with the results of this chapter. For example one can combine Theorem 3E.1 with itself to obtain the determinacy of games of the kind Gcont+cont . (The rules of Gcont+cont are phrased in the obvious way.) While we are on the subject of variations let us also mention the following result, on games where the payoff is homogeneously Suslin (see Martin–Steel [18, §2] for the definition). The result has nothing to do with the methods of the current section. Rather it involves an adaptation of the argument in Sections 3B and 3C. We give it as an exercise, meant for readers who are familiar with homogeneously Suslin trees and their uses in proofs of determinacy. Exercise 3E.6. Suppose that τ < λ are such that (a) λ is a Woodin cardinal in V; and (b) τ is strong to λ + 1, again in V. Let ν be 02 and let C be (λ + 1)-homogeneously Suslin in the codes. Prove that Gcont (ν, C) is determined. Hint. Let C ∗ witness that C is (λ + 1)-homogeneously Suslin in the codes. Precisely y α, yα % ∈ C ∗ iff y ∈ C for any run this means that C ∗ is a subset of R × R; $ ∗ y = $yξ | ξ < α + 1%; and C is (λ + 1)-homogeneously Suslin. Let the tree T and the countable collection of measures µ witness that C ∗ is (λ + 1)homogeneously Suslin. (In particular the projection of T is equal to C ∗ .) Let Vθ be a rank initial segment of V, large enough that λ, T , and µ all belong to Vθ . Let H be a all in H . Let M be the countable elementary substructure of Vθ , with τ , λ, T , and µ transitive collapse of H , let π : M → Vθ be the anti-collapse embedding, and let κ, δ, S, and µ¯ be such that π(κ) = τ , π(δ) = λ, π(S) = T , and π(µ) ¯ = µ. Let D be the map which assigns to each pair $x, y% ∈ R × R the following game D[x, y]: player I plays a sequence u(0), u(1), . . . so that $x, y, u% ∈ [S], thereby witnessing that $x, y% belongs to the projection of S, and player II does nothing. (This game is taken from the determinacy proof for homogeneously Suslin sets, see for example [18, Theorem 2.3].)
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Now follow the argument in Sections 3B and 3C, but modify it to use D instead of Martin’s games map C. First modify the definitions in Sections 3B (1) and 3C (1), to use D in the games G∗ (. . . ) and H ∗ (. . . ), instead of C. You will then have to modify the relevant part of the successor stage in Section 3B (3), to use D. On the one hand the argument becomes simpler, since only player I has moves in D[. . . ]. But on the other hand there is an extra detail: Your argument will show that the code $xβ , yβ % belongs to the projection of (kβ ◦ j0,β )(S). Through uses of Theorem 12 and Claim 14 in Appendix A you should be able to make sure that there is an embedding σ : Qβ → Vθ so that σ ◦ kβ ◦ j0,β = π . Use this to conclude that $xβ , yβ % belongs to the projection of T . Finally you will have to modify the argument made in the proof of Lemma 3C.2, ¯ which witness the homogeneity of (kβ ◦ j0,β )(S) in to use the measures (kβ ◦ j0,β )(µ) Qβ , instead of Silver indiscernibles for Qβ . Your argument will show that the tower of measures associated $xβ , yβ % is illfounded over Qβ . Through uses of Theorem 12 and Claim 14 in Appendix A you should again be able to make sure that there is an embedding σ : Qβ → Vθ so that σ ◦ kβ ◦ j0,β = π . Use σ and the illfoundedness of the tower associated to $xβ , yβ % over Qβ to conclude that the tower associated to $xβ , yβ % over V is also illfounded, and therefore $xβ , yβ % does not belong to the projection of T . # 3E (1) A sketch of the proof. Let M be a model of ZFC∗ . Let δ∞ be a Woodin cardinal of M, and suppose that M&δ∞ + 1 is countable in V. Let A˙ and B˙ be two names in M col(ω,δ∞ ) for subsets of R × R. We work to prove Theorem 3E.1 for these objects. Let κ < δ < δ∞ be cardinals of M so that δ is a Woodin cardinal, and κ is strong to δ + 1 in M. Our tactic is to run the argument of Sections 3B and 3C—or Section 3D (1) rather, when ν is 02 and not continuous—but replacing Martin’s game C with the ˙ auxiliary games maps associated to the names A˙ and B. ω col(ω,δ ) ∞ ˙ ˙ . Define B˙ ∞ similarly. Let A∞ = Let A∞ name (M&δ∞ ) × A in M (x, y → A∞ [x, y]) be the auxiliary games map associated to A˙ ∞ , δ∞ , and X = M&δ. Let B∞ = (x, y → B∞ [x, y]) be the mirrored auxiliary games map associated to B˙ ∞ , δ∞ , and X = M&δ. Remark 3E.7. The use of X = M&δ in these definitions is meant to ensure that the corresponding pivots later on reside above δ. Note the implicit use here of the fact that δ < δ∞ . For simplicity suppose that ν is continuous. We shall follow the argument of Sections 3B and 3C. The case of ν in the pointclass 02 is similar, following the modifications of Section 3D(1). ˙ ], and G∗ following SecDefine G∗main (. . . , γ ), G∗ (. . . , γ ), the names A[γ ini tion 3B (1) and Section 3B(3), except that the definition of G∗ ( a , x, γ ) is modified. a , x, γ ) is now played according to Diagram 3.17. Players I and II collaborate G∗ ( as usual to produce a real yα = $yα (i) | i < ω%. In addition they play auxiliary moves
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3 Games of continuously coded length
I II
yα (0)
∞ a0−I
∞ a0−II
∞ a1−I
yα (1)
∞ a1−II
yα (2)
... ...
Diagram 3.17. Modified G∗ ( a , x, γ ), so long at ν˜ (yα i) is not defined or belongs to {nξ | ξ < α}.
in the game A∞ [x, yα ]. They continue this way until, if ever, reaching an i < ω so that: (a) ν˜ (yα i) is defined; (b) ν˜ (yα i) ∈ {nξ | ξ < α}; and (c) i ≥ ν˜ (yα i). The first player to violate any of the rules of A∞ [x, yα ] loses. If an i < ω satisfying conditions (a)–(c) is reached, the game ends. As in Section 3B(1) we set s = yα i and P = a ˜ν (s). We let N = Ult(M, Eord(x) ) and let π be the ultrapower map. Player I wins iff N [g] |= φmain (π(A))[x, s, π(P ), π(γ )]. Other than that I is the open player; infinite runs where an i < ω satisfying conditions (a)–(c) was not reached are won by player II. The definition here differs from that in Section 3B (1) in two ways. First, the map A∞ is used instead of the map C. Secondly, infinite runs of G∗ (. . . , γ ) are now won by player II. cont (M, δ∞ , ν, A). ˙ Lemma 3E.8. Suppose that I wins G∗ . Then I wins G ini
Proof sketch. Fix an imaginary opponent willing to play for II in the game cont (M, δ∞ , ν, A). ˙ We work with the imaginary opponent to construct a run of this G game. The construction is a variant of the argument in Section 3B(3). The use of the iteration strategy in Section 3B (3) is replaced here by a use of the imaginary opponent. Note how the structure of the iteration trees in the argument of Section 3B (3) corresponds precisely to the structure of the trees in Diagram 3.16 up to Qα . The moves supplied by the imaginary opponent under rules (L2) and (S4) are therefore an adequate replacement for the branches supplied by the iteration strategy in Section 3B (3). Other than that only the successor stage in Section 3B (3) requires modification. So suppose we are at a successor stage β + 1. We have the model Qβ and the embedding kβ ◦ j0,β : M → Qβ . We have xβ coding the current position $yξ | ξ < β%. We have aβ and γβ given by the construction so far, see the successor stage in Section 3B (3). We know that conditions (S1) and (S2) of Section 3B (3) hold true. Thus player I wins aβ , xβ , γβ ). Let σβ∗ be a strategy witnessing this. We the game G∗β = (kβ ◦ j0,β )(G∗ )( ∗ may pick σβ in the model Qβ [hβ ]. Recall that hβ is generic for col(ω, (kβ ◦ j0,β )(δ)). Let Apiv−∞ and σpiv−∞ be the pivot games and strategies maps associated to A˙ ∞ , δ∞ , and X = M&δ. In Section 3B(3) we used the method of ascribing indiscernibles to form a run of G∗β played according to σβ∗ . Here we use σpiv−∞ instead.
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Using the initial assumption that M&δ∞ + 1 is countable in V, it is easy to check that Qβ &(kβ ◦ j0,β )(δ∞ ) + 1 is countable in V. Fix in V some surjection : ω → Qβ &(kβ ◦ j0,β )(δ∞ ) + 1. We work to form a real yβ and a run a∞,β , T∞,β of the pivot game (kβ ◦ j0,β )(Apiv−∞ )[xβ , yβ ]. The construction proceeds along the usual lines: (P1) the imaginary opponent produces yβ (i) for odd i; ∞,β
(P2) (kβ ◦ j0,β )(σpiv )[, xβ , yβ ] produces the moves ai−II for all i, and the extenders which give rise to T∞,β ; (P3) σβ∗ and its shifts along the even branch of T∞,β produce yβ (i) for even i, and the ∞,β
moves ai−I for all i. Remark 3E.9. As usual the construction requires shifting σβ∗ along the even branch of T∞,β . It is important for this shifting that σβ∗ belongs to a generic extension of Qβ of size (kβ ◦ j0,β )(δ), and that the extenders used in T∞,β all have critical points above (kβ ◦ j0,β )(δ). The latter fact traces back to Remark 3E.7, and to Lemma 1C.7. If an i < ω which satisfies conditions (a)–(c) in the definition of G∗ (. . . ) is reached, we must end the construction; the shift of G∗β ends at the first such i, and the shift of σβ∗ ceases to supply moves. We know in this case that condition (S3) in Section 3B (3) holds true. (Note the use of the fact that the extenders in T∞,β have critical points above (kβ ◦ j0,β )(δ). Initially we only know that a version of condition (S3) shifted to the 2i-th model of T∞,β holds true. But the critical point involved in the shift is high enough that this can be pulled back to Qβ .) Once we have condition (S3) we can simply return to the construction of Section 3B (3). So suppose that an i < ω which satisfies conditions (a)–(c) in the definition of ˙ in G∗ (. . . ) is not reached. Note that β + 1 is the final mega-round of G(M, δ∞ , ν, A) this case. We have so far produced the real yβ . To complete this final mega-round we must follow rules (F1) and (F2). We must then verify that the run constructed is won by player I. This will complete the proof of Lemma 3E.8. The construction in items (P1)–(P3) produces an infinite run a∞,β , T∞,β of (kβ ◦ j0,β )(Apiv−∞ )[xβ , yβ ], played according to (kβ ◦ j0,β )(σpiv−∞ )[, xβ , yβ ]. Play T∞,β as the move T∞ for player I in rule (F1). The imaginary opponent, playing according to rule (F2), produces a cofinal branch b∞ through this tree. Let M∞ be the direct limit along this branch, and let k∞ : Qβ → M∞ be the direct limit embedding. If M∞ or any other model along the way is illfounded, then player I wins and our task is complete. So suppose that M∞ and all the models along the way are wellfounded. Now σβ∗ is a strategy for player I, the open player, in G∗β . There are thus no infinite runs according to σβ∗ . But a∞,β is an infinite run according to the image of σβ∗ along the even branch of T∞ . It follows that the even branch leads to an illfounded direct limit. b∞ must therefore be an odd branch.
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I II
yα (0)
∞ b0−II
∞ b0−I
yα (1)
∞ b1−II
∞ b1−I
yα (2)
... ...
x, γ ), so long at ν˜ (yα i) is not defined or belongs to {nξ | ξ < α}. Diagram 3.18. Modified H ∗ (b,
a∞,β , T∞,β is a (kβ ◦ j0,β )(A∞ )-pivot for $xβ , yβ % by Lemma 1C.5. By Definition 1C.1 and since b∞ is an odd branch it follows that there must exist an h so that: (1) h is col(ω, (k∞ ◦ kβ ◦ j0,β )(δ∞ ))-generic/M∞ ; and (2) $ a∞,β , xβ , yβ % ∈ (k∞ ◦ kβ ◦ j0,∞ )(A˙ ∞ )[h]. The definition of A˙ ∞ is such that condition (2) can be re-written as: ˙ (2 ) $xβ , yβ % ∈ (k∞ ◦ kβ ◦ j0,∞ )(A)[h]. ˙ This precisely is the winning condition in G(M, δ∞ , ν, A).
#
∗ (. . . , γ ), H ∗ (. . . , γ ), the names B[γ ˙ ], and H ∗ by mirroring the defDefine Hmain ini initions given before Lemma 3E.8. The mirrored games H ∗ (. . . , γ ) use the map B∞ . Player II is the open player in these games. Other than that the mirroring process is similar to the one in Section 3C(1). By mirroring the proof of Lemma 3E.8 we get: ∗ . Then II wins H cont (M, δ∞ , ν, B). ˙ Lemma 3E.10. Suppose that II wins Hini
#
Lemmas 3E.8 and 3E.10 cover cases (1) and (2) in Theorem 3E.1. Let us set the ground for a proof of case (3) when the hypotheses of these lemmas fail. We intend in this case to follow a modification of the argument in Section 3C (2). Let γL and γH be as in Section 3C(2). Fix g∞ ∈ V which is col(ω, δ∞ )-generic/M. Fix g ∈ M[g∞ ] which is col(ω, δ)-generic/M. We work below with reference to these fixed objects. Recall that the definition of G∗main (. . . , γ ), modified to the current settings from ˙ ]. Recall further that Section 3B (1), involves the definition of col(ω, δ) names A[γ ˙ A[γ ] is the auxiliary games map associated to A[γ ], δ, and M&κ. The definition ∗ (. . . , γ ) similarly involves names B[γ ˙ ] and the associated mirrored auxiliary of Hmain ˙ L ], δ, games maps B[γ ]. Let σgen [γL ] be the generic strategies map associated to A[γ and M&κ. Similarly let τgen [γL ] be the mirrored generic strategies map associated to ˙ L ], δ, and M&κ. Note that both σgen [γL ] and τgen [γL ] belong to M[g]. B[γ Let σgen−∞ be the generic strategies map associated to A˙ ∞ , δ∞ , and X = M&δ. σgen−∞ is defined using the generic g∞ , and belongs to M[g∞ ]. Let τgen−∞ ∈ M[g∞ ] be the mirrored generic strategies map associated to B˙ ∞ , δ∞ , and X = M&δ. The next claim is an adaptation of Claim 3C.7 to the current definitions of A and B, definitions which rest on the modified games G∗ (. . . ) and H ∗ (. . . ) described above. Conditions (1)–(3) in the conclusion of Claim 3C.7 are replicated here in conditions (B1)–(B3). But here there is an extra clause, clause (A) in the conclusion of Claim 3E.11.
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Claim 3E.11. Suppose x ∗ , a ∗ , b∗ ∈ M[g] are such that: (1) x ∗ is a code, x ∗ = yξ | ξ < λ say; (2) a ∗ is an infinite run of A[γL , x ∗ ], played according to σgen [γL , x ∗ ]; (3) b∗ is an infinite run of B[γL , x ∗ ], played according to τgen [γL , x ∗ ]. Then either (A) there exists some yλ ∈ M[g∞ ] so that $yξ | ξ < λ% is a complete run of the game of continuously coded length associated to ν and so that $x ∗ , yλ % belongs ˙ ∞ ]; or else there exists some s ∗ ∈ ω<ω so that: ˙ ∞ ] nor B[g to neither A[g (B1) ν˜ (s ∗ ) is defined, ν˜ (s ∗ ) ∈ {ν(yξ ) | ξ < λ}, and lh(s ∗ ) ≥ ν˜ (s ∗ ); (B2) player II wins π(G∗main )(x ∗ , s ∗ , π(P ∗ ), π(γL )) in N[g], where P ∗ denotes a ∗ ˜ν (s ∗ ); ∗ )(x ∗ , s ∗ , π(Q∗ ), π(γ )) in N[g], where Q∗ denotes (B3) player I wins π(Hmain L ∗ ∗ b ˜ν (s ).
In conditions (B2) and (B3), N denotes Ult(M, Eord(x ∗ ) ) and π : M → N denotes the ultrapower embedding. Note then that x ∗ belongs to N[g]. Proof. We adapt the proof of Claim 3C.7 to the current situation. ˙ L ][g]. Since both a ∗ and By condition (2) and Lemma 1B.2, $ a ∗ , x ∗ % ∈ A[γ ∗ x belong to M[g], this means that I does not win G∗ ( a ∗ , x ∗ , γL ) in M[g]. Since ∗ ∗ ∗ a , x , γL ) is an open game, it must be that II wins. Fix then σ ∗ ∈ M[g], a winning G ( a ∗ , x ∗ , γL ). Working similarly with condition (3) fix τ ∗ ∈ M[g], strategy for II in G∗ ( a winning strategy for I in H ∗ (b∗ , x ∗ , γL ). a ∗ , x ∗ , γL ), and that As a side remark let us note that II is the closed player in G∗ ( I is the closed player in H ∗ (b∗ , x ∗ , γL ). σ ∗ and τ ∗ are thus strategies for the closed players in their respective games. Working inside M[g∞ ] we construct a real yλ , an infinite run a∞ of A∞ [x ∗ , yλ ], and an infinite run b∞ of B∞ [x ∗ , yλ ]. We construct these objects by combining moves given by σgen−∞ , σ ∗ , τgen−∞ , and τ ∗ : ∞ for all i; • σgen−∞ [x ∗ , yλ ] produces the auxiliary moves ai−I ∞ for all i and the natural number moves • σ ∗ produces the auxiliary moves ai−II yλ (i) for odd i; ∞ for all i; and • τgen−∞ [x ∗ , yλ ] produces the auxiliary moves bi−II ∞ for all i and the natural number moves y (i) • τ ∗ produces the auxiliary moves bi−I λ for even i.
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3 Games of continuously coded length
The reader may consult Diagrams 3.17 and 3.18 to check that these conditions cover all the moves in the games G∗ ( a ∗ , x ∗ , γL ) and H ∗ (b∗ , x ∗ , γL ). Note that σgen−∞ , τgen−∞ , ∗ ∗ σ , and τ all belong to M[g∞ ]. (σ ∗ and τ ∗ in fact belong to M[g] ⊂ M[g∞ ].) The construction can thus be carried out inside M[g∞ ]. If ever an i < ω is reached which satisfies conditions (a)–(c) in the definition of a ∗ , x ∗ , γL ), we must stop the construction. The reason is that both G∗ ( a ∗ , x ∗ , γL ) G∗ ( and H ∗ (b∗ , x ∗ , γL ) end at the first such i, and so both σ ∗ and τ ∗ cease to supply moves for the construction at the first such i. But if an i as above is reached, we can simply imitate the argument in Claim 3C.7 to find an s ∗ which satisfies conditions (B1)–(B3). So suppose an i < ω which satisfies conditions (a)–(c) is not reached. Lemma 1B.2 a∞ , x ∗ , yλ % ∈ A˙ ∞ [g∞ ]. and the use of σgen−∞ during the construction guarantee that $ ˙ ∞ ]. Similarly $b∞ , x ∗ , yλ % ∈ B˙ ∞ [g∞ ], so $x ∗ , yλ % ∈ In other words $x ∗ , yλ % ∈ A[g ˙ B[g∞ ]. The construction takes place inside M[g∞ ], so certainly yλ ∈ M[g∞ ]. We have obtained clause (A) of the claim. # Lemma 3E.12. Suppose that both the hypothesis of Lemmas 3E.8 and the hypothesis of Lemma 3E.10 fail. Then in M[g∞ ] there is a run $yξ | ξ ≤ α% so that $yξ | ξ < α, yα % ˙ ∞ ] nor B[g ˙ ∞ ]. belongs to neither A[g Proof. Suppose for contradiction that a run $yξ | ξ ≤ α% ∈ M[g∞ ] which satisfies the conclusion of the lemma does not exist. Note that in this case clause (A) of Claim 3E.11 can never actually occur. Claim 3E.11 therefore becomes an exact replica of Claim 3C.7. We can now follow the argument of Section 3C(2), using Claim 3E.11 as a replacement for Claim 3C.7, to obtain a contradiction. # Remark 3E.13. The arrangement of the proof of Lemma 3E.12 as an argument by contradiction is just a matter of typographical convenience. In fact it is a constructive proof. One follows the argument of Section 3C (2) with Claim 3E.11 substituting for Claim 3C.7, until reaching a point where an application of Claim 3E.11 yields clause (A). Lemmas 3E.8, 3E.10, and 3E.12 together show that at least one of the cases in Theorem 3E.1 must hold. To complete the proof of Theorem 3E.1 we must check that the hypotheses of Lemmas 3E.8 and 3E.10 (or at least something sufficiently close to these hypotheses) can be expressed in M using as parameters only δ∞ , ν, and the ˙ appropriate name A˙ or B. ∗ ˙ our The game Gini is definable over M using the parameters κ, δ, δ∞ , ν, and A; ∗ ˙ description earlier gives the definition. Let ψI [κ, δ, δ∞ , ν, A, G ] be a formula which ˙ expresses the statement “G∗ is the game G∗ini defined using κ, δ, δ∞ , ν, and A.” ˙ stand for the statement “there exist some κ, δ, and G∗ so that: Let φI [δ∞ , ν, A] ˙ G∗ ] holds; δ < δ∞ is a Woodin cardinal; κ < δ is strong to δ + 1; ψI [κ, δ, δ∞ , ν, A, and I wins G∗ .” Define ψII and φII similarly. It is clear from Lemmas 3E.8, 3E.10, and 3E.12 that φI and φII witness the conclusion of Theorem 3E.1.
Chapter 4
Pullbacks
The determinacy proofs in Chapter 3 may be viewed as combining the basic methods of Chapter 1 with linear iterations. This is most clearly evident in Diagram 3.8; the diagram folds the methods of Section 1F into a linear iteration by the extenders Fξ . Here we begin the process of upgrading the combination. Our intention is to fold the methods of Chapter 1 into non-linear iterations. We have in mind very specific designs for the non-linearity. We intend to fold the methods of Chapter 1 into iterations dictated by the extender algebra of Woodin [45]. Our hope is to reach a point where runs of long games are made generic, not over collapse algebras as in the previous chapters, but over Woodin algebras. The method we begin to engineer here, a combination of the techniques of Chapter 1 with non-linear iterations governed by Woodin’s extender algebra, will eventually allow us to handle long games which run to local cardinals. But it will be a while before we get there. This chapter only sets the grounds. Much of this setting of the grounds involves definitions of names; we have to arrange all sorts of auxiliary games in all sorts of particular manners. The reader may find these arrangements, presented in Sections 4C, 4D, and 4E, a little difficult to digest in a first reading. Our advice is to skim through them, and then return to go over them more carefully while reading Chapter 5.
4A Codes Fix a transitive ZFC∗ model M. Let W denote the set (or class) of δ ∈ M which are Woodin cardinals in M, but not limits of Woodin cardinals in M. We say that δ ∈ W is a relative successor if δ is the first Woodin cardinal of M, or there is a largest Woodin cardinal of M below δ. Otherwise δ ∈ W is a relative limit. For δ ∈ W let e(δ) denote sup{τ + 1 | τ ∈ W ∧ τ < δ}. Note that e(δ) < δ strictly; otherwise δ would be a limit of Woodin cardinals in M, but limits of Woodin cardinals were excluded from W . Definition 4A.1. A position is a function F which satisfies: (1) The domain of F is an initial segment of W . Precisely, dom(F ) = λ ∩ W for some ordinal λ. (2) For δ ∈ dom(F ) a relative successor, F (δ) is a real. (3) For δ ∈ dom(F ) a relative limit, F (δ) is an element of (M& e(δ) + 1)ω . The least λ witnessing condition (1) is the relative domain of F , denoted rdm(F ).
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Formally we think of F (δ) as a sequence $F (δ)(n) | n < ω%. F (δ)(n) is a natural number if δ is a relative successor, and an element of M& e(δ) + 1 if δ is a relative limit. In the case of a relative limit δ it is convenient to split F (δ) into two sequences, $F (δ)(2n) | n < ω%, and $F (δ)(2n + 1) | n < ω%. Each of these sequences is an element of (M& e(δ) + 1)ω . We refer to them as FI (δ) and FII (δ) respectively. Remark 4A.2. Let F be a position of relative domain λ. Suppose there are Woodin cardinals of M above λ and let δ be the first one. Let y be a real if δ is a relative successor, and an element of (M&λ + 1)ω if δ is a relative limit. F and y together determine a position of relative domain δ + 1, namely the position F ∗ which extends F with the value F ∗ (δ) = y. We use F −−, y to denote this extended position. We would like to “code” positions using sequences of length ω. To do this we need an enumeration in order type ω of the domain of the position. Let us start by describing a specific method of witnessing that the domain of a position is countable in V. Let L denote the collection of ordinals λ so that λ ∩ W is cofinal in λ. Remark 4A.3. When we say that a set B is cofinal in a successor ordinal λ, we mean that λ−1 belongs to B. We further adopt the convention that ∅ is cofinal in the ordinal 0. Under our definition L contains 0, all ordinals in {δ + 1 | δ ∈ W }, and all limit points of W . Note that L contains precisely those ordinals which may serve as relative domains (see Definition 4A.1). Definition 4A.4. A witness for an ordinal λ ∈ L is a set w ⊂ ω × (λ ∩ W ) which satisfies the following conditions: (1) for every n ∈ ω there exists at most one τ ∈ W so that $n, τ % ∈ w; (2) for every τ ∈ λ ∩ W there exists at most one n ∈ ω so that $n, τ % ∈ w; (3) the set {τ | ∃n ∈ ω so that $n, τ % ∈ w} is cofinal in λ. A natural number n is used in w if there exists some τ so that $n, τ % ∈ w. We write dom(w) to denote the set of numbers which are used in w. For n ∈ dom(w) we write w(n) for the unique τ so that $n, τ % ∈ w. Uniqueness is given by condition (1). We write range(w) to denote the set {w(n) | n ∈ dom(w)}. This is a subset of λ ∩ W . By condition (3) it is cofinal in λ. range(w) is countable, so the existence of a witness for λ witnesses that λ has countable (or finite) cofinality in V. We say that a witness w is cofinal in an ordinal κ just in case that range(w) ∩ κ is cofinal in κ. (In particular κ must belong to L.) We allow for the case that κ is a successor ordinal or 0; see Remark 4A.3. We use L(w) to denote the set of κ so that w is cofinal in κ. For a witness w and an ordinal κ we write w&κ to denote w ∩ (ω × κ). If κ belongs to L(w) then w&κ is a witness for κ. The converse is also true; if w&κ is a witness for κ then it is cofinal in κ by condition (3) in Definition 4A.4, and from this it follows that κ belongs to L(w).
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Definition 4A.5. Let w and w ∗ be witnesses, for λ and λ∗ say. w∗ is said to extend w just in case that w ∗ &λ = w. There is another way to look at witnesses which may be intuitively clearer. Let w be a witness for an ordinal λ ∈ L. range(w) is a countable set of ordinals. Let αw be the order type of this set (with the natural order on ordinals). For ξ < αw let τξw be the ξ -th ordinal in range(w), so that ξ → τξw is an order preserving bijection from αw w onto range(w). For ξ < αw let nw ξ be the unique n so that w(n) = τξ . Uniqueness is given by condition (2) in Definition 4A.4. The sequences τw = $τξw | ξ < αw % and nw = $nw ξ | ξ < αw % satisfy the following conditions: (1) $τξw | ξ < αw % is a strictly increasing sequence of elements of λ ∩ W ; (2) $τξw | ξ < αw % is cofinal in λ; and (3) $nw ξ | ξ < αw % is a sequence of distinct natural numbers. Condition (3), which corresponds to condition (1) in Definition 4A.4, should remind the reader of the coding used in Chapter 3. We think of τw and nw as an alternative presentation for the witness w. It is useful to keep this alternative presentation in mind. For example Definition 4A.5 is intuitively clearer when phrased for the alternative presentation. It is easy to check that w∗ extends w iff the sequences τw∗ and nw∗ extend the sequences τw and nw . Definition 4A.6. Let λ ∈ L. An annotated position of relative domain λ is a function t which satisfies: (1) The domain of t is λ ∩ (W ∪ L). (2) t(λ ∩ W ) is a position (in the sense of Definition 4A.1). Precisely: if δ ∈ λ ∩ W is a relative successor then t (δ) is real, and if δ ∈ λ ∩ W is a relative limit then t (δ) is an element of (M& e(δ) + 1)ω . (3) For each κ ∈ λ ∩ L, t (κ) is a witness for κ. There are no conflicts between the last two clauses since W and L are disjoint. We write rdm(t) to denote the relative domain of t, namely λ. An annotated position t ∗ is said to extend t just in case that t ∗ rdm(t) is equal to t. An annotated position thus divides into two parts: a position F = t(λ ∩ W ), and a sequence of witnesses $t (κ) | κ ∈ λ ∩ L%. Letting wκ = t (κ) we often abuse notation and refer to the second part as a sequence w = $wκ | κ < λ%, instead of w = $wκ | κ ∈ λ ∩ L%. Remark 4A.7. Let t be an annotated position of relative domain λ. Suppose there are Woodin cardinals of M above λ and let δ be the first one. Let w be a witness for λ. Let y be a real if δ is a relative successor, and an element of (M&λ + 1)ω if δ is a relative limit.
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t, w, and y together determine an annotated position of relative domain δ + 1, namely the annotated position t ∗ which extends t with the assignments t ∗ (λ) = w and t ∗ (δ) = y. We use t−−, w, y to denote this extended annotated position. Let t be an annotated position of relative domain λ. Let F denote the position t(λ ∩ W ), and for each κ ∈ λ ∩ L let wκ denote t (κ), a witness for κ. Let w be a witness for λ. The witnesses $wκ | κ < λ% and w together witness that the domain of F is countable. They can thus be used to code F by a sequence of length ω. We describe this coding more precisely below. We define a code t, w for the pair $t, w%. We shall make sure that: Property 4A.8. t, w is an element of (M& rdm(t))ω . Property 4A.9. t and w can be recovered from the code t, w . The definition of t, w proceeds by induction on rdm(t). We assume knowledge of t¯, w ¯ for t¯ of relative domain smaller than rdm(t). We also assume Properties 4A.8 and 4A.9 for t¯ of relative domain smaller than rdm(t). Given these assumptions we make the following definitions: • Let q(w) be the sequence $ q (w)n | n < ω% given by: w(n) if n ∈ dom(w); q(w)n = ↑ otherwise. • For n ∈ dom(w) let κn = e(w(n)) and let xn = tκn , wκn . (w(n) is an element of W smaller than λ. e(w(n)) is therefore an element of L smaller than λ = rdm(t). The definition of xn is thus meaningful by induction.) • Let x(t, w) be the sequence $ x (t, w)n | n < ω% given by: xn if n ∈ dom(w); x(t, w)n = ↑ otherwise. • Let f(t, w) be the sequence $f(t, w)n | n < ω% given by: F (w(n)) if n ∈ dom(w); f(t, w)n = ↑ otherwise. We think of “↑” as some low rank element of M ∩ M ω standing for “undefined.” It is clear that t and w can be recovered from $ q (w), x(t, w), f(t, w)%. This uses the induction hypothesis which implies that tκn and wκn can be recovered from xn (Property 4A.9), the fact that range(w) is cofinal in λ, and the fact that w can be recovered from q(w).
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The induction hypothesis (specifically Property 4A.8) implies that xn is an element of (M&κn )ω and so certainly an element of (M&λ)ω . It follows that x(t, w) is an element of (M&λ)ω×ω . q(w) is clearly an element of (M&λ)ω . By Definition 4A.1, F (w(n)) is at worst an element of (M&κn + 1)ω . κn = e(w(n)) is smaller than w(n) which in turn is smaller than λ. So F (w(n)) is certainly an element of (M&λ)ω . It follows from this q (w), x(t, w), f(t, w)% is that f(t, w) is an element of (M&λ)ω×ω . The entire triple $ thus an element of (M&λ)ω×(1+ω+ω) . Using some recursive bijection b : ω ↔ ω×ω×2 we may convert $ q (w), x(t, w), f(t, w)% to an element of (M&λ)ω . Define t, w to be this element. It is clear that Properties 4A.8 and 4A.9 hold for $t, w%. The precise bijection b used to convert $ q (w), x(t, w), f(t, w)% into the code t, w is not so important, so long as it yields the following property: Property 4A.10. For any natural number l, t, wl depends only on q(w)l, x(t, w)0 , . . . , x(t, w)l−1 , and f(t, w)0 , . . . , f(t, w)l−1 . Any reasonable choice of the bijection b will give rise to a coding which satisfies Property 4A.10. Definition 4A.11. A λ-code is a code x = t, w where t is an annotated position of relative domain λ (and w is a witness for λ). Remark 4A.12. Suppose x = t, w is a λ-code. Suppose there are Woodin cardinals of M above λ and let δ be the first one. Let y be a real if δ is a relative successor, and an element of (M&λ + 1)ω if δ is a relative limit. t−−, w, y is then an annotated position of relative domain δ + 1 (see Remarks 4A.7 and 4A.2). We use x y to denote this annotated position. Note that t and w can be recovered from x, so the notation x y makes sense. So far we saw how a position can be coded as a sequence of length ω, granted enough witnesses. Let us next see how to generate witnesses. Let t be an annotated position of relative domain λ. Recall that t splits into two parts, a position F = t(λ ∩ W ) and a sequence of witnesses $wκ = t (κ) | κ ∈ λ ∩ L%. We show how to generate an additional witness, for λ. We consider two cases: successor; and limit. Successor case. Suppose that λ is a successor, say λ = δ +1. Suppose we are handed κ and n so that: (1) κ is an element of L, smaller than δ; and (2) n is a natural number which is not used in wκ = t (κ). Let w = wκ ∪ {$n, δ%}. w is then a witness for λ = δ + 1.
# (Successor case)
Definition 4A.13. We use wκ + $n, δ% to denote wκ ∪ {$n, δ%}. Definition 4A.14 (for an annotated position t of successor relative domain λ = δ + 1). A witness w for λ is amenable to t if it is equal to t (κ) + $n, δ% for some κ and n which satisfy the conditions of the successor case above.
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Remark 4A.15. Among the conditions placed on κ and n in the successor case let us specifically underline the requirement that n is not used in wκ . This requirement is crucial. Without this requirement wκ + $n, δ% would not be a witness; it would fail to satisfy condition (1) in Definition 4A.4. Claim 4A.16 (under the conditions of the successor case). The codes tκ, wκ and t, wκ + $n, δ% agree to n. Proof. Let w denote wκ + $n, δ%. It is easy to see that q(w) and q(wκ ) agree to n. (In fact q(w)i = q(wκ )i for all i except i = n.) It is easy to see that x(t, w) and x(tκ, wκ ) agree to n. (In fact x(t, w)i = x(tκ, wκ )i for all i except i = n.) It is easy to see that f(t, w) and f(tκ, wκ ) agree to n. (In fact f(t, w)i = f(tκ, wκ )i for all i except # i = n.) Using Property 4A.10 it follows that t, w and tκ, wκ agree to n. Definition 4A.17. Let w and w∗ be witnesses. Let n be a natural number. w∗ is said to n-extend w just in case that: (1) w ∗ extends w; and (2) dom(w ∗ ) ∩ n = dom(w) ∩ n. Note that the two conditions in Definition 4A.17 together imply that w∗ (i) = w(i) for all i < n in the relevant domains. Using this and an argument similar to the proof of Claim 4A.16 one can show that: Claim 4A.18. Let t and t ∗ be annotated positions, of relative domains λ and λ∗ say. Suppose that t ∗ extends t. Let w and w∗ be witnesses for λ and λ∗ respectively. Let n be a natural number. Suppose that w ∗ is an n-extension of w. Then t ∗ , w∗ agrees with t, w to n. # Limit case. Let t be an annotated position of relative domain λ. For each κ ∈ λ ∩ L let wκ denote t (κ). Suppose that λ is a limit ordinal. Suppose we are handed a set A so that: (1) A is contained in λ ∩ L; (2) A is cofinal in λ; and (3) for all κ¯ < κ both in A, wκ extends wκ¯ . Let w = κ∈A wκ = κ∈A t (κ). w is then a witness for λ.
# (Limit case)
Definition 4A.19 (for an annotated position t of limit relative domain λ). A witness w for λ is amenable to t if it is equal to κ∈A t (κ) for some A which satisfies the conditions of the limit case above. Claim 4A.20 (under theconditions of the limit case above). The codes tκ, wκ , for κ ∈ A, converge to t, κ∈A wκ as κ approaches λ.
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Proof. Let w denote κ∈A wκ . Fix n < ω. We must find some λn < λ so that for all κ ∈ A ∩ [λn , λ), t, w agrees with tκ, wκ to n. Note that the union κ∈A wκ is increasing by condition (3) in the limit case. So $dom(wκ ) ∩ n | κ ∈ A% is an increasing sequence of subsets of n. Since λ is a limit ordinal and since n is finite, there must exist some point on which this sequence reaches a terminal value. More precisely there must exist some λn ∈ A so that dom(wκ ) ∩ n = dom(w) ∩ n for all κ ∈ A ∩ [λn , λ). But then w is an n-extension of wκ for all κ ∈ A ∩ [λn , λ). Using Claim 4A.18 it follows that t, w agrees with tκ, wκ to n, # for all κ ∈ A ∩ [λn , λ). Claims 4A.16, 4A.18, and 4A.20 are similar to the agreement claims of Chapter 3, specifically to Claims 3A.3, 3A.4, and 3A.5. The similarities become more evident if one works with the alternative presentation described following Definition 4A.5. The diligent reader may wish to phrase Claim 4A.20 in the language of alternative presentations, to see more clearly the connection to Chapter 3. We end this section with a few words on the connection of our definitions to sequences of reals, and on the dependence of our definitions on M. Definition 4A.21. Let t be an annotated position of relative domain λ. Define z(t) to be the sequence $t (δξ ) | ξ < α%, where α is the order type of the set of relative successors below λ and $δξ | ξ < α% enumerates this set in increasing order. z(t) is a sequence of real numbers, by condition (2) in Definition 4A.6. We refer to it as the real part of the annotated position t. Remark 4A.22. All the notions defined in this section are dependent on M. When we wish to emphasize this dependence we write: “M-position” instead of “position,” “annotated M-position” instead of “annotated position,” etc.
4B Woodin’s extender algebra This section is expository, presenting a forcing notion due to Woodin. Work as usual with a transitive ZFC∗ model M. A basic identity of type 1 is any expression of the kind t˙(δ)(n) = a where: (1) δ belongs to W ; (2) n is a natural number; (3) if δ is a relative successor then a is a natural number; and (4) if δ is a relative limit then a belongs to M& e(δ) + 1. t˙ is just a decorative symbol which we use for aesthetic reasons. The parameters which define a basic identity of type 1 are δ, n, and a. A basic identity of type 2 is any expression of the kind a ∈ t˙(κ) where:
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(1) κ belongs to L; and (2) a belongs to ω × (κ ∩ W ). Again t˙ is just a decorative symbol which we use for aesthetic reasons. The parameters which define a basic identity of type 2 are κ and a. A basic identity is any expression fitting one of the two types above. An identity is anything generated from basic identities using negations and transfinite disjunctions. Formally we represent identities as follows: • The basic identity t˙(δ)(n) = a is represented by the triple $δ, n, a%. • The basic identity a ∈ t˙(κ) is represented by the pair $κ, a%. • The identity ¬σ is represented by the pair $1, r% where r is the formal representation of σ . • The identity ξ <α σξ is represented by the sequence $rξ | ξ < α% where for each ξ < α, rξ is the formal representation of σξ . One can check easily that the map which assigns to identities their formal representations is injective. When we say that the identity σ belongs to the model Q we mean to say that the formal representation of σ belongs to Q. The height of an identity σ , denoted ht(σ ), is defined by induction as follows: • The height of the basic identity t˙(δ)(n) = a is δ + 1. • The height of the basic identity a ∈ t˙(κ) is κ + 1. • The height of ¬σ is equal to the height of σ . • The height of ξ <α σξ is equal to sup{ht(σξ ) | ξ < α}. If σ is an identity of height λ, then the only expressions of the kinds “t˙(δ)” and ˙ “t (κ)” which appear in σ are ones where δ and κ are smaller than λ. Given an annotated position t of relative domain at least λ we can then evaluate the truth value of σ under t. Let us be more precise. For an identity σ and an annotated position t with rdm(t) ≥ ht(σ ) we define the truth value of σ under t. The definition is by induction on the complexity of σ , following the natural rules: (T1) Suppose σ is a basic identity of type 1, say the expression t˙(δ)(n) = a. Then σ is true under t just in case that t (δ)(n) = a. (T2) Suppose σ is a basic identity of type 2, say the expression a ∈ t˙(κ). Then σ is true under t just in case that a ∈ t (κ).
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(T3) Suppose σ = ¬σ¯ . Then σ is true under t just in case that σ¯ is not true under t. (T4) Suppose σ = ξ <α σξ . Then σ is true under t just in case that there exists some ξ < α so that σξ is true under t. For condition (T1) recall that t (δ) is a sequence of length ω, see Definitions 4A.1 and 4A.6. For condition (T2) recall that t (κ) is a subset of ω ×(κ ∩W ), see Definitions 4A.4 and 4A.6. In both conditions we are using the assumption that rdm(t) ≥ ht(σ ); to make sense of t (δ) in the case of condition (T1) and to make sense of t (κ) in the case of condition (T2). We say that σ is false under t just in case that ht(σ ) ≤ rdm(t) and σ is not true under t. We write t |= σ to indicate that σ is true under t, and t |= σ to indicate that σ is false under t. We only use this notation when rdm(t) ≥ ht(σ ). Trying to evaluate the truth value of σ under t when rdm(t) < ht(σ ) creates difficulties with conditions (T1) and (T2). So we shall not talk about the truth value at all in such cases. Remark 4B.1. Note that t |= σ and t |= ¬σ can never both hold true. We write t |= A just in case that A is a set of identities, each with height at most the relative domain of t, and each true under t. It is convenient to use some standard abbreviations when discussing identities. We use ∨ ¬τ ). We use σ → τ to abbreviate ¬(σ ∧ ¬τ ). We use σ ∧ τ to abbreviate ¬(¬σ ˙ ˙ σ to abbreviate ¬ ξ ξ <α ξ <α ¬σξ . We use t (δ)(n) = a to abbreviate ¬t (δ)(n) = a. Finally we use a ∈ t˙(κ) to abbreviate ¬a ∈ t˙(κ). Fix now some θ which is a Woodin limit of Woodin cardinals in M. We aim to define a forcing notion for introducing an annotated M-position of relative domain θ. This forcing notion is a version of Woodin’s extender algebra, with θ generators, customized to fit our needs. Woodin’s extender algebra is a refinement of the Lindenbaum algebra on identities in M&θ . To start let us work toward a definition of the Lindenbaum algebra. We work with identities which belong to M&θ. Note that each identity in M&θ certainly has height smaller than θ. So its truth value can be evaluated under any annotated position of relative domain θ. By inference relation we mean some relation relating sets of identities with identities. We write A σ , where A is a set of identities and σ is an identity, to indicate that $A, σ % belongs to the relation . Definition 4B.2. Let be an inference relation. Define an inference relation + by throwing into + all the pairs in , and all the pairs needed to satisfy conditions (I1)–(I6) below, and nothing more. (I1) If A σ and A ∪ {σ } τ , then A + τ . (I2) If A σ , then A + ¬¬σ . (I3) If A ¬¬σ , then A + σ .
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(I4) If $σξ | ξ < α% ∈ M is a sequence of identities of length α < θ, and if there + exists some ξ < α so that A σξ , then A ξ <α σξ . of identities of length α < θ, and if A ¬σξ (I5) If $σξ | ξ < α% ∈ M is a sequence for all ξ < α, then A + ¬ ξ <α σξ . (I6) If τ and σ are identities so that A ∪ {τ } σ and A ∪ {τ } ¬σ , then A + ¬τ . We refer to conditions (I1)–(I6) as inference rules. Rule (I6) corresponds to proof by contradiction. Remark 4B.3. If belongs to M then so does + . In fact the map →+ (acting on inference relations in M) belongs to M. Working by induction on γ ≤ θ define a sequence of inference relations "γ as follows: • A "0 σ iff A ∈ M is a set of identities, A ⊂ M&θ, and σ ∈ A; • for each γ < θ, "γ +1 = ("γ )+ ; and • for limit γ ≤ θ, "γ = γ¯ <γ "γ¯ . Note that the entire definition can be carried out inside the model M. Claim 4B.4 (for γ ≤ θ ). Let t be an annotated position of relative domain θ. Suppose that t |= A and A "γ σ . Then t |= σ . Proof. The proof is by induction on γ . The limit case and the case of γ = 0 are trivial. The successor case too is quite easy. Let us only remark that the inference rules (I2) and (I3) correspond to condition (T3), the inference rules (I4) and (I5) correspond to condition (T4), and the inference rule (I6) corresponds to Remark 4B.1. # Let " denote the relation "θ . Note that " belongs to M. We have: Claim 4B.5. "+ is equal to ". Proof. This is immediate using the restriction to sequences of length smaller than θ in the inference rules (I4) and (I5). # The relation " therefore satisfies the inference rules (I1)–(I6), with both and + replaced by ". Because " contains "0 it satisfies the following additional rule: (I0) If A ⊂ M&θ in M is a set of identities and σ belongs to A, then A " σ . We say that A infers σ just in case that A " σ . We say that an identity τ infers σ (and write τ " σ ) just in case that the singleton set {τ } infers σ . Let us prove some simple claims concerning this inference relation. In these claims σ and τ always stand for identities in M&θ, and A and B always stand for sets of identities, in M and contained in M&θ .
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Claim 4B.6. Suppose A " σ and B ⊃ A. Then B " σ . Proof. Prove the claim for each "γ , working by induction on γ ≤ θ.
#
Claim 4B.7. Suppose τ " σ . Then ¬σ " ¬τ . Proof. We have {¬σ, τ } " ¬σ by the inference rule (I0). Since τ " σ we have {¬σ, τ } " σ by Claim 4B.6. Now by the inference rule (I6) it follows that {¬σ } " ¬τ . # Claim 4B.8. σ ∧ τ " σ and σ ∧ τ " τ . Proof. We show that σ ∧τ " σ . By the inference rule (I4) we know that ¬σ " ¬σ ∨¬τ . Applying Claim 4B.7 we get σ ∧ τ " ¬¬σ . By the inference rule (I3) it follows that σ ∧ τ " σ. # Claim 4B.9. Suppose A " ¬(τ ∧ σ ). Then A ∪ {τ } " ¬σ . Proof. Using Claim 4B.6 and the assumption that A " ¬(τ ∧ σ ) we get A ∪ {τ, σ } " ¬(τ ∧σ ). On the other hand using the inference rules (I2) and (I5) we get {τ, σ } " τ ∧σ , so certainly A ∪ {τ, σ } " τ ∧ σ . Applying the inference rule (I6) we get A ∪ {τ } " ¬σ . # Claim 4B.10. Suppose A " τ → σ . Then A ∪ {τ } " σ . Proof. Recall that τ → σ abbreviates ¬(τ ∧ ¬σ ). Suppose A " τ → σ . Using Claim 4B.9 we see that A ∪ {τ } " ¬¬σ . By the inference rule (I3), A ∪ {τ } " σ . # Let us now proceed with the definition of Woodin’s extender algebra. The starting point is the Lindenbaum algebra on identities in M&θ . Definition 4B.11. For two identities τ, σ ∈ M&θ, put τ σ iff τ " σ . Claim 4B.12. The relation is reflexive and transitive. Proof. Reflexivity follows from the inference rule (I0). Transitivity is proved using the inference rule (I1). # Definition 4B.13. For identities τ, σ ∈ M&θ, put τ ∼ σ iff τ σ and σ τ . The relation ∼ is an equivalence relation. Both the relations and ∼ belong to M, because " belongs to M. For each identity σ ∈ M&θ let [σ ] denote the equivalence class of σ . More precisely [σ ] = {τ ∈ M&θ | τ ∼ σ } ∩ (M&ν) where ν is least so that this intersection is not empty. (We are employing here the usual Scott trick to convert from an equivalence class which is potentially an unbounded subset of M&θ, to an element of M&θ which represents this equivalence class in a canonical manner.) Let A ∈ M be the set of equivalence classes. For a, b ∈ A put a ≤A b iff σ τ for some (any) σ ∈ a and some (any) τ ∈ b. Which σ and τ we pick does not matter, because of the transitivity of .
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There is a dependence on θ in our definitions, which begins already in Definition 4B.2. When we wish to take note of this dependence we write Aθ instead of A, and write [σ ]θ instead of [σ ]. $A, ≤A % is a partial ordering in the model M. It is a Lindenbaum algebra, computed in M on identities in M&θ. Woodin’s extender algebra is the result of restricting this partial order to some subset W of A. W is the set of equivalence classes which are consistent with certain axioms. Let us describe these axioms. Definition 4B.14. A basic axiom is any identity which fits one of the descriptions below: (B1) Suppose δ ∈ θ ∩ W is a relative successor and n is a natural number. Then ˙ t (δ)(n) = i is a basic axiom. i<ω (B2) Suppose δ ∈ θ ∩ W is a relative successor, n is a natural number, and i, j are two distinct elements of ω. Then t˙(δ)(n) = i ∨ t˙(δ)(n) = j is a basic axiom. (B3) Suppose δ ∈ θ ∩ W is a relative limit, n is a natural number, α is an ordinal smaller than θ, and $aξ | ξ < α% is an enumeration of M& e(δ) + 1 which belongs to M. Then ξ <α t˙(δ)(n) = aξ is a basic axiom. (B4) Suppose δ ∈ θ ∩ W is a relative limit, n is a natural number, and a, b are two distinct elements of M& e(δ) + 1. Then t˙(δ)(n) = a ∨ t˙(δ)(n) = b is a basic axiom. (B5) Suppose κ belongs to θ ∩ L, n is a natural number, and τ1 , τ2 are two distinct elements of κ ∩W . Let a1 = $n, τ1 % and a2 = $n, τ2 %. Then a1 ∈ t˙(κ)∨a2 ∈ t˙(κ) is a basic axiom. (B6) Suppose κ belongs to θ ∩ L, τ is an element of κ ∩ W , and n1 , n2 are two distinct natural numbers. Let a1 = $n1 , τ % and a2 = $n2 , τ %. Then a1 ∈ t˙(κ) ∨ a2 ∈ t˙(κ) is a basic axiom. (B7) Suppose κ belongs to θ ∩ L, and τ¯ is an element of κ ∩ W . Suppose α is an ordinal smaller than θ and $a ξ | ξ < α% is an enumeration in M of {$n, τ % ∈ ω × (κ ∩ W ) | τ ≥ τ¯ }. Then ξ <α aξ ∈ t˙(κ) is a basic axiom. Note how axioms (B5)–(B7) correspond precisely to the conditions in Definition 4A.4. Axioms (B1)–(B4) correspond to the conditions in Definition 4A.1. Claim 4B.15. Suppose t is an annotated position of relative domain θ . Then all the basic axioms are true under t. # To generate Woodin’s algebra W we need some additional axioms, more restrictive than the basic axioms, and slightly more complicated. Suppose that κ and δ are cardinals of M, both smaller than θ , with δ ∈ W and κ < δ. Suppose that κ is not a Woodin cardinal in M.
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Suppose that E is an extender in M, with crit(E) = κ and strong enough that M&δ + 1 ⊂ Ult(M, E). Suppose further that E belongs to M&δ + ω. Let π : M → Ult(M, E) be the ultrapower embedding by E. Suppose that σ = $ σ (ξ ) | ξ < κ% ∈ M is a sequence of identities, of length κ. Suppose that for each ξ , σ (ξ ) belongs to M&κ. Applying π to σ produces a sequence of identities π( σ ), of length π(κ) > κ. Let τ be the κ-th identity in this sequence. Precisely, let τ = π( σ )(κ). Suppose that the height of τ is less than or equal to δ + 1. Under these circumstances, and only under these circumstances, we define an identity χ (κ, δ, E, σ ). We define:
σ (ξ ). χ(κ, δ, E, σ ) = τ → ξ <κ
Definition 4B.16. An extender axiom is any identity χ (κ, δ, E, σ ) of the kind described above. Remark 4B.17. The following list summarizes precisely the conditions under which χ (κ, δ, E, σ ) is defined: (1) δ ∈ θ ∩ W , κ < δ, and κ is not a Woodin cardinal in M; (2) E ∈ M&δ + ω is an extender of M, crit(E) = κ, and M&δ + 1 ⊂ Ult(M, E); (3) σ = $ σ (ξ ) | ξ < κ% ∈ M is a sequence of identities, each one in M&κ; (4) the identity π( σ )(κ), where π is the ultrapower embedding by E, has height at most δ + 1. Let us try to explain the motivation behind extender axioms. Work with κ, δ, E, and σ as above. Let π be the ultrapower embedding by E and let τ = π( σ )(κ). Suppose for a moment that we are given an annotated position t (with relative domain above all σ )(ξ ). Since π is relevant heights). If t |= π( σ )(κ) then certainly t |= ξ <π(κ) π( an elementary embedding we feel that morally we should be able to take the fact that π( σ )(ξ ) holds true, pull it down by π , and conclude that (ξ ) holds ξ <π(κ) ξ <κ σ true. Thus we feelthat morally t |= τ should imply t |= ξ <κ σ (ξ ). In other words we feel that τ → ξ <κ σ (ξ ) should be true. Given these feelings it seems reasonable to take this last identity as an axiom. The moral argument of the previous paragraph seems to indicate that any annotated position should satisfy the extender axioms. Of course this is false; the flawed reasoning above fails to take account of the possible effect of π on t. But something close enough is true. By passing to ultrapowers of M we can somehow eliminate those extender axioms which fail under t. (A precise formulation of this process is given by Lemma 4B.32.) There is therefore no harm in restricting our attention to annotated positions which satisfy the extender axioms. In other words there is no harm in restricting our attention to identities which are consistent with all the extender axioms.
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Let us carry out this restriction. Let Ax be the set of all the axioms described in Definitions 4B.14 and 4B.16. Each identity in Ax is either a basic axiom or an extender axiom, and all identities of this form are in Ax. Note that Ax belongs to M. This is because both Definition 4B.14 and Definition 4B.16 can be phrased inside M. An identity σ ∈ M&θ is bad if Ax " ¬σ . An identity σ is good if σ ∈ M&θ and σ is not bad. Since both Ax and the relation " belong to M, these notions can be (and are) defined inside M. Note that if σ ∼ τ then σ is bad iff τ is bad. (This uses Claim 4B.7.) Working in M define W ⊂ A by setting: [σ ] ∈ W
⇐⇒
σ is a good identity.
The definition makes sense because σ ∼ τ implies that σ is good iff τ is good. Let ≤W be the restriction of ≤A to W. Woodin’s extender algebra is the partial order $W, ≤W %. Remark 4B.18. As noted this is a particular version of the extender algebra, specifically customized to our own needs. We comment more on the relationship with the standard version of Woodin’s extender algebra in the historical remark at the end of the section. We use W alternately to denote the set W defined above, or the partial order $W, ≤W %, or the set of good identities. Which one we mean should be clear from the context. We think of W as a forcing notion, and of elements of W as conditions. Note that W is defined in M relative to the parameter θ. When we wish to emphasize which parameter is used we write Wθ instead of W. Let us develop some properties of the forcing notion W. We follow the ideas of Woodin [45], only adapted and customized to our context. We work throughout under the assumption that θ is a Woodin limit of Woodin cardinals in M. This assumption remains valid to the end of the section, even when not explicitly mentioned. Lemma 4B.19 (assuming θ is a Woodin limit of Woodin cardinals). Wθ has the θ chain condition in M. Proof. Work in M. By induction we may assume that the lemma holds for θ¯ < θ. Precisely: (1) For every θ¯ < θ, if θ¯ is a Woodin limit of Woodin cardinals in M then Wθ¯ has the θ¯ chain condition in M. Suppose for contradiction that T ⊂ W = Wθ is an anti-chain of length θ . We view T as a sequence $σξ | ξ < θ% where each σξ is a good identity. The fact that T is an anti-chain tells us that for any ξ < κ < θ , σκ and σξ are incompatible in W. This means that there is no good identity ρ with both ρ σκ and ρ σξ . Using Claim 4B.8 it follows that σκ ∧ σξ is not good. Thus: (2) (For any ξ < κ < θ.) Ax " ¬(σκ ∧ σξ ).
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Let H be the set H = {$ξ, σξ % | ξ < θ}. We view H alternately as a sequence and as a function, writing H (ξ ) for σξ . Each identity σξ is an element of M&θ. So H is a subset of M&θ . Using the assumption that θ is a Woodin cardinal fix a cardinal κ < θ which (in M) is <θ-strong wrt H . (See Definition 1A.11.) Let ν < θ be large enough that for each ξ ≤ κ, $ξ, σξ % belongs to M&ν. Using the assumption that θ is a limit of Woodin cardinals in M fix δ < θ so that δ ∈ W and δ ≥ ν. (Hence in particular δ > κ.) Using the fact that κ is <θ-strong wrt H fix an extender E in M so that: (i) crit(E) = κ; (ii) M&δ + 1 ⊂ Ult(M, E); and (iii) π(H ) ∩ M&δ = H ∩ M&δ where π is the ultrapower embedding by E. By passing to a restriction of E if needed we may further assume that: (iv) E belongs to M&δ + ω. It is implicit in condition (ii) that π(κ) > δ (see Fact 2 in Appendix A). Thus: (v) κ < δ < π(κ). Claim 4B.20. For each ξ ≤ κ, π(H )(ξ ) = σξ . Proof. Note to begin with that π(H ) is a function (in other word for each ξ there is at most one object z so that $ξ, z% ∈ π(H )). This follows from the fact that H is a function using the elementarity of π . Fix now ξ ≤ κ. From condition (iii) and the fact that $ξ, σξ % belongs to H ∩ M&δ if follows that $ξ, σξ % is an element of π(H ). But this # means that π(H )(ξ ) = σξ . Claim 4B.21. For each ξ < κ, σξ belongs to M&κ. Proof. Fix ξ < κ. By the elementarity of π , π(H )(π(ξ )) = π(σξ ). Since ξ is smaller than κ it is not moved by π. So π(H )(ξ ) = π(σξ ). We already know, by the previous claim, that π(H )(ξ ) = σξ . So π(σξ ) = σξ . In particular π(σξ ) belongs to M&δ. M&δ = Ult(M, E)&δ by condition (ii), and δ < π(κ) by condition (v). So π(σξ ) belongs to Ult(M, E)&π(κ). Pulling this statement back via π we get σξ ∈ M&κ. # Claim 4B.22. κ is not a Woodin cardinal in M. Proof. Condition (v) and the fact that δ is a Woodin cardinal in Ult(M, E) together imply that κ is a limit of Woodin cardinals in M. Suppose for contradiction that κ is itself a Woodin cardinal in M. Thus κ is a Woodin limit of Woodin cardinals in M. Using the fact that κ is <θ-strong in M one can verify that Wκ simply equals Wθ ∩ (M&κ). But then the fact that $σξ | ξ < κ% is a sequence of incompatible identities in Wθ ∩ (M&κ) implies that it is an anti-chain in Wκ . This contradicts condition (1) above with θ¯ = κ. #
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Let σ = H κ. We have π( σ )(κ) = π(H )(κ) = σκ . (The last equality uses Claim 4B.20.) δ was chosen large enough that σκ ∈ M&δ. Thus π( σ )(κ) ∈ M&δ. This certainly implies that the height of π( σ )(κ) is at most δ + 1. We have verified by now that the tuple $κ, δ, E, σ % falls under the circumstances required for the definition of an extender axiom χ (κ, δ, E, σ ). Since π( σ )(κ) = σκ the extender axiom χ(κ, δ, E, σ ) is simply the identity σκ → ξ <κ σξ . We have therefore: (3) σκ → ξ <κ σξ belongs to Ax. Using this and Claim 4B.10 we see that: (4) Ax ∪ {σκ } " ξ <κ σξ . On the other hand using condition (2) above and Claim 4B.9 we see that Ax ∪ {σκ } " ¬σξ for all ξ < κ. Using the inference rule (I5) we get: (5) Ax ∪ {σκ } " ¬ ξ <κ σξ . Using conditions (4), (5), and the inference rule (I6) we conclude that Ax " ¬σκ . But then σκ is a bad identity, contradicting the fact that it (or rather its equivalence class) is a condition in W. # (Lemma 4B.19) Definition 4B.23. Given an annotated M-position t ∈ V with relative domain θ let G(t) be defined by: G(t) = {[σ ] | σ is an identity in M&θ and t |= σ }.
Lemma 4B.24 (assuming θ is a Woodin limit of Woodin cardinals). Suppose t ∈ V is an annotated M-position with relative domain θ , and suppose t |= Ax. Then G(t) is W-generic/M. Proof. From the fact that t |= Ax it follows that G(t) ⊂ W. Using Claims 4B.4 and 4B.8 it is easy to see that G(t) is a filter. Let us check that G(t) is generic over M. Let T = $[σξ ] | ξ < α% be a maximal anti-chain in W, with T ∈ M. We must find some ξ < α so that [σξ ] belongs to G(t). By Lemma 4B.19 we know that α, the length of T , is strictly smaller than θ. τ = ξ <α σξ is therefore an identity in M&θ. Claim 4B.25. Ax " τ . Proof. Suppose not. Then ¬τ is a good identity, and [¬τ ] is a condition in W. Since T is a maximal anti-chain there must be some ξ < α so that ¬τ and σξ are compatible. So there must be some good identity ρ so that ρ " ¬τ and ρ " σξ . From ρ " σξ we get ρ " τ using the inference rule (I4) and the definition of τ . Then using the inference rule (I6) we get ∅ " ¬ρ. But then certainly Ax " ¬ρ, contradicting the fact that ρ is good. #
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Using the last claim and the initial assumption that t |= Ax we get t |= τ . This means that there exists some ξ < α so that t |= σξ . We have [σξ ] ∈ G(t) for that ξ . # (Lemma 4B.24) Definition 4B.26. Suppose δ is an element of W and suppose t ∈ V is an annotated M-position of relative domain δ + 1. An M-obstruction for t is a pair $E, σ % which satisfies conditions (1)–(6) below: (1) E is a δ + 1-strong extender in M&δ + ω, with crit(E) < δ. Let κ < δ denote crit(E) and let π denote the ultrapower embedding by E. (2) κ is not a Woodin cardinal in M; (3) σ ∈ M is a sequence of identities of length κ, σ = $ σ (ξ ) | ξ < κ% say, and each identity σ (ξ ) belongs to M&κ; (4) for each ξ < κ, t |= ¬ σ (ξ ); (5) the height of the identity π( σ )(κ) is at most δ + 1; and (6) t |= π( σ )(κ). t is obstruction free over M if it has no M-obstructions. Note that Definition 4B.26 is made with no reference to θ . This is important; we will later use the definition in general contexts with no specific θ in mind. Remark 4B.27. By a κ-iseq (κ identities sequence) we mean a sequence σ of the kind described in condition (3) of Definition 4B.26. A κ-iseq σ = $ σ (ξ ) | ξ < κ% is realized by t just in case that there is some ξ < κ so that t |= σ (ξ ). Remark 4B.28. Suppose $E, σ % is an M-obstruction for t. We use τ (E) to denote the first Woodin cardinal of M above crit(E). Since δ itself is a Woodin cardinal of M we have certainly τ (E) ≤ δ. Note that τ (E) is a relative limit in W . This is because crit(E) is a limit of Woodin cardinals of M by condition (1) in Definition 4B.26, but not a Woodin cardinal of M by condition (2). Definition 4B.29. An annotated position t is M-clear if tδ + 1 is obstruction free over M for all δ ∈ rdm(t) ∩ W . Corollary 4B.30. Suppose t ∈ V is an annotated M-position of relative domain θ , where θ is a Woodin limit of Woodin cardinals. Suppose further that t is M-clear. Then G(t) is Wθ -generic/M. Proof. From the assumption that t is clear it follows easily that all extender axioms are true under t. (Note that if an extender axiom χ(κ, δ, E, σ ) fails under t then $E, σ % is an obstruction for tδ + 1.) Certainly all position axioms are true under t, see Claim 4B.15. # So t |= Ax. Now apply Lemma 4B.24.
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Definition 4B.31. An M-obstruction $E, σ % for t is called minimal if it satisfies the following condition in addition to the conditions of Definition 4B.26: (7) t is obstruction free over Ult(M, E). Lemma 4B.32. Suppose δ belongs to W and suppose t ∈ V is an annotated M-position of relative domain δ + 1. Then either t is obstruction free over M, or else there exists a minimal M-obstruction for t. Lemma 4B.32 says that either t is obstruction free, or else it can be made obstruction free by passing to a certain ultrapower of M. Proof (of Lemma 4B.32) . If t is obstruction free over M we are done. So suppose t is not obstruction free. Fix an obstruction $E0 , σ0 % ∈ M. We continue to pick pairs $En+1 , σn+1 % working by induction on n < ω. If ever an n < ω is reached so that t is obstruction free over Ult(M, En ) we end the construction. Otherwise we let $En+1 , σn+1 % be some Ult(M, En )-obstruction for t and continue the construction. Note that an Ult(M, En )-obstruction for t is also an M-obstruction for t. It is thus enough to show that our construction ends at some finite n. Suppose for contradiction the construction continues through all n < ω. We have En+1 ∈ Ult(M, En ) by construction. In other words, $En | n < ω% is an infinite descending chain in the Mitchell order on short extenders in M. But this order is wellfounded by the results of Steel [42]. # So far we considered how an annotated position t can induce a generic G(t). Let us now consider the opposite direction. Suppose for the rest of the section that G is W-generic/M. Claim 4B.33. Let σ be an identity in M&θ. Suppose [σ ] belongs to G. Then [¬σ ] does not belong to G. Proof. Suppose for contradiction that both [σ ] and [¬σ ] belong to G. Since G is a filter there must exist some [ρ] ∈ G which is below both [σ ] and [¬σ ]. So ρ " σ and ρ " ¬σ . From this and the inference rule (I6) it follows that ∅ " ¬ρ. So certainly Ax " ¬ρ. But then [ρ] ∈ W. This contradicts the fact that [ρ] ∈ G ⊂ W. # Claim 4B.34. Let $σξ | ξ < α% be a sequence of identities in M&θ , of length less than θ. Suppose that ξ <α σξ is one of the basic axioms. Then there exists some ξ < α so that [σξ ] ∈ G. Proof. Let D ⊂ W be defined by: [ρ] ∈ D iff ρ is a good identity and ρ σξ for some ξ < α. It is enough to check that D is dense in W. So fix some condition [τ ] ∈ W. We wish to find [ρ] ∈ W so that ρ τ and [ρ] ∈ D. If for some ξ < α, τ ∧ σξ is good, then ρ = τ ∧ σξ works. So suppose for contradiction that for each ξ < α, τ ∧ σξ is bad. In
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other words Ax infers ¬(τ ∧ σξ ) for each ξ < α. By Claim 4B.9 then Ax ∪ {τ} " ¬σξ for each ξ < α.Using the inference rule (I5) it follows that Ax ∪{τ } " ¬ ξ <α σξ . By assumption ξ <α σξ is a basic axiom, so certainly Ax ∪ {τ } " ξ <α σξ . Using the inference rule (I6) it follows that Ax " ¬τ . So τ is a bad identity. But this contradicts the fact that [τ ] is a condition in W. # Claim 4B.35. Suppose n is a natural number and suppose δ ∈ θ ∩ W is a relative successor. Then there exists some natural number i so that [t˙(δ)(n) = i] belongs to G. Proof. This is a direct application of Claim 4B.34 with the basic axiom (B1) of Definition 4B.14. # Claim 4B.36. Suppose n is a natural number and suppose δ ∈ θ ∩ W is a relative successor. Then there exists at most one natural number i so that [t˙(δ)(n) = i] belongs to G. Proof. Suppose for contradiction that both [t˙(δ)(n) = i] and [t˙(δ)(n) = j ] belong to G, where i and j are distinct natural numbers. By Claim 4B.33, neither [t˙(δ)(n) = i] nor [t˙(δ)(n) = j ] belongs to G. Now apply Claim 4B.34 with the basic axiom (B2) to get a contradiction. # Claim 4B.37. Suppose n is a natural number and suppose δ ∈ θ ∩ W is a relative limit. Then there exists exactly one a ∈ M& e(δ) + 1 so that [t˙(δ)(n) = a] belongs to G. Proof. Similar to the last two claims, but using the basic axioms (B3) and (B4).
#
Claim 4B.38. Suppose κ is an element of θ ∩ L. Then: (1) For every n ∈ ω there is at most one τ ∈ κ ∩ W so that [$n, τ % ∈ t˙(κ)] belongs to G. (2) For every τ ∈ κ ∩ W there is at most one n ∈ ω so that [$n, τ % ∈ t˙(κ)] belongs to G. (3) The set {τ | ∃n ∈ ω so that [$n, τ % ∈ t˙(κ)] belongs to G} is cofinal in κ. Proof. Similar to the previous claims, using the basic axioms (B5)–(B7).
#
For δ ∈ θ ∩ W a relative successor define a real yδ = $yδ (n) | n < ω% by letting yδ (n) be the unique natural number i so that [t˙(δ)(n) = i] belongs to G. Existence and uniqueness are given by Claims 4B.35 and 4B.36. For δ ∈ θ ∩ W a relative limit define yδ = $yδ (n) | n < ω% in (M& e(δ) + 1)ω by letting yδ (n) be the unique element a of M& e(δ) + 1 so that [t˙(δ)(n) = a] belongs to G. Existence and uniqueness are given by Claim 4B.37. For κ ∈ θ ∩ L define wκ ⊂ ω × (κ ∩ W ) by putting a ∈ wκ iff [a ∈ t˙(κ)] belongs to G. By Claim 4B.38, wκ is a witness for κ.
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Working inside the generic extension M[G] let t (G) be the annotated position, of relative domain θ, defined by: yβ if β belongs to θ ∩ W ; and t (G)(β) = wβ if β belongs to θ ∩ L. Definition 4B.39. Let t˙ ∈ M be the canonical name, in the forcing W, for the annotated position t (G) defined above. Remark 4B.40. We are using the symbol t˙ for two purposes. On the one hand it is used as a decorative symbol in the notation for identities. On the other hand it is used as the canonical name for t (G). The two uses are related. For example: the condition ˇ n) [t˙(δ)(n) = a] forces the statement “t˙(δ)( ˇ = a,” ˇ the condition [a ∈ t˙(κ)] forces the statement “aˇ ∈ t˙(κ),” ˇ etc. To each M-clear annotated position t of relative domain θ we associated earlier a generic filter G(t). To each generic filter G we now have associated an annotated position t˙[G] = t (G) of relative domain θ. The following claim relates these associations to one another. Claim 4B.41. Suppose t ∈ V is an M-clear annotated position of relative domain θ . Then t˙[G(t)] = t. In particular t belongs to M[G(t)]. # To end let us record the following claim: Claim 4B.42. Suppose that Y˙ ∈ M is a W-name for a set of annotated positions of relative domain θ. Then there is some ordinal α, strictly smaller than θ , so that: (∗) For any G which is W-generic/M, t˙[G] ∈ Y˙ [G] iff this is forced by some condition in G ∩ (M&α). Proof. Let K ⊂ W be the set of conditions which force “t˙ ∈ Y˙ .” Let T ⊂ K be a maximal anti-chain in K. By Lemma 4B.19, T has size strictly smaller than θ . Since each element of T belongs to M&θ, and since θ is a Woodin cardinal and hence inaccessible in M, it follows that T is contained in M&α for some α strictly smaller than θ . It is easy to see that this α witnesses (∗). # Definition 4B.43. An ordinal α < θ which satisfies (∗) of Claim 4B.42 is said to seal Y˙ . Historical Remark. It was Woodin [45] who introduced extender axioms; used them to refine the Lindenbaum algebra on identities in M&θ to the point of having the θ chain condition; and noted that any extra hardship imposed by the extender axioms can be removed through the use of iterated ultrapowers. The end result was a forcing notion in M which could, through the formation of an iteration tree on M, be made to admit any real as generic.
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All the definitions and claims presented in Section 4B trace back directly to the work of Woodin [45]. Wθ is essentially Woodin’s algebra for identities on θ generators, using only extenders which overlap Woodin cardinals and have non-Woodin critical points. We introduced some cosmetic changes to the definitions. For example our notion of an identity is specifically tailored to create a forcing notion which adds an annotated position, rather than an arbitrary sequence. Our basic axioms are also specifically tailored for this purpose. We made one change which is a little more substantial. A direct adaptation of Woodin [45] would define an extender axiom χ(κ, δ, E, σ ) under the conditions summarized in Remark 4B.17, except that condition (4) would have an additional clause: (4)(b) π( σ )(κ) has hereditary cardinality at most δ (so that it is not affected by elementary embeddings with critical points above δ). Our ability to manage without this clause ultimately traces to the fact that δ is a Woodin cardinal.
4C Names, part I Fix a transitive ZFC∗ model M. We work in M and in generic extensions of M throughout this section. When we say a Woodin cardinal we mean a Woodin cardinal of M. When we say generic we mean generic over M. By annotated position we always mean an annotated M-position. We occasionally write “position” instead of “annotated position.” But we always mean annotated position. Definition 4C.1. Suppose δ belongs to W . A δ-sequence is an M-clear annotated position of relative domain δ + 1. A δ-name is a col(ω, δ)-name C˙ for a set of δsequences. We intend to define a certain “pullback” operation on names. Given δ < δ ∗ both in W and a δ ∗ -name C˙ ∗ , we intend to define a δ-name which we refer to as the (δ, δ ∗ )-pullback of C˙ ∗ , denoted Back(δ, δ ∗ )(C˙ ∗ ) (read “back δ, δ ∗ of C˙ ∗ ”). The complete definition of the pullback operation stretches over Sections 4C and 4D. Roughly speaking we intend Back(δ, δ ∗ )(C˙ ∗ ) to name the set of δ-sequences from which player I can win to produce a δ ∗ -sequence which belongs to (an interpretation of a shift of) C˙ ∗ . But this intuition is not entirely accurate; we fold into the definition of Back(δ, δ ∗ )(C˙ ∗ ) some extra ingredient, intended to remove obstructions to the positions encountered along the way to C˙ ∗ . 4C (1) Removing obstructions. We begin by making precise our plans for dealing with obstructions. Suppose δ ∈ W , and suppose t is an annotated position of relative domain δ + 1, which exists in some generic extension of M. We work to define a property of t which tells us that obstructions for t can be removed while making t “winning for I.” We will later fold this property into the definition of the pullback.
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Definition 4C.2. Let Q ∈ M be a forcing notion. Let u˙ ∈ M be a name in Q. For a set a we write a ∈ u[∗] ˙ to mean that a belongs to u[G] ˙ for some G which is Qgeneric/M. We write a ∈ u[∗] ˙ to mean that a does not belong to u[G] ˙ for any G which is Q-generic/M (possibly for the trivial reason that a does not belong to M[G]). Remark 4C.3. Any extension N of M which has a as an element can figure out whether a ∈ u[∗] ˙ or not. To see this first use absoluteness to note that a ∈ u[∗] ˙ iff a G witnessing ˙ a)|). Then use the this exists in the generic extension of N by col(ω, 2|Q| + |trcl(u, homogeneity of the collapse. (trcl(u, ˙ a) here is the transitive closure of {u, ˙ a}.) Definition 4C.4. Let κ be an element of L. A κ-functor is a function c which satisfies the following conditions: ¯ ∈ (κ ∩ L) × (κ ∩ W ) | κ¯ < δ} ¯ into M&κ; and (1) c is a function from I = {$κ, ¯ δ% ¯ in I , c(κ, ¯ is a col(ω, δ)-name. ¯ (2) for each pair $κ, ¯ δ% ¯ δ) Recall that we are working with some δ ∈ W and some annotated position t of relative domain δ + 1. Let τ ≤ δ be a relative limit in W . By Definition 4A.6, t (τ ) = $t (τ )(n) | n < ω% is an element of (M& e(τ ) + 1)ω . We split t (τ ) into two sequences, denoted tI (τ ) and tII (τ ), by setting tI (τ )(n) = t (τ )(2n) and tII (τ )(n) = t (τ )(2n + 1). We only use tI in this section, leaving tII for the mirrored definitions of Section 4E. tI (τ ) is an element of (M& e(τ ) + 1)ω . Definition 4C.5 (for t an annotated position of relative domain δ + 1, and τ ≤ δ a relative limit in W ). t is suitable at τ just in case that for every n < ω, tI (τ )(n) is an e(τ )-functor. An e(τ )-functor is formally a set of pairs taken from ((κ ∩ L) × (κ ∩ W )) × (M&κ) where κ = e(τ ). In particular it is a subset of M&κ = M& e(τ ). Definition 4C.5 is thus compatible with the fact that tI (τ )(n) is an element of M& e(τ ) + 1. It places some structural restrictions on the format of tI (τ )(n), saying that it is not just any element of M& e(τ ) + 1, but that in fact it is a function of a particular kind. Definition 4C.6 (for t an annotated position of relative domain δ + 1). $E, σ % is an acceptable obstruction for t just in case that: (1) $E, σ % is a minimal M-obstruction for t, see Definition 4B.31; (2) t is suitable at τ (E); and (3) there exists a natural number n so that t belongs to π(c)(κ, δ)[∗] where π stands for the ultrapower embedding by E, c stands for tI (τ (E))(n), and κ stands for crit(E). Some words are needed to explain Definition 4C.6. Let us start with the structural aspects. Recall that τ (E) is the first Woodin cardinal of M above crit(E). τ (E) ∈ W is a relative limit by Remark 4B.28. The reference to “suitable” in condition (2) therefore
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makes sense. Having demanded that t be suitable at τ (E) we know that c = tI (τ (E))(n) is a crit(E)-functor. In particular the domain of c is equal to the set I of Definition 4C.4 with κ = crit(E). Now π, the ultrapower embedding by a δ + 1-strong extender, sends κ above δ which remains a Woodin cardinal in the ultrapower. From this it follows that $κ, δ% belongs to π(κ ∩ L) × π(κ ∩ W ), and hence to the domain of π(c). The restrictions placed on c by Definition 4C.4 imply that π(c)(κ, δ) is a name in col(ω, δ). The reference to π(c)(κ, δ)[∗] in condition (3) of Definition 4C.6 therefore makes sense. Definition 4C.6 says that t belongs to a shifted interpretation of a name recorded as part of t. The “record” referred to is tI (τ (E)). This record is shifted by E. A particular name π(c)(κ, δ) in the shifted record is taken. Definition 4C.6 says that t belongs to an interpretation of this name. As it stands Definition 4C.6 says very little; we know nothing of what kind of names are recorded in tI (τ (E)). To make the definition useful when reaching t of relative domain δ + 1 we should make sure to have recorded useful information in tI (τ ), at relative limits τ earlier on in the construction of t. In Section 4C(3) we shall say precisely what kind of names our constructions record in tI (τ ). Roughly speaking we intend to use tI (τ ) to record names for “winning positions for I” (whatever that means). Passing to Ult(M, E) then makes t into such a winning position; this is the point of condition (3) in Definition 4C.6. Now passing to Ult(M, E) also removes all obstructions for t; this is the point of the minimality demand in condition (1). So passing to Ult(M, E) makes t both obstruction free and winning for I. (This is assuming that E is part of an acceptable obstruction for t, and t has “useful” information recorded at relative limits.) Definition 4C.7 (for t an annotated position of relative domain δ + 1). t is acceptably obstructed if there exists an acceptable obstruction for t. From now on we intuitively think of acceptably obstructed positions t as “winning for I.” Our justification for this intuition is the paragraph preceding Definition 4C.7. Remark 4C.8. Conditions (1) and (2) in Definition 4C.6 can clearly be phrased over any extension of M which has t as an element. Condition (3) can also be phrased over any extension of M which has t as an element, using Remark 4C.3. So any extension of M which has t as an element can figure out whether t is acceptably obstructed. 4C (2) Relative successors. Let δ † be a relative successor in W . Fix a δ † -name C˙ † . In other words fix a col(ω, δ † )-name C˙ † ∈ M for a set of M-clear annotated positions of relative domain δ † + 1. Let µ = e(δ † ). Fix (in some generic extension of M) an annotated position t of relative domain µ, and a witness w for µ. For expository simplicity fix g † which is col(ω, δ † )-generic/M. Working in M[g † ] define A† ⊂ (M&µ)ω × ωω by: A† = {$x, y% ∈ M[g † ] | x is a µ-code, y is a real, and x y is either: (a) an element of C˙ † [g † ]; or (b) acceptably obstructed}.
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(See Remark 4A.12 for the definition of x y.) A† is a subset of (M&µ)ω × ωω . Note that clause (b) in the definition of A† can be phrased over M[g † ] using Remark 4C.8. Let A˙ † ∈ M be the canonical name for (M&δ † )ω × A† . We view A˙ † as naming a subset of (M&δ † )ω × ((M&µ) × ω)ω . Let A† = (x, y → A† [x, y]) be the auxiliary games map associated to A˙ † , δ † , and X = (M&µ) × ω. Remark 4C.9. For future reference let us record the fact that we are using the definitions of Chapter 1 with the set X = (M&µ) × ω, where µ = e(δ † ). Define the game G∗suc (t, w, C˙ † ) to be played as follows: I and II play natural numbers y(i), creating together the real y = $y(i) | i < ω%. In addition they play moves in the auxiliary game A† [t, w, y]. Infinite runs of G∗suc (t, w, C˙ † ) are won by player II. I II
y(0)
a0−I
a1−I a0−II
y(1)
y(2) a1−II
... ...
Diagram 4.1. The game G∗suc (t, w, C˙ † ).
As usual we use the fact that A† is Lipschitz continuous. Knowledge of (t, w, and) yi suffices to determine the rules for round i of A† [t, w, y]. The definition of G∗suc (t, w, C˙ † ) therefore makes sense. Moves in G∗suc (t, w, C˙ † ) are elements of M (in fact of M&δ † ). But the game itself is defined modulo knowledge of t and w. So the game itself only exists in models which have t and w as elements. We shall refer to the games G∗suc (. . . ) in our future definitions of the pullback operation. Informally G∗suc (t, w, C˙ † ) can be viewed as a game where players I and II collaborate to produce y, and in addition player I works against the auxiliary moves played by II to try and witness that the annotated positions t−−, w, y is either: (a) an element of C˙ † [h† ] for some col(ω, δ † )-generic h† ; or (b) acceptably obstructed. Let ϕsuc (t, w, C˙ † ) be the statement “player I has a winning strategy in ∗ Gsuc (t, w, C˙ † ).” ϕsuc (t, w, C˙ † ) can be evaluated in any model which contains M and has t and w as elements. Since G∗suc (t, w, C˙ † ) is an open game, and since winning an open game is absolute, we get: Claim 4C.10. ϕsuc (t, w, C˙ † ) is absolute between any two generic extensions of M which have t and w as elements. # 4C (3) Relative limits and records. Fix a relative limit τ in W . We work to define the pullback operation on τ -names. More precisely we define the (δ, τ )-pullback operations for our fixed τ and all relevant δ. We assume complete knowledge of the pullback operation below τ . More precisely we assume knowledge of the (δ, τ¯ )-pullback operations for all τ¯ < τ in W , and all relevant δ.
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Remark 4C.11. This inductive assumption encompasses all τ¯ ∈ τ ∩ W , relative limits and relative successors alike. But the definition of the pullback is limited here to the case of a relative limit τ ∈ W . We shall define the pullback operation in the case of relative successors later on, in Section 4D. Fix a τ -name Y˙ . Let λ = e(τ ). We make various supporting definitions before defining the pullback operation on Y˙ : ˙ ] for a subset of (M&τ )ω × • For each ordinal γ we define a col(ω, τ )-name R[γ ω (M&λ) . • For each ordinal γ , each P ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ we define a δ-name C(κ, δ, P , γ ). These objects are defined with reference to Y˙ . Really we should call them R˙ Y˙ [γ ] and C˙ Y˙ (κ, δ, P , γ ). But Y˙ is fixed here, and so we suppress its mention in the notation. The supporting definitions are made by induction on the ordinal γ . Let us fix γ and ˙ . . , γ ) and R[γ ˙ ]. We assume knowledge of C(. ˙ . . , γ¯ ) and R[ ˙ γ¯ ] for work to define C(. all γ¯ < γ . ˙ . . , γ ). Fix P ∈ (M&τ )<ω . Fix δ ∈ τ ∩ W . Fix 4C(3)(a) The definition of C(. ˙ κ ∈ δ ∩ L. We work to define C(κ, δ, P , γ ). ˙ γ¯ ] for each γ¯ < γ . R[ ˙ γ¯ ] is a col(ω, τ )-name Recall that by induction we know R[ ω ω for a subset of (M&τ ) × (M&λ) . Let R[γ¯ ] be the auxiliary games map associated ˙ γ¯ ], τ , and X = M&λ. to R[ Remark 4C.12. For future reference let us record the fact that we are using the definitions of Chapter 1 with the set X = M&λ, where λ = e(τ ). For expository simplicity fix some g which is col(ω, δ)-generic/M. Fix in M[g] an M-clear annotated position t of relative domain δ + 1. Let n = lh(P ). Suppose that n is not used in t (κ). Let w = t (κ) + $n, δ% (see Definition 4A.13). Let x = t, w. Let δ † be the first Woodin cardinal above δ. Define G∗lim (t, κ, δ, P , γ ) to be the game played according to Diagram 4.2, rules (1)–(3), and the payoff condition (P) below. I II
δ ∗ , n∗
γn an−I
... an−II
γ(n∗ −1) a(n∗ −1)−I ...
Diagram 4.2. The game G∗lim (t, κ, δ, P , γ ).
(1) (Rule for I) (a) δ ∗ is an element of W , greater than δ † and smaller than τ .
a(n∗ −1)−II
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(b) n∗ is a natural number which is strictly greater than n, and not used in w. After I plays δ ∗ and n∗ subject to rule (1), the game continues with n∗ −n rounds subject to rules (2) and (3) below. We use i to index these rounds, starting with i = n. We set ai = $ai−I , ai−II % following round i. We use Pn to denote P and inductively define Pi+1 = Pi −−, ai as the game proceeds. (2) (Rule for I) γi is an ordinal chosen so that: (a) Pi is a legal position in R[γi , x]; (b) γi < γ if i = n; and (c) γi < γi−1 if i > n. (3) (Rule for I and II) ai−I and ai−II are legal moves in the game R[γi , x], following the position Pi . (We remind the reader that in the above rules, δ † is the first Woodin cardinal above δ, n = lh(P ), w = t (κ) + $n, δ%, and x = t, w.) Once the game is over we let P ∗ = Pn∗ and let γ ∗ = γn∗ −1 . We set κ ∗ = δ + 1 ˙ ∗ , δ ∗ , P ∗ , γ ∗ ). We finally set C˙ † = Back(δ † , δ ∗ )(C˙ ∗ ). and C˙ ∗ = C(κ (P) The run of G∗lim (t, κ, δ, P , γ ) described above is won by player I just in case that ϕsuc (t, w, C˙ † ) holds true. Remark 4C.13. We made various uses of the inductive assumptions in the definition of ˙ γ¯ ] for γ¯ < γ in rules (2a) and G∗lim (t, κ, δ, P , γ ). We used knowledge of the names R[ (3). (Note that each γi is smaller than γ by rules (2b) and (2c).) We used knowledge of ˙ . . , γ¯ ) for γ¯ < γ to define the name C˙ ∗ in preparing to phrase the payoff the names C(. condition. (Note that γ ∗ < γ .) Finally we used knowledge of the pullback operation below τ to define the name C˙ † . (We used knowledge of the (δ † , δ ∗ )-pullback operation. Note that δ ∗ < τ by rule (1a).) ˙ Definition 4C.14. Let C(κ, δ, P , γ ) be the canonical name for the set of t ∈ M[g] so that: (1) t is an M-clear annotated position or relative domain δ + 1; (2) n = lh(P ) is not used in t (κ); and (3) I wins the game G∗lim (t, κ, δ, P , γ ). Remark 4C.15. For future reference let us observe that the dependence of ˙ ˙ C(κ, δ, P , γ )[g] on γ is monotone increasing: if γ ≤ γ then C(κ, δ, P , γ )[g] ⊂ ∗ ˙ C(κ, δ, P , γ )[g]. To see this simply note that Glim (t, κ, δ, P , γ ) becomes easier for I as γ increases.
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˙ ]. Continue to work with a fixed ordinal γ . We have 4C(3)(b) The definition of R[γ ˙ ˙ ]. at our disposal the names C(. . . , γ ), defined above. Our next goal is to define R[γ ˙ Recall that all our definitions here are made with reference to a τ -name Y fixed at the outset of Section 4C(3). We shall refer to Y˙ in condition (3a) of Definition 4C.18 below. For expository purposes fix h which is col(ω, τ )-generic/M. We define a set R[γ ] ⊂ ˙ ] be the canonical name (M&τ )ω ×(M&λ)ω working inside M[h]. We shall later let R[γ for this set. Definition 4C.16. Given two sequences u = $u(n) | n < ω% and v = $v(n) | n < ω%, let u⊕v be the length ω sequence defined by (u⊕v)(2n) = u(n) and (u⊕v)(2n+1) = v(n) for each n < ω. Let I = {$κ, δ% ∈ (λ ∩ L) × (λ ∩ W ) | κ < δ}. Definition 4C.17 (for a ∈ (M&τ )ω ). For each n < ω let ua,γ (n) : I → M&λ be the ˙ function defined by ua,γ (n)(κ, δ) = C(κ, δ, a n, γ ). Let ua,γ denote the sequence $ua,γ (n) | n < ω%. ua,γ is the record we attach to a and γ . It records, in the manner of Definition 4C.4, ˙ . . , P , γ ) for P -s which are initial segments of a . the names C(. Definition 4C.18. Working in M[h] let R[γ ] be the set of all pairs $ a , x% so that: (1) a belongs to (M&τ )ω ; (2) x ∈ (M&λ)ω is a λ-code; and (3) there exists some v ∈ (M&λ + 1)ω so that x (ua,γ ⊕ v) is either: (a) an element of Y˙ [h], or (b) acceptably obstructed. ˙ ] be the canonical (Clause (3b) can be phrased over M[h] using Remark 4C.8.) Let R[γ name for R[γ ]. Remark 4C.19. Let t denote the position x (ua,γ ⊕ v) of condition (3) in Definition 4C.18. This is an annotated position of relative domain τ + 1. Note how tI (τ ), ˙ δ, a n, γ ) ranging over κ, δ, and n. which is equal to ua,γ , records the names C(κ, This is in line with our stated intention in Section 4C (1) to use tI (τ ) to somehow record winning positions for player I. 4C(3)(c) The pullback operation on Y˙ . We defined so far the following objects: ˙ ] for a subset of (M&τ )ω × • For each ordinal γ , we defined a col(ω, τ )-name R[γ ω (M&λ) . • For each ordinal γ , each P ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ we defined a δ-name C(κ, δ, P , γ ).
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These objects were defined with reference to a fixed τ -name Y˙ . The dependence of the definitions on Y˙ came in through condition (3a) of Definition 4C.18. Directly it may ˙ ] depends on the fixed τ -name Y˙ . But remember seem that only the definition of R[γ ˙ ˙ γ¯ ] for γ¯ < γ , that the definition of C(. . . , γ ) is made with reference to the names R[ ˙ and hence with indirect dependence on Y . Let $γL , γH % be the least pair of local indiscernibles of M relative to τ , see Definition 1A.15 for details. ˙ Definition 4C.20. Define Back(δ, τ )(Y˙ ) to be the name C(κ, δ, P , γ ) taken with κ = 0, P = ∅, and γ = γL .
4D Names, part II We continue to work with a fixed transitive ZFC∗ model M, as we did in Section 4C. As before “Woodin” means Woodin in M. Definition 4D.1. Suppose δ is a Woodin limit of Woodin cardinals. A δ-sequence is an M-clear annotated position of relative domain δ. A δ-name is a Wδ -name C˙ for a set of δ-sequences. Definitions 4C.1 and 4D.1 together give the notions of δ-sequences and δ-names for all Woodin cardinals δ, whether limits of Woodin cardinals or elements of W . Note that there is no conflict between the two definitions, since Woodin limits of Woodin cardinals are excluded from W . In Section 4C we initiated the definition of a pullback operation on names. We already defined the (δ, τ )-pullback operations (for δ ∈ τ ∩ W ) in the case that τ is a relative limit in W . We have yet to handle the case of relative successors. But let us already enlarge our goals. We wish to define a (δ, τ )-pullback operation not only for τ ∈ W , but also in the case that τ is a Woodin limit of Woodin cardinals. 4D (1) Woodin limits of Woodin cardinals. Let θ be a Woodin limit of Woodin cardinals. Fix a θ-name Y˙ . We work to define the (χ , θ )-pullback of Y˙ for each χ ∈ θ ∩ W . We assume knowledge of the pullback operation below θ during the definition. Let α < θ be the least ordinal which seals Y˙ , see Definition 4B.43. Definition 4D.2 (for δ ∈ [α, θ) ∩ W , where α < θ is the least ordinal which seals Y˙ ). Let s be an M-clear annotated position of relative domain δ + 1. s is hopeful wrt Y˙ if there exists some good identity σ ∈ M&α so that: (1) s |= σ ; and (2) [σ ] Wθ “t˙ ∈ Y˙ .” (We are employing here the notation and terminology of Section 4B.)
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Definition 4D.2 is restricted to δ ≥ α. Note that condition (1) makes sense granted this restriction: σ belongs to M&α and therefore has height less than α. The relative domain of s is δ + 1, which is greater than α. So the truth value of σ can be evaluated under s. Claim 4D.3. Suppose δ ∈ θ ∩W is greater than δ. Suppose s is an M-clear annotated position of relative domain δ + 1 which extends s. Then s is hopeful wrt Y˙ iff s is hopeful wrt Y˙ . Proof. Since s extends s and since δ ≥ α, we have s α = sα. Using this we see that s |= σ ⇐⇒ s |= σ for any σ ∈ M&α. The claim follows. # Continue to work with the fixed θ-name Y˙ , and with α standing for the least ordinal which seals Y˙ . We make the following supporting definition before defining the pullback operation on Y˙ : • For each ordinal γ , each δ ∈ [α, θ) ∩ W , each κ ∈ δ ∩ L, and each n < ω we ˙ define a δ-name C(κ, δ, n, γ ). The name is defined with reference to Y˙ and really we should call it C˙ Y˙ (κ, δ, n, γ ). But Y˙ is fixed and so we suppress its mention in the notation. As usual the definition is by induction on γ . Let us begin the definition. Fix an ordinal γ . Fix δ ∈ [α, θ )∩W . Fix κ ∈ δ ∩L. Fix n < ω. For expository simplicity fix some g which is col(ω, δ)-generic/M. Fix in M[g] an M-clear annotated position s of relative domain δ + 1. Suppose that n is not used in s(κ). Let w denote the witness s(κ) + $n, δ%. We define a game G∗wl (s, κ, δ, n, γ ) as follows: Players I and II play according to Diagram 4.3 and rules (1)–(3) below. I II
δ ∗ , n∗ ν
γ∗
(If s is hopeful wrt Y˙ )
I II
δ ∗ , n∗
γ∗
ν
(If s is not hopeful wrt Y˙ )
Diagram 4.3. The game G∗wl (s, κ, δ, n, γ ).
(1) (Rule for II) ν < θ. (2) (Rule for I) δ ∗ ∈ θ ∩ W is greater than ν, and greater than the first Woodin cardinal above δ. n∗ is a natural number which is not used in w. (3) (Rule for I or II) γ ∗ < γ . ˙ ∗ , δ ∗ , n∗ , γ ∗ ). We let δ † be Once the game is over we let κ ∗ = δ + 1 and let C˙ ∗ = C(κ the first Woodin cardinal above δ, and let C˙ † = Back(δ † , δ ∗ )(C˙ ∗ ).
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(P) The run of G∗wl (s, κ, δ, n, γ ) described above is won by player I just in case that ϕsuc (s, w, C˙ † ) holds true. Remark 4D.4. The move γ ∗ corresponding to rule (3) is played by II if s is hopeful with respect to Y˙ , and by I otherwise. The game G∗wl (s, κ, δ, n, γ ) is very simple. The two players together come up with ˙ + 1, δ ∗ , n∗ , γ ∗ ) ∈ [α, θ ) ∩ W , n∗ ∈ ω − dom(w), and γ ∗ < γ . They look at C(δ and essentially ask whether I can win to enter this name. The moves δ ∗ and n∗ are made by player I but player II can force δ ∗ to be arbitrarily large below θ . The move γ ∗ we think of as a step towards a descending chain. Which player is burdened with this step depends on whether or not s is hopeful with respect to Y˙ . The placement of the burden is in line with our intuitive view of Y˙ as a payoff set for I. If s is hopeful wrt Y˙ then I is making good progress towards this payoff set, so we punish II with the burden of the descending ordinal. If s is not hopeful wrt Y˙ then we feel that I is making very bad progress towards the payoff set, and so we punish I. δ∗
˙ Definition 4D.5. Define C(κ, δ, n, γ ) to be the canonical name for the set of s ∈ M[g] so that: (1) s is an M-clear annotated position of relative domain δ + 1; (2) n is not used in s(κ); and (3) I wins the game G∗wl (s, κ, δ, n, γ ). Recall that α < θ is the least ordinal which seals Y˙ . Definition 4D.5 determines ˙ a δ-name C(κ, δ, n, γ ) for each δ ∈ [α, θ) ∩ W , each κ ∈ δ ∩ L, each n < ω, and each ordinal γ . The definition is inductive in two ways. It assumes knowledge of the ˙ . . , γ ∗ ) for pullback operation below θ, and it assumes knowledge of the names C(. ∗ γ < γ . Both assumptions were used in preparing to phrase the payoff condition (P); the first inductive assumption was used in the definition of C˙ † , and the second inductive assumption was used in the definition of C˙ ∗ . Let us continue to work with the fixed θ-name Y˙ . We continue to use α to denote the least ordinal which seals Y˙ . Fix now some χ ∈ θ ∩ W . χ may be an element of [α, θ ) or it may be smaller than α. For expository simplicity fix some q which is col(ω, χ )-generic/M. Let $γL , γH % be the least pair of local indiscernibles of M relative to θ. Let χ † be the ˙ first Woodin cardinal above χ. For each δ > χ † in [α, θ ) ∩ W let C˙ δ = C(κ, δ, n, γ ) † ˙ ˙ where κ = 0, n = 0, and γ = γL . Let Aδ = Back(χ , δ)(Cδ ). (This last assignment requires knowledge of the (χ † , δ)-pullback operation. We have this knowledge by induction since δ is smaller than θ.) Definition 4D.6. Define Back(χ, θ)(Y˙ ) to be the canonical name in col(ω, χ ) for the set of r ∈ M[q] so that: (1) r is an M-clear annotated position of relative domain χ + 1; and
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(2) in M[q] there exists some witness w for χ + 1 and there exists some δ > χ † in [α, θ ) ∩ W so that ϕsuc (r, w, A˙ δ ) holds true. 4D (2) Relative successors. Let us now begin to discharge our obligation to define the pullback operation in the case of relative successors in W . Fix a relative successor δ † ∈ W . Let δ be the largest Woodin cardinal below δ † . Remark 4D.7. Recall that the first Woodin cardinal is considered a relative successor. When we say “let δ be the largest Woodin cardinal below δ † ” we make the implicit assumption that δ † is not the first Woodin cardinal. Let C˙ † be a δ † -name. We work to define Back(δ, δ † )(C˙ † ). We divide the definition into two cases, depending on whether δ is a Woodin limit of Woodin cardinals, or an element of W . Case 1. If δ belongs to W . For expository simplicity fix some g which is col(ω, δ)generic/M. Define Back(δ, δ † )(C˙ † ) to be the canonical name for the set of t ∈ M[g] so that: (A1) t is an M-clear annotated position of relative domain δ + 1; and (A2) in M[g] there exists some witness w for δ + 1 so that ϕsuc (t, w, C˙ † ) holds true. We refer the reader to Section 4C(2) for the definition of ϕsuc .
# (Case 1)
Case 2. If δ is a Woodin limit of Woodin cardinals. A δ-name in this case is a name in the forcing Wδ , see Definition 4D.1. Fix for expository simplicity some G which is Wδ generic/M. Define Back(δ, δ † )(C˙ † ) to be the canonical name for the set of t ∈ M[G] so that: (B1) t is an M-clear annotated position of relative domain δ; and (B2) the statement “there exists some witness w for δ so that ϕsuc (t, w, C˙ † ) holds true” is forced in col(ω, δ) over M[G]. “Forced” in condition (B2) means forced by some condition. But this is the same as forced by the empty condition, since col(ω, δ) is homogeneous. # (Case 2) The differences between the two cases above stem from the differences between Definitions 4C.1 and 4D.1. In condition (A1) we talk about t of relative domain δ + 1; while in condition (B1) we talk about t of relative domain δ. In condition (A2) we work over the model M[g] where g collapses δ; while in condition (B2) we work over the model M[G] where G is generic for Wδ . In the case of condition (B2) note that we cannot hope to find w inside the model M[G]. This is because Wδ has the δ chain condition, and so δ maintains uncountable cofinality in M[G]. Thus instead of searching for w in M[G] we search for it in M[G]col(ω,δ) .
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4D (3) Compositions. So far we defined the pullback operation in the following three basic cases: (I) We defined the (δ, τ )-pullback operation in the case that τ is a relative limit in W and δ < τ is an element of W . (II) We defined the (χ, θ)-pullback operation in the case that θ is a Woodin limit of Woodin cardinals and χ < θ is an element of W . (III) We defined the (δ, δ † )-pullback operation in the case that δ † is a relative successor in W and δ is the largest Woodin cardinal below δ † . We deal with the remaining cases by composing the basic cases. The exact compositions are defined in cases (IV) and (V) below. (IV) From successor. Suppose δ † is a relative successor in W and C˙ † is a δ † -name. Let δ be the largest Woodin cardinal below δ † . Suppose χ < δ is an element of W . Let C˙ = Back(δ, δ † )(C˙ † ). This assignment uses the definitions of Section 4D (2), ˙ This assignment is possible by listed above as case (III). Let A˙ = Back(χ, δ)(C). ˙ induction, since δ is smaller than δ † . Define Back(χ , δ † )(C˙ † ) to be A. (V) To Woodin limit. Suppose δ is a Woodin limit of Woodin cardinals. Let δ † be the first Woodin cardinal above δ. Suppose δ ∗ is a Woodin cardinal greater than δ † . Suppose C˙ ∗ is a δ ∗ -name. Let C˙ † = Back(δ † , δ ∗ )(C˙ ∗ ). Since δ † belongs to W this assignment can be made using one of the cases (I)–(IV) listed so far. Which case is used depends on whether δ ∗ is a relative limit, a relative successor, or a Woodin limit of Woodin cardinals, and on the distance between δ † and δ ∗ . Let C˙ = Back(δ, δ † )(C˙ † ). This assignment uses the definitions of Section 4D(2), listed above as case (III). Define Back(δ, δ ∗ )(C˙ ∗ ) to ˙ be C. 4D (4) Summary. Our work in Sections 4C and 4D defines a general pullback operation on names. Suppose δ ∗ is a Woodin cardinal, either a Woodin limit of Woodin cardinals or an element of W . Suppose C˙ ∗ is a δ ∗ -name. (The notion of a δ ∗ -name is given by Definition 4D.1 when δ ∗ is a Woodin limit of Woodin cardinals, and by Definition 4C.1 when δ ∗ belongs to W .) Suppose that δ is a Woodin cardinal, again either a Woodin limit of Woodin cardinals or an element of W , and smaller than δ ∗ . Back(δ, δ ∗ )(C˙ ∗ ), the (δ, δ ∗ )-pullback of C˙ ∗ , is defined under these general assumptions. The definition is by induction on δ ∗ , using cases (I)–(V) above. The reader may easily verify that exactly one of the cases fits each given pair $δ, δ ∗ %. The particular case used defines Back(δ, δ ∗ )(C˙ ∗ ), assuming knowledge of the pullback below δ ∗ . Intuitively speaking, Back(δ, δ ∗ )(C˙ ∗ ) names the set of δ-sequences from which player I can win to either enter a shift of C˙ ∗ , or at least reach an acceptably obstructed position along the way to C˙ ∗ . Acceptably obstructed positions are themselves winning for I, not in the game for C˙ ∗ but in a different game, recorded roughly at the critical
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point of the acceptable obstruction. This intuition will be given precise meaning in Chapter 6. Remark 4D.8. It is convenient sometimes to have Back(δ, δ ∗ )(C˙ ∗ ) defined not only for δ < δ ∗ , but also for δ = δ ∗ . Let us set Back(δ, δ ∗ )(C˙ ∗ ) = C˙ ∗ in the case that δ = δ∗.
4E Mirror images We work as before with a fixed transitive ZFC∗ model M. Given two Woodin cardinals δ < δ ∗ , and given a δ ∗ -name D˙ ∗ , we intend to define the δ-name Back II (δ, δ ∗ )(D˙ ∗ ). We refer to this name as the mirrored (δ, δ ∗ )-pullback of D˙ ∗ . Our definitions here precisely mirror the definitions in Sections 4C and 4D. Remark 4E.1. For the sake of symmetry we sometimes write Back I rather than plain Back for the pullback operations defined in Sections 4C and 4D. 4E (1) Removing obstructions. We start by mirroring the definitions of Section 4C (1). Definition 4E.2 (for δ ∈ W , t an annotated position of relative domain δ + 1, and τ ≤ δ a relative limit in W ). t is II-suitable at τ just in case that for every n < ω, tII (τ )(n) is an e(τ )-functor. Note the use of tII (τ ) in Definition 4E.2, compared to the use of tI (τ ) in Definition 4C.5. This is the only difference between the two definitions. Thus Definition 4E.2 places on tII (τ ) the same structural requirements that were placed on tI (τ ) in Definition 4C.5. Definition 4E.3 (for δ ∈ W and t an annotated position of relative domain δ + 1). $E, σ % is a II-acceptable obstruction for t just in case that: (1) $E, σ % is a minimal M-obstruction for t, see Definition 4B.31; (2) t is II-suitable at τ (E); and (3) there exists a natural number n so that t belongs to π(d)(κ, δ)[∗] where π stands for the ultrapower embedding by E, d stands for tII (τ (E))(n), and κ stands for crit(E). Definition 4E.3 is similar to Definition 4C.6, but uses tII instead of tI . It says that t belongs to a shifted interpretation of a name recorded as part of t. But the record consulted here is tII (τ (E)) rather than tI (τ (E)). Later on when we define the mirrored pullback operation in the case of relative limits τ ∈ W we shall use tII (τ ) to record names for winning positions for II, just as in Section 4C (3) we used tI (τ ) to record winning positions for I.
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Definition 4E.4 (for δ ∈ W and t an annotated position of relative domain δ + 1). t is II-acceptably obstructed if there exists a II-acceptable obstruction for t. We intuitively think of II-acceptably obstructed positions t as “winning for II,” just as in the previous sections we thought of I-acceptably obstructed positions as winning for I. Remark 4E.5. We occasionally attach the prefix “I-” to the terms of Sections 4C and 4D, to distinguish them more clearly from the mirrored terms of the current section. For example we say I-acceptably obstructed instead of acceptably obstructed, to differentiate more clearly references to Definition 4C.7 from references to Definition 4E.4. 4E (2) Mirrored successor game. Let us next mirror the definitions of Section 4C (2). Let δ † be a relative successor in W . Fix a δ † -name D˙ † . Let µ = e(δ † ). Fix (in some generic extension of M) an annotated position t of relative domain µ and a witness w for µ. For expository simplicity fix g † which is col(ω, δ † )-generic/M. Working in M[g † ] define B † ⊂ (M&µ)ω × ωω by: B † = {$x, y% ∈ M[g † ] | x is a µ-code, y is a real, and x y is either: (a) an element of D˙ † [g † ]; or (b) II-acceptably obstructed}. Let B˙ † ∈ M be the canonical name for (M&δ † )ω × B † . We view B˙ † as naming a subset of (M&δ † )ω × ((M&µ) × ω)ω . Let B † = (x, y → B † [x, y]) be the mirrored auxiliary games map associated to B˙ † , δ † , and X = (M&µ) × ω. ∗ (t, w, D ˙ † ) to be played as follows: I and II play natural Define the game Hsuc numbers y(i), creating together the real y = $y(i) | i < ω%. In addition they play ∗ (t, w, C ˙ † ) are won by player I. moves in B † [t, w, y]. Infinite runs of Hsuc I II
y(0)
b0−I b0−II
b1−I y(1)
y(2)
b1−II
... ...
∗ (t, w, D ˙ † ). Diagram 4.4. The game Hsuc ∗ (t, w, D ˙ † ) differs from the game defined in Section 4C(2) in several ways: Hsuc infinite runs here are won by player I; the auxiliary moves here are in the mirrored auxiliary games associated to the name B˙ † ; and clause (b) of the definition of B † asks for II-acceptable obstructions rather than I-acceptable obstructions. ∗ (t, w, D ˙ † ) can be viewed as a game where players I and II collabInformally Hsuc orate to produce y, and in addition player II works against the auxiliary moves played by I to try and witness that the annotated positions t−−, w, y is either:
(a) an element of D˙ † [h† ] for some col(ω, δ † )-generic h† ; or (b) II-acceptably obstructed.
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Let ψsuc (t, w, D˙ † ) be the statement “player II has a winning strategy in ∗ (t, w, D ˙ † ).” ψsuc (t, w, D˙ † ) is the natural mirror to the formula ϕsuc (t, w, C˙ † ) Hsuc of Section 4C (2). We have: Claim 4E.6. ψsuc (t, w, D˙ † ) is absolute between any two generic extensions of M which have t and w as elements. # 4E (3) Relative limits and records. Let τ be a relative limit in W . We work to define the mirrored (δ, τ )-pullback operations for this fixed τ and for δ ∈ τ ∩ W . Our definitions here mirror the definitions in Section 4C (3). ˙ Let λ = e(τ ). As in Section 4C (3) we start with some supporting Fix a τ -name Z. definitions: ˙ ] for a subset of (M&τ )ω × • For each ordinal γ we define a col(ω, τ )-name S[γ ω (M&λ) . • For each ordinal γ , each Q ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ we define a δ-name D(κ, δ, Q, γ ). These objects are defined with reference to Z˙ and really we should call them S˙Z˙ [γ ] and D˙ Z˙ (κ, δ, Q, γ ). But Z˙ is fixed here, and so we suppress its mention in the notation. ˙ . . , γ ). Fix Q ∈ (M&τ )<ω . Fix δ ∈ τ ∩ W . Fix 4E(3)(a) The definition of D(. ˙ κ ∈ δ ∩ L. We work to define D(κ, δ, Q, γ ). ˙ γ¯ ], For each γ¯ < γ let S[γ¯ ] be the mirrored auxiliary games map associated to S[ τ , and X = M&λ. For expository simplicity fix some g which is col(ω, δ)-generic/M. Fix in M[g] an M-clear annotated position t of relative domain δ + 1. Let n = lh(Q). Suppose that n is not used in t (κ) and let w = t (κ) + $n, δ%. Let x = t, w. Let δ † be the first Woodin cardinal above δ. ∗ (t, κ, δ, Q, γ ) to be the game played according to Diagram 4.5, rules Define Hlim (1)–(3), and the payoff condition (P) below. I II
δ ∗ , n∗
bn−I γn
bn−II
... ...
b(n∗ −1)−I γ(n∗ −1)
b(n∗ −1)−II
∗ (t, κ, δ, Q, γ ). Diagram 4.5. The game Hlim
(1) (Rule for II) (a) δ ∗ is an element of W , greater than δ † and smaller than τ . (b) n∗ is a natural number which is strictly greater than n, and not used in w.
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After II plays δ ∗ and n∗ subject to rule (1), the game continues with n∗ − n rounds subject to rules (2) and (3) below. We use i to index these rounds, starting with i = n. We set bi = $bi−II , bi−I % following round i. We use Qn to denote Q and inductively define Qi+1 = Qi −−, bi as the game proceeds. (2) (Rule for II) γi is an ordinal chosen so that: (a) Qi is a legal position in S[γi , x]; (b) γi < γ if i = n; and (c) γi < γi−1 if i > n. (3) (Rule for I and II) bi−II and bi−I are legal moves in the game S[γi , x], following the position Qi . Once the game is over we let Q∗ = Qn∗ and let γ ∗ = γn∗ −1 . We set κ ∗ = δ + 1 ˙ ∗ , δ ∗ , Q∗ , γ ∗ ). We finally set D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ). and D˙ ∗ = D(κ ∗ (t, κ, δ, Q, γ ) described above is won by player II just in case (P) The run of Hlim that ψsuc (t, w, D˙ † ) holds true. ∗ (. . . ) is the natural mirror to the definition of G∗ (. . . ) in The definition of Hlim lim Section 4C (3). The auxiliary moves here are in the mirrored auxiliary games associated ˙ γ¯ ]; the roles of the players in Diagram 4.5 are precisely dual to their to the names S[ roles in Diagram 4.2; the name D˙ † is defined using the mirrored pullback operation; and the payoff condition is phrased with reference to the formula ψsuc of Section 4E (2) rather than the formula ϕsuc of Section 4C(2).
˙ Definition 4E.7. Let D(κ, δ, Q, γ ) be the canonical name for the set of t ∈ M[g] so that: (1) t is an M-clear annotated position of relative domain δ + 1; (2) n = lh(Q) is not used in t (κ); and ∗ (t, κ, δ, Q, γ ). (3) II wins the game Hlim
˙ ]. Continue to work with a fixed ordinal γ . We have 4E(3)(b) The definition of S[γ ˙ ˙ ]. at our disposal the names D(. . . , γ ), defined above. Our next goal is to define S[γ Let I = {$κ, δ% ∈ (λ ∩ L) × (λ ∩ W ) | κ < δ}. Definition 4E.8 (for b ∈ (M&τ )ω ). For each n < ω let vb,γ (n) : I → M&λ be the ˙ function defined by vb,γ denote the sequence (n)(κ, δ) = D(κ, δ, bn, γ ). Let vb,γ (n) | n < ω%. $vb,γ vb,γ is the II-record we attach to b and γ . It records, in the manner of Definition 4C.4, ˙ . . , Q, γ ) for Q-s which are initial segments of b. the names D(.
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Definition 4E.9. For expository purposes fix h which is col(ω, τ )-generic/M. Working x% so that: in M[h] let S[γ ] be the set of all pairs $b, (1) b belongs to (M&τ )ω ; (2) x ∈ (M&λ)ω is a λ-code; and (3) there exists some u ∈ (M&λ + 1)ω so that x (u ⊕ vb,γ ) is either: ˙ (a) an element of Z[h], or (b) II-acceptably obstructed. ˙ ] be the canonical name for S[γ ]. Let S[γ Definition 4E.9 is the natural dual to Definition 4C.18. Note particularly how condition (3a) here talks about x (u ⊕ vb,γ ), while in Definition 4C.18 the condition talked about x (ua,γ ⊕ v). For t = x (u ⊕ v) we have tI (τ ) = u and tII (τ ) = v. So in Definition 4C.18 the record ua,γ was attached on tI (τ ), while here the record is attached on tII (τ ). This distinction is in line with our declared intention in vb,γ Section 4E (1), to use tII for the mirrored definitions just as previously we used tI . Remark 4E.10. The definitions above are inductive in several ways. They assume knowledge of the pullback operation below τ , they assume knowledge of the names ˙ . . , γ ∗ ) for γ ∗ < γ , and they assume knowledge of the mirrored auxiliary games D(. maps S[γ¯ ] for γ¯ < γ . The inductive assumptions were used in the definition of the ∗ (. . . , γ ). The rules of the game are phrased with reference to the maps S[γ¯ ] game Hwl ˙ . . , γ ∗) for γ¯ < γ . The payoff condition (P) is phrased with reference to a name D(. ∗ † ∗ for γ < γ and with reference to the (δ , δ )-pullback operation for some δ ∗ < τ . ˙ We defined so far the following objects: 4E(3)(c) The pullback operation on Z. ˙ ] for a subset of (M&τ )ω × • For each ordinal γ , we defined a col(ω, τ )-name S[γ ω (M&λ) . • For each ordinal γ , each Q ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ we defined a δ-name D(κ, δ, P , γ ). ˙ (The dependence on Z˙ The definitions were made with reference to a fixed τ -name Z. came in through condition (3b) of Definition 4E.9.) Let $γL , γH % be the least pair of local indiscernibles of M relative to τ . ˙ to be the name D(κ, ˙ δ, Q, γ ) taken with Definition 4E.11. Define Back II (δ, τ )(Z) κ = 0, Q = ∅, and γ = γL . This definition mirrors Definition 4C.20.
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4E (4) Woodin limits of Woodin cardinals. Let θ be a Woodin limit of Woodin car˙ We work to define Back II (χ , θ )(Z) ˙ for χ ∈ θ ∩ W . Our dinals. Fix a θ -name Z. definitions here mirror those of Section 4D(1). ˙ We make the following supporting Let β < θ be the least ordinal which seals Z. ˙ definition before defining the pullback operation on Z. • For each ordinal γ , each δ ∈ [β, θ) ∩ W , each κ ∈ δ ∩ L, and each n < ω we ˙ define a δ-name D(κ, δ, n, γ ). ˙ Since Z˙ is fixed we suppress it in the notation. The name is defined with reference to Z. Let us begin the definition. Fix an ordinal γ . Fix δ ∈ [β, θ)∩W . Fix κ ∈ δ ∩L. Fix n < ω. For expository simplicity fix some g which is col(ω, δ)-generic/M. Fix in M[g] an M-clear annotated position s of relative domain δ + 1. Suppose that n is not used in ∗ (s, κ, δ, n, γ ) as s(κ) and let w denote the witness s(κ) + $n, δ%. We define a game Hwl follows: Players I and II play according to Diagram 4.6 and rules (1)–(3) below. I II
ν
δ ∗ , n∗
γ∗
˙ (If s is hopeful wrt Z)
I II
ν
δ ∗ , n∗
γ∗
˙ (If s is not hopeful wrt Z)
∗ (s, κ, δ, n, γ ). Diagram 4.6. The game Hwl
(1) (Rule for I) ν < θ. (2) (Rule for II) δ ∗ ∈ θ ∩ W is greater than ν, and greater than the first Woodin cardinal above δ. n∗ is a natural number which is not used in w. (3) (Rule for I or II) γ ∗ < γ . ˙ ∗ , δ ∗ , n∗ , γ ∗ ). We let δ † be Once the game is over we let κ ∗ = δ + 1 and let D˙ ∗ = D(κ the first Woodin cardinal above δ, and let D˙ † = Back II (δ † )(D˙ ∗ ). ∗ (s, κ, δ, n, γ ) described above is won by player II just in case that (P) The run of Hwl † ˙ ψsuc (s, w, D ) holds true.
Remark 4E.12. The move γ ∗ corresponding to rule (3) is played by I if s is hopeful ˙ and by II otherwise. with respect to Z, ∗ (s, κ, δ, n, γ ) is the natural dual to the game G∗ (s, κ, δ, n, γ ) of The game Hwl wl Section 4D (1). Note how the roles of the players in Diagram 4.6 are precisely dual to their roles in Diagram 4.3. Note how Remark 4E.12 is precisely dual to Remark 4D.4; Z˙ is intuitively regarded as a payoff for II, and so it is player II who is punished here with the burden of γ ∗ if s is not hopeful. Note finally how the payoff condition (P) uses the formula ψsuc of Section 4E(2) rather than the formula ϕsuc of Section 4C (2).
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˙ Definition 4E.13. Define D(κ, δ, n, γ ) to be the canonical name for the set of s ∈ M[g] so that: (1) s is an M-clear annotated position of relative domain δ + 1; (2) n is not used in s(κ); and ∗ (s, κ, δ, n, γ ). (3) II wins the game Hwl
˙ Definition 4E.13 determines Recall that β < θ is the least ordinal which seals Z. ˙ D(κ, δ, n, γ ) for any δ ∈ [β, θ) ∩ W , any κ ∈ δ ∩ L, any n < ω, and any ordinal γ . As usual the definition is inductive. It assumes knowledge of the mirrored pullback ˙ . . , γ ∗ ) for γ ∗ < γ . operation below θ, and it assumes knowledge of the names D(. Both assumptions were used in preparing to phrase the payoff condition (P); the first in the definition of D˙ † , and the second in the definition of D˙ ∗ . ˙ and continue to use β to denote Let us continue to work with the fixed θ-name Z, ˙ the least ordinal which seals Z. Fix now some χ ∈ W smaller than θ . For expository simplicity fix some q which is col(ω, χ)-generic/M. Let $γL , γH % be the least pair of local indiscernibles of M relative to θ. Let χ † be the first Woodin cardinal above χ . For ˙ δ, n, γ ) where κ = 0, n = 0, and γ = γL . each δ > χ † in [β, θ) ∩ W let D˙ δ = D(κ, II † Let B˙ δ = Back (χ , δ)(D˙ δ ). ˙ to be the canonical name for the set of r ∈ M[q] Definition 4E.14. Define Back II (χ)(Z) so that: (1) r is an M-clear annotated position of relative domain χ + 1; and (2) in M[q] there exists some witness w for χ + 1 and there exists some δ > χ † in [β, θ ) ∩ W so that ψsuc (r, w, B˙ δ ) holds true. Definition 4E.14 establishes the action of the mirrored (χ , θ )-pullback operation ˙ It mirrors Definition 4D.6. on Z. 4E (5) Relative successors. Fix a relative successor δ † ∈ W . Let D˙ † be a δ † -name. Let δ be the largest Woodin cardinal below δ † . We work to define Back II (δ, δ † )(D˙ † ). Our definitions here mirror the definitions in Section 4D (2). Case 1. If δ belongs to W . For expository simplicity fix some g which is col(ω, δ)generic/M. Define Back(δ, δ † )(D˙ † ) to be the canonical name for the set of t ∈ M[g] so that: (A1) t is an M-clear annotated position of relative domain δ + 1; and (A2) in M[g] there exists some witness w for δ + 1 so that ψsuc (t, w, D˙ † ) holds true. # (Case 1) Case 2. If δ is a Woodin limit of Woodin cardinals. Fix for expository simplicity some G which is Wδ -generic/M. Define Back(δ, δ † )(C˙ † ) to be the canonical name for the set of t ∈ M[G] so that:
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(B1) t is an M-clear annotated position of relative domain δ; and (B2) the statement “there exists some witness w for δ so that ψsuc (t, w, D˙ † ) holds true” is forced in col(ω, δ) over M[G]. # (Case 2) ψsuc in conditions (A2) and (B2) is the formula defined in Section 4E (2). 4E (6) Compositions. So far we defined the mirrored pullback operation in the following three basic cases: (I) We defined the mirrored (δ, τ )-pullback operation in the case that τ is a relative limit in W and δ < τ is an element of W . (II) We defined the mirrored (χ, θ)-pullback operation in the case that θ is a Woodin limit of Woodin cardinals and χ < θ is an element of W . (III) We defined the mirrored (δ, δ † )-pullback operation in the case that δ † is a relative successor in W and δ is the largest Woodin cardinal below δ † . The remaining cases are defined by composition using the basic cases. Let us just say that the compositions trivially mirror those of Section 4D (3), and omit the exact phrasing. 4E (7) Summary. Our work in Section 4E defines a mirrored pullback operation on names. Suppose δ ∗ is a Woodin cardinal, either a Woodin limit of Woodin cardinals or an element of W . Suppose D˙ ∗ is a δ ∗ -name. Suppose that δ is a Woodin cardinal, again either a Woodin limit of Woodin cardinals or an element of W , and smaller than δ ∗ . Back II (δ, δ ∗ )(D˙ ∗ ), the mirrored (δ, δ ∗ )-pullback of D˙ ∗ , is defined under these assumptions. The definition is by induction on δ ∗ via a process which precisely mirrors that of Sections 4C and 4D. Both the pullback operation of Sections 4C and 4D and the mirrored pullback operation of the current section will be used in Chapters 5, 6, and 7. The nature of these operations should become clearer with use.
Chapter 5
When both players lose Fix a transitive ZFC∗ model M. We work in M and in generic extensions of M throughout this chapter. We follow the notational conventions of Sections 4C, 4D, and 4E. In particular “Woodin” means Woodin in M. ϕsuc and ψsuc below are the formulae defined in Sections 4C (2) and 4E(2) respectively. ˙ Definition (5G.2). Let δ be a Woodin cardinal. Let C˙ be a δ-name. Define ϕini (δ, C) to be the formula “ϕsuc (t0 , w0 , C˙ 0 ) holds true with: • t0 equal to the empty annotated position of relative domain 0; • w0 equal to the empty witness for 0; and ˙ where δ0 is the first Woodin cardinal of M.” • C˙ 0 equal to Back(δ0 , δ)(C) ˙ be defined similarly, but using ψsuc and the mirrored For a δ-name D˙ let ψini (δ, D) pullback. ˙ holds then we intuitively expect player I to be able to win to enter a If ϕini (δ, C) ˙ where δ0 is the first Woodin cardinal of M, and from there we shift of Back(δ0 , δ)(C) ˙ holds then ˙ Similarly if ψini (δ, D) expect player I to be able to win to enter a shift of C. ˙ we expect player II to be able to win to enter a shift of D. We will turn these intuitive expectations into precise arguments in Chapter 6. Here we handle the remaining case, ˙ and ψini (δ, D) ˙ fail. This is the analogue to case (3) in the case that both ϕini (δ, C) Theorems 2A.2 and 3E.1, and we produce in this case a δ-sequence t which avoids C˙ ˙ and D. We construct t working by induction on δ. The induction divides into five cases corresponding to the ones listed in Section 4D(3) and mirrored in Section 4E (6). The three main cases are: relative limits, to be handled in Section 5C; Woodin limits of Woodin cardinals, to be handled in Section 5D; and relative successors, to be handled in Section 5E.
5A Saturation Let us recall some definitions from Section 4B. A κ-iseq is a κ-length sequence of identities σ = $ σ (ξ ) | ξ < κ% so that σ (ξ ) belongs to M&κ for each ξ < κ, and so that the entire sequence σ belongs to M. A κ-iseq σ is realized by an annotated position t just in case that there exists some ξ < κ so that t |= σ (ξ ). It is implicit in the statement t |= σ (ξ ) that the relative domain of t is at least ht( σ (ξ )). But other than this we make
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no assumptions on the relative domain of t. It may be smaller than κ, and it may be smaller than ht( σ (ζ )) for many ζ < κ. The terminology reviewed above is derived from Definition 4B.26. Without reproducing that definition let us point out that the more iseqs t realizes, the less likely it is to be obstructed. Our main aim in this section is to phrase a definition which states that t realizes enough iseqs so as to avoid minimal unacceptable obstructions. Definition 5A.1. An annotated position t is nice if it satisfies the following conditions: (N1) For each δ ∈ rdm(t) ∩ W , tδ + 1 belongs to a generic extension of M by col(ω, δ). (N2) For each relative limit τ ∈ rdm(t)∩W and each i < ω, t (τ )(i) is an e(τ )-functor. (N3) For each κ ∈ rdm(t) ∩ L and each ρ ∈ L(t (κ)), t (κ) extends t (ρ). (N4) For each κ ∈ rdm(t) ∩ L, there are infinitely many n ∈ ω which are not used in t (κ). Condition (N2) should be read in conjunction with definition 4C.4. It implies that t is both I-suitable and II-suitable at all relative limits τ in its domain. Conditions (N3) and (N4) should be read in conjunction with the definitions of Section 4A. Condition (N3) should be viewed as a coherence requirement on the witnesses in t. Condition (N4) says that the domain of t (κ) leaves enough free room for extension. Remark 5A.2. Definition 5A.1 depends on M. (Condition (N1) is phrased with reference to M, and the very notion of a position depends on M through the definition of W and L.) When we wish to take note of this dependence we write “nice over M” instead of nice. Claim 5A.3. Suppose t is nice over M. Let M ∗ be a transitive model of ZFC∗ and suppose that M ∗ & rdm(t) is equal to M& rdm(t). Then t is nice also over M ∗ . Proof. Only condition (N1) requires some work. Note that if tδ + 1 belongs to a generic extension of M by col(ω, δ) then the canonical name for tδ + 1 can be coded as a subset of M&δ. The agreement between M and M ∗ is sufficient that this name # belongs also to M ∗ . Definition 5A.4. Suppose δ belongs to W . Let C˙ and D˙ be δ-names. A δ-sequence t is said to avoid C˙ and D˙ if there exists some g so that: (1) g is col(ω, δ)-generic/M; (2) t belongs to M[g]; and ˙ ˙ (3) t belongs to neither C[g] nor D[g].
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We shall refer to Definition 5A.4 often throughout this chapter. It should be read in conjunction with Definition 4C.1. Recall that a δ-sequence, in the case that δ belongs to W , is an M-clear annotated position of relative domain δ + 1. A δ-name is a col(ω, δ)name for a set of such sequences. ¯ t¯% where: Definition 5A.5. A step is a triple $κ, ¯ δ, ¯ (1) δ¯ ∈ W , κ¯ ∈ L, and κ¯ < δ; ¯ (2) t¯ is a δ-sequence; (3) t¯ is nice; and (4) t¯ is saturated (see Definition 5A.8). ¯ t¯% a κ-step if δ¯ + 1 (the relative domain of t¯) is smaller than κ. We call $κ, ¯ δ, Definition 5A.6. Let s be a nice annotated position, and let w be a witness for rdm(s). ¯ t¯% is said to extend the pair $s, w% if: Let a be a finite subset of ω. A step $κ, ¯ δ, (1) t¯ extends s strictly; (2) κ¯ ≥ rdm(s); (3) t¯(rdm(s)) = w; and (4) t¯(κ) ¯ extends w. ¯ t¯% is said to a-extend $s, w% if in addition: $κ, ¯ δ, (5) no number in a is used in t¯(κ). ¯ Note particularly the last three conditions in Definition 5A.6. a-extending the pair $s, w% requires not only finding some t¯ which extends s, but also making sure that t¯(rdm(s)) = w, and finding some κ¯ ≥ rdm(s) so that t¯(κ) ¯ extends w and does not use any number in a. We are now ready to phrase the main definition of this section: Definition 5A.7 (for a nice annotated position t). Let τ be a relative limit in rdm(t)∩W . Let κ denote e(τ ). t is saturated at τ just in case that: for every κ-iseq σ which is not realized by t, there is an m < ω, a ρ < κ in L(t (κ)), and a finite a ⊂ ω − dom(t (κ)), ¯ t¯% which satisfies the following conditions: so that there is no κ-step $κ, ¯ δ, ¯ t¯% is an a-extension of $tρ, t (ρ)%; (1) $κ, ¯ δ, ¯ and d(κ, ¯ where c = tI (τ )(m) and d = tII (τ )(m); and (2) t¯ avoids c(κ, ¯ δ) ¯ δ), (3) t¯ realizes σ .
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Roughly speaking Definition 5A.7 states that if a κ-iseq σ is not realized by t, then it is also not realized by any “smaller” t¯ which is “sufficiently similar” to t and “appropriate.” Smaller means of relative domain δ¯ + 1 for some δ¯ < κ. Similar means satisfying the conditions of Definition 5A.6 with $s, w% given by $tρ, t (ρ)% for some ρ ∈ κ ∩ L(t (κ)) and with a disjoint from dom(t (κ)). Note how these conditions translate in the context of Definition 5A.7 to conditions on t¯ which are satisfied by t. Appropriate means satisfying conditions (2), (3), and (4) in Definition 5A.5, and condition (2) in Definition 5A.7. With these interpretations of smaller, similar, and appropriate, Definition 5A.7 precisely states that if a κ-iseq σ in not realized by t then there is some level of similarity (given by ρ and a) and some notion of appropriateness (given by m) so that σ is also not realized by any t¯ which is smaller than t, similar to t to the degree given by ρ and a, and appropriate. Definition 5A.7 is made for a nice annotated position t. Since t is nice it certainly satisfies condition (N2). From this it follows that c = tI (τ )(m) = t (τ )(2m) and d = tII (τ )(m) = t (τ )(2m + 1) are functors of the kind described in Definition 4C.4. ¯ ∈ (κ ∩ L) × (κ ∩ W ) | κ¯ < δ}. ¯ It The domain of these functors is I = {$κ, ¯ δ% ¯ is easy to check that the pairs $κ, ¯ δ% which come up in Definition 5A.7 belong to ¯ and d(κ, ¯ are this I . From the fact that c and d are functors it follows that c(κ, ¯ δ) ¯ δ) ¯δ-names. Condition (2) in Definition 5A.7, which talks about avoiding c(κ, ¯ ¯ δ) and ¯ therefore makes sense. d(κ, ¯ δ), Definition 5A.8 (for a nice annotated position t). t is saturated if it is saturated at all relative limits τ in rdm(t) ∩ W . Definition 5A.8 at last gives meaning to condition (4) in Definition 5A.5. Definitions 5A.7 and 5A.8 are made together by induction. To define saturation at τ we require, through Definition 5A.5, knowledge of Definition 5A.8 for nice annotated positions t¯ of relative domain δ¯ smaller than κ = e(τ ). For this we require knowledge of saturation at τ¯ , but only for τ¯ < κ = e(τ ) < τ . Remark 5A.9. The definition of saturation is dependent on M. When we wish to take note of this dependence we write “saturated over M” instead of saturated. We similarly talk about “steps over M” when we wish to take note of the dependence of Definition 5A.5 on M. We write “avoids over M” when we wish to take note of the dependence of Definition 5A.4 on M. Claim 5A.10. Let t be nice, and saturated at τ over M. Let M ∗ be a transitive model of ZFC∗ . Suppose that t is nice also over M ∗ . Let κ denote e(τ ). Suppose that M ∗ &κ + 1 is equal to M&κ + 1. Then t is saturated at τ also over M ∗ . Proof. Work by induction on τ . Using the claim with τ¯ < τ it follows that any κ-step over M is also a κ-step over M ∗ and (by symmetry) vice versa. Using this and the fact that all quantifiers in Definition 5A.7 range over M&κ + 1 at worst, it is easy to see that τ -saturation over M is the same as τ -saturation over M ∗ . Let us only comment that the most sizable quantifier in Definition 5A.7 is the first one, ranging over κ-iseqs. A κ-iseq is a κ-sequence of identities in M&κ. The agreement between M and M ∗ is sufficient to imply that M ∗ and M have the same κ-iseqs. #
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Recall the intuition behind the definition of saturation: saturation is meant to say that t realizes enough iseqs so as to avoid minimal unacceptable obstructions. The next lemma gives this intuition a precise meaning. It states that a minimal obstruction to a saturated position cannot be unacceptable. Lemma 5A.11. Let δ belong to W , let t be a nice annotated position of relative domain δ + 1, and suppose that t e(δ) is M-clear. Let $E, σ % be a minimal obstruction for t. Suppose that t is saturated. Then the obstruction $E, σ % is either I-acceptable, or II-acceptable, or both. Proof. Let τ = τ (E). (Recall that this is the first Woodin cardinal above crit(E).) Let κ = e(τ ). Let M ∗ = Ult(M, E) and let π : M → M ∗ be the ultrapower embedding. It is easy to see that e(τ ) = crit(E). So the critical point of π is κ. Since t is nice, t = tδ + 1 belongs to a generic extension of M by col(ω, δ). Fix some g, col(ω, δ)-generic/M, so that t ∈ M[g]. t is a function with domain contained in δ + 1 and range contained in (M&δ)ω . From this it follows easily that the canonical name for t can be coded as a subset of M&δ. Since M ∗ agrees with M on subsets of M&δ, t belongs also to M ∗ [g]. Since $E, σ % is an obstruction for t we have: (i) σ is a κ-iseq; (ii) t does not realize σ ; and σ ). (iii) (in M ∗ [g]) t realizes π( In the case of condition (iii) note that π( σ ) is a π(κ)-iseq over M ∗ [g]. To see that it is realized by t note that t |= π( σ )(κ), by condition (6) of Definition 4B.26. By Definition 4B.26, E is δ + 1 strong. Implicit in this is the following condition (see Fact 2 in Appendix A): (iv) π(κ) > δ. Since t is nice over M we get using the strength of E and Claim 5A.3: (v) t is nice over M ∗ . Since t is saturated over M we get by Claim 5A.10: (vi) t is saturated over M ∗ . Since $E, σ % is a minimal obstruction for t we get by Definition 4B.31: (a) t is obstruction free over M ∗ . Combining this with the assumption that t e(δ) is M-clear, and therefore M ∗ -clear since M and M ∗ agree past e(δ), we get: (vii) t is M ∗ -clear.
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t is assumed saturated, and so it is certainly saturated at τ . Applying Definition 5A.7 with the κ-iseq σ we obtain an m < ω, a ρ < κ in L(t (κ)) and a finite a ⊂ ω − dom(t (κ)) so that: ¯ t¯%, at least one of the conditions (1)–(3) in Definition 5A.7 (A) For each κ-step $κ, ¯ δ, fails. The fact that a ⊂ ω − dom(t (κ)) implies that: (viii) No number in a is used in t (κ). From condition (N3) and the fact that ρ ∈ L(t (κ)) it follows that: (ix) t (κ) extends t (ρ). For the record let us note that: (x) κ ≥ ρ. We trivially have the following condition: (xi) t extends tρ strictly and t (ρ) = t (ρ). Suppose now for contradiction that $E, σ % is not I-acceptable, and not II-acceptable. Let c = tI (τ )(m) and let d = tII (τ )(m). (Recall that m is the number given by Definition 5A.7 applied with the iseq σ .) Looking at Definitions 4C.6 and 4E.3 we see that: (b) t ∈ π(c)(κ, δ)[∗] and t ∈ π(d)(κ, δ)[∗]. (Note that t is both I-suitable and II-suitable at τ . This follows from condition (N2) in Definition 5A.1.) Recall that we fixed earlier some col(ω, δ)-generic g so that t belongs to M ∗ [g]. We did this using condition (N1) and the agreement between M and M ∗ . Combining the fact that t belongs to M ∗ [g] with condition (b) we see that: (xii) t avoids π(c)(κ, δ) and π(d)(κ, δ) over M ∗ . Conditions (iii)–(xii) combined directly give: (B) Over M ∗ , $κ, δ, t% is a π(κ)-step which satisfies each of the following conditions: (1) $κ, δ, t% is an a-extension of $tρ, t (ρ)%; (2) t avoids π(c)(κ, δ) and π(d)(κ, δ); and (3) t realizes π( σ ). But condition (B) contradicts the shift of condition (A) to M ∗ via the elementary embedding π . (For the sake of precision let us note that condition (A) can be stated over any extension of M which has tρ and t (ρ) as elements. Since t is nice, and since ρ is smaller than κ, tρ and t (ρ) belong to a generic extension of M of size less than κ. π can be extended to act on this generic extension, and used to shift condition (A) to # the corresponding generic extension of M ∗ .)
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Corollary 5A.12. Let δ belong to W and let t be a nice annotated position of relative domain δ + 1. Suppose that t e(δ) is M-clear. Suppose that t itself is not I-acceptably obstructed and not II-acceptably obstructed. Suppose that t is saturated. Then t is M-clear. Proof. Suppose for contradiction that t is not M-clear. Since t e(δ) is M-clear this must mean that t itself is obstructed. Using Lemma 4B.32 fix a minimal obstruction $E, σ % for t. Now apply Lemma 5A.11. The lemma says that $E, σ % is I-acceptable, or II-acceptable, or both. But this contradicts the assumption that t is not acceptably obstructed. #
5B Successors, basic step Let δ † be a relative successor in W . Fix δ † names C˙ † and D˙ † . Let µ = e(δ † ). Fix, in some generic extension of M, an annotated position t of relative domain µ. Suppose that: (S1) t is nice, saturated, and M-clear. Fix a witness w for µ. Suppose that: (S2) w is amenable to t (see Definitions 4A.14 and 4A.19), and there are infinitely many numbers which are not used in w. Let g † be col(ω, δ † )-generic/M. Suppose that: (S3) t and w belong to M[g † ]. We work under these assumptions to prove the following lemma: Lemma 5B.1 (under the assumptions (S1)–(S3)). Suppose that ϕsuc (t, w, C˙ † ) and ψsuc (t, w, D˙ † ) both fail. Then there exists some real y ∈ M[g † ] so that t−−, w, y is nice, saturated, M-clear, and belongs to neither C˙ † [g † ] nor D˙ † [g † ]. Regarding the statement of Lemma 5B.1 let us note that ϕsuc (t, w, C˙ † ) and ψsuc (t, w, D˙ † ) are absolute for generic extensions of M which have t and w as elements. We can therefore simply state that ϕsuc (t, w, C˙ † ) and ψsuc (t, w, D˙ † ) fail, without having to say where. We will use the fact that they fail in M[g † ]. Proof of Lemma 5B.1. We follow the notation of Sections 4C (2) and 4E (2). The assumption that ϕsuc (t, w, C˙ † ) fails implies that I does not have a winning strategy in G∗suc (t, w, C˙ † ). G∗suc (t, w, C˙ † ) is an open game, and hence determined. Since the game is not won by I, it must be won by II (the closed player). Let σ ∗ be a winning strategy for II in G∗suc (t, w, C˙ † ). Since winning an open game is absolute we may pick σ ∗ ∈ M[g † ]. Working similarly with the assumption that ψsuc (t, w, D˙ † ) fails let us ∗ (t, w, D ˙ † ). pick a strategy τ ∗ ∈ M[g † ] which is winning for I (the closed player) in Hsuc
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Let σgen ∈ M[g † ] be the generic strategies map associated to the name A˙ † of Section 4C (2). Let τgen ∈ M[g † ] be the mirrored generic strategies map associated to the name B˙ † of Section 4E(2). Working inside M[g † ] we construct y, a , and b so that: (1) y = $y(n) | n < ω% is a real; (2) y and a form an infinite play of G∗suc (t, w, C˙ † ) played according to both σ ∗ and σgen [t, w, y]; ∗ (t, w, D ˙ † ) played according to both τ ∗ and (3) y and b form an infinite play of Hsuc τgen [t, w, y].
The construction follows the usual lines. σgen [y] produces the moves an−I for all n; σ ∗ produces the numbers y(n) for odd n and the moves an−II for all n; τ ∗ produces the numbers y(n) for even n and the moves bn−I for all n; and τgen [y] produces the moves bn−II for all n. As always the construction uses the Lipschitz continuity of the maps σgen and τgen . From condition (2) above and Lemma 1B.2 we get $t, w, y% ∈ A† . From this and the definition of A† in Section 4C(2) it follows that: (i) t−−, w, y = t, w y is not an element of C˙ † [g † ]; and (ii) t−−, w, y is not I-acceptably obstructed. Working similarly with condition (3), Lemma 1D.2, and the definition of B † in Section 4E (2) we get: (iii) t−−, w, y is not an element of D˙ † [g † ]; and (iv) t−−, w, y is not II-acceptably obstructed. Claim 5B.2. t−−, w, y is nice. Proof. Conditions (N1)–(N4) of Definition 5A.1 are easy to verify using assumptions (S1)–(S3) and the fact that y belongs to M[g † ]. Let us only note that condition (N3) in the case κ = µ is proved using the assumption that w is amenable to t in condition (S2), and using the assumption that t itself is nice in condition (S1). # Claim 5B.3. t−−, w, y is saturated. Proof. Saturation only imposes conditions on relative limits (see Definition 5A.8). Since δ † is a relative successor, the set of relative limits below rdm(t−−, w, y) is precisely equal to the set of relative limits below rdm(t). The saturation of t therefore directly implies the saturation of t−−, w, y. # Claim 5B.4. t−−, w, y is M-clear.
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Proof. This follows from the assumption that t is M-clear in condition (S1), conditions (ii) and (iv) above, and Claim 5B.3, using Corollary 5A.12 on the annotated position t−−, w, y. # We established now that t−−, w, y is nice, saturated, and M-clear. Conditions (i) # (Lemma 5B.1) and (iii) above state that it belongs to neither C˙ † [g † ] nor D˙ † [g † ].
5C Relative limits Fix a relative limit τ ∈ W . Let Y˙ and Z˙ be τ -names. We plan to prove the following lemma: Lemma (5C.9). Let δ < τ be an element of W and let t be a nice, saturated δ-sequence. ˙ Then there exists a nice, Suppose that t avoids Back I (δ, τ )(Y˙ ) and Back II (δ, τ )(Z). ˙ ˙ saturated τ -sequence s which extends t and avoids Y and Z. Remark 5C.1. By a nice, saturated δ-sequence we of course mean a δ-sequence t which is nice (Definition 5A.1) and saturated (Definition 5A.8) as an annotated position. The complete proof of Lemma 5C.9 is an induction which stretches to Section 5G. For the time being let us just say that we use the following hypothesis during the proof, but only for χ ∗ < τ : Hypothesis 5C.2. Let χ ∗ belong to W . Let C˙ ∗ and D˙ ∗ be χ ∗ -names. Let χ < χ ∗ belong to W . Let r be a nice, saturated χ-sequence. Suppose that r avoids Back I (χ, χ ∗ )(C˙ ∗ ) and Back II (χ , χ ∗ )(D˙ ∗ ). Then there exists a nice, saturated χ ∗ -sequence r ∗ which extends r and avoids C˙ ∗ and D˙ ∗ . 5C (1) Basic step. Recall that we are working with a fixed relative limit τ ∈ W ˙ We follow the notation of Sections 4C (3) and 4E (3). and fixed τ -names Y˙ and Z. ˙ ] and associated auxiliary Section 4C (3) defines (among other things) a list of names R[γ ˙ games maps R[γ ]. Section 4E(3) defines names S[γ ] and mirrored auxiliary games maps S[γ ]. Let $γL , γH % be the least pair of local indiscernibles of M relative to τ . ˙ L ] is equal to R[γ ˙ H ]. Claim 5C.3. R[γ ˙ ] is uniformly definable in M&γ + ω from the parameters: Proof. Note that R[γ (1) γ ; (2) c0 coding the canonical equivalent to Y˙ (by this we mean the canonical name for the set named by Y˙ ); (3) c1 coding the pullback operation below τ ;
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(4) c2 coding the map which assigns to each canonical col(ω, τ )-name R˙ for a subset of (M&τ )ω × (M& e(τ ))ω its corresponding auxiliary games map R; and (5) c3 coding the map which assigns to each relative successor δ † < τ and each canonical δ † -name C˙ † , the canonical name in the collapse to the largest Woodin cardinal below δ † for the set of pairs $t, w% so that ϕsuc (t, w, C˙ † ) holds true. ˙ ] from these parameters is the formalization of The uniform definition of R[γ ˙ Section 4C (3). Y is a col(ω, τ )-name for a set of τ -sequences. Its canonical equivalent can be coded as an element of M&τ + 2. The parameters c1 and c3 are subsets of M&τ . The parameter c2 is a map from M&τ + 2 into M&τ + 1. Thus all the parameters c0 , . . . , c3 belong to M&τ + ω. The claim now follows from the local indiscernibility of γL and γH . # Claim 5C.4. For each P ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ ˙ δ, P , γH ). C(κ, δ, P , γL ) is equal to C(κ, Proof. This again follows from the local indiscernibility of γL and γH .
#
Claim 5C.5. Let k < ω be given. There exists a sequence $γi | i < k% so that: (1) γi < γi−1 for each i ∈ (0, k − 1], and γ0 < γL ; ˙ L ]; and ˙ i ] is equal to R[γ (2) for each i ≤ k − 1, R[γ (3) for each i ≤ k − 1, each P ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and each κ ∈ δ ∩ L, ˙ ˙ C(κ, δ, P , γi ) is equal to C(κ, δ, P , γL ). ˙ L ]. Let c5 be the function κ, δ, P → C(κ, ˙ δ, P , γL ), defined Proof. Let c4 equal R[γ <ω on P ∈ (M&τ ) , δ ∈ τ ∩ W , and κ ∈ δ ∩ L. Note that c4 and c5 are both elements of M&τ + ω. Let c0 , . . . , c3 be the parameters used in the proof of Claim 5C.3. Let φk (γ , c0 , . . . , c5 ) be the formula which says: “there exists below γ a descending se˙ i ] is equal to c4 for each i ≤ k − 1, and the quence γ0 > γ1 > · · · > γk−1 so that R[γ ˙ map κ, δ, P → C(κ, δ, P , γi ) is equal to c5 for each i ≤ k − 1.” (The references to ˙ . . , γi ) can be made precise using the parameters c0 , . . . , c3 .) ˙ i ] and C(. R[γ From the local indiscernibility of γL and γH it follows that φk [γ , c0 , . . . , c5 ] holds for γ = γL iff it holds for γ = γH . It is easy from this to prove Claim 5C.5 by induction on k. # Remark 5C.6. Indiscernibility claims similar to Claims 5C.3 and 5C.5 can be made ˙ ] and D(. ˙ . . , γ ) of Section 4E (3). for the mirrored names S[γ Fix some h which is col(ω, τ )-generic/M. Let σgen ∈ M[h] be the generic strategies ˙ L ]. Let τgen ∈ M[h] (not to be confused with the fixed map associated to the name R[γ Woodin cardinal τ ∈ W ) be the mirrored generic strategies map associated to the ˙ L ]. name S[γ Fix δ ∈ τ ∩ W . Fix κ ∈ δ ∩ W . Fix n < ω. Fix P , Q ∈ (M&τ )<ω , both of length n, say P = $a0 , . . . , an−1 % and Q = $b0 , . . . , bn−1 %. Fix an annotated position t. Suppose that:
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(L1) t is a nice, saturated δ-sequence; and (L2) t (κ) does not use n. Let w denote t (κ) + $n, δ%. This is a witness for δ + 1 = rdm(t). Suppose that: (L3) P is a position in R[γL , t, w], played according to σgen [t, w]; (L4) Q is a position in S[γL , t, w], played according to τgen [t, w]; and ˙ ˙ δ, Q, γL ). (L5) t avoids C(κ, δ, P , γL ) and D(κ, We work under these assumptions to prove the following claim, which we will use later on in the proof of Lemma 5C.9. Claim 5C.7 (under the assumptions (L1)–(L5)). Let ζ < e(τ ) and m ∈ ω be given. Suppose that m is greater than n and not used in w. Then there is a δ ∗ ∈ τ ∩ W greater than ζ , an n∗ ∈ ω greater than m, a κ ∗ ∈ ∗ δ ∩ L, an annotated position t ∗ , and finite sequences P ∗ and Q∗ , so that the following conditions, (1)–(9), hold: (1) t ∗ is a nice, saturated δ ∗ -sequence; (2) t ∗ extends t strictly; (3) t ∗ (δ + 1) = w; (4) t ∗ (κ ∗ ) is an m-extension of t ∗ (δ + 1) (in particular κ ∗ ≥ δ + 1); (5) t ∗ (κ ∗ ) does not use m; (6) t ∗ (κ ∗ ) does not use n∗ . Let w ∗ denote t ∗ (κ ∗ ) + $n∗ , δ ∗ %. This is a witness for δ ∗ + 1 = rdm(t ∗ ). (7) P ∗ is a position of length n∗ in R[γL , t ∗ , w∗ ], extending P , and played according to σgen [t ∗ , w∗ ]; (8) Q∗ is a position of length n∗ in S[γL , t ∗ , w∗ ], extending Q, and played according to τgen [t ∗ , w∗ ]; and ˙ ∗ , δ ∗ , P ∗ , γL ) and D(κ ˙ ∗ , δ ∗ , Q∗ , γL ). (9) t ∗ avoids C(κ Conditions (1) and (6)–(9) should be viewed as parallels of conditions (L1)–(L5), setting the grounds for an iterated use of Claim 5C.7. Conditions (2)–(5) should be viewed as stating the particular way in which the objects κ ∗ , δ ∗ , and t ∗ extend κ, δ, and t.
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Proof of Claim 5C.7. Fix some g so that: g is col(ω, δ)-generic/M; t belongs to M[g]; ˙ ˙ and t belongs to neither C(κ, δ, P , γL )[g] nor D(κ, δ, P , γL )[g]. This is possible using ˙ condition (L5). Using the fact that t belongs to M[g] but not to C(κ, δ, P , γL )[g], Definition 4C.14, the fact that t is a δ-sequence hence in particular M-clear, and assumption (L2), we see that I does not win G∗lim (t, κ, δ, P , γL ). But the game is clopen, hence determined. So II must win. Fix a winning strategy σ for player II in G∗lim (t, κ, δ, P , γL ). Using condition (N4) in Definition 5A.1 fix some n∗ > m which is not used in t (κ). Let $γi | i < n∗ % be given by Claim 5C.5 applied with k = n∗ . Fix some δ ∗ in τ ∩ W , larger than the first Woodin cardinal above δ and larger than ζ . This is possibly since δ and ζ are both smaller than e(τ ), and τ is a relative limit. δ ∗ , n∗ , $γi | n ≤ i < n∗ %, σgen [t, w] and σ combine to produce a run of ∗ Glim (t, κ, δ, P , γL ) as presented in Diagram 4.2. We leave it to the reader to verify that this combination satisfies all the necessary rules stated in Section 4C (3). Let us only note that one uses condition (L3) to start the construction off, and uses the fact that R[γi ] = R[γL ] as the construction progresses through rounds i ≥ n. Let $ai | n ≤ i < n∗ % be the auxiliary moves in the run of G∗lim (t, κ, δ, P , γL ) constructed above. Following the notation of Section 4C (3) let Pn denote P , let Pi+1 = Pi −−, ai , and let P ∗ = Pn∗ . We have: (ii) P ∗ is a position of length n∗ in R[γL , t, w], extending P , and played according to σgen [t, w]. In phrasing this condition we are making use the indiscernibility given by Claim 5C.5. The indiscernibility allows us to switch from R[γn∗ −1 , . . . ] to R[γL , . . . ]. Continuing to follow the notation of Section 4C (3) set κ ∗ = δ + 1 and let δ † denote the first Woodin cardinal above δ. Since P ∗ is part of a run of G∗lim (. . . ) played according to σ , and since σ is a winning strategy for II, we have: (iii) ϕsuc (t, w, C˙ † ) fails with the assignment C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ), where C˙ ∗ = ˙ ∗ , δ ∗ , P ∗ , γL ). C(κ This condition is essentially the negation of the payoff condition (P) in Section 4C (3). By a strict reading of that payoff condition we should have set C˙ ∗ = ˙ ∗ , δ ∗ , P ∗ , γn∗ −1 ) in condition (iii). But Claim 5C.5 states that this is the same as C(κ ˙ ∗ , δ ∗ , P ∗ , γL ). C(κ Working as we did above but with Q, the condition (L4), and the fact that t ∈ ˙ D(κ, δ, Q, γL )[g], we obtain Q∗ so that: (iv) Q∗ is a position of length n∗ in S[γL , t, w], extending Q, and played according to τgen [t, w]; and (v) ψsuc (t, w, D˙ † ) fails with the assignment D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ), where D˙ ∗ = ˙ ∗ , δ ∗ , Q∗ , γL ). D(κ The collapse to δ is absorbed by the collapse to δ † . So we may fix g † which is col(ω, δ † )-generic/M, with g ∈ M[g † ]. Since t belongs to M[g] we have t ∈ M[g † ].
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Note that w then also belongs to M[g † ], since it is easily obtained from t (κ). By conditions (iii) and (v) we are in a position to use Lemma 5B.1. An application of the lemma produces a nice, saturated δ † -sequence t † ∈ M[g † ] so that: (vi) t † extends t and t † (δ + 1) = w; and (vii) t † ∈ M[g † ] belongs to neither C˙ † [g † ] nor D˙ † [g † ]. Rephrasing condition (vii) we see that t † avoids C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ) and D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ). From this an application of Hypothesis 5C.2 produces a nice, saturated δ ∗ -sequence t ∗ so that: (viii) t ∗ extends t † ; and (ix) t ∗ avoids C˙ ∗ and D˙ ∗ . δ ∗ , n∗ , κ ∗ = δ + 1, t ∗ , P ∗ , and Q∗ are easily seen to satisfy the requirements in the conclusion of Claim 5C.7. Let us only comment on conditions (7) and (8). w∗ by definition is equal to t ∗ (κ ∗ ) + $n∗ , δ ∗ %. It follows from this that w∗ is an n∗ -extension of t ∗ (κ ∗ ). t ∗ (κ ∗ ) is just t ∗ (δ + 1), which is equal to t † (δ + 1), which in turn is equal to w. w∗ is thus an n∗ -extension of w. Using Claim 4A.18 we now see that t, w and t ∗ , w∗ agree to n∗ . Conditions (7) and (8) therefore follow from conditions (ii) and (iv). # Remark 5C.8. The proof of Claim 5C.7 used Hypothesis 5C.2. We used the hypothesis with χ = δ † , r = t † , and χ ∗ = δ ∗ . δ ∗ was chosen smaller than τ . So we only needed the hypothesis for χ ∗ < τ . 5C (2) Construction. We continue to work with a fixed relative limit τ ∈ W , and ˙ The next lemma achieves the intention stated at the start of fixed τ -names Y˙ and Z. Section 5C: Lemma 5C.9 (using Hypothesis 5C.2, but only for χ ∗ < τ ). Let δ < τ be an element of W . Let t be a nice, saturated δ-sequence. Suppose that t avoids Back I (δ, τ )(Y˙ ) and ˙ Then there exists a nice, saturated τ -sequence s which extends t and Back II (δ, τ )(Z). ˙ ˙ avoids Y and Z. Proof. By condition (N1) in Definition 5A.1, t belongs to a generic extension of M by col(ω, δ). col(ω, δ) is absorbed into col(ω, τ ). We can therefore fix some h which is col(ω, τ )-generic/M with t ∈ M[h]. We work inside M[h] throughout the proof. We follow the notation of Section 5C(1). σgen ∈ M[h] is the generic strategies ˙ L ]. τgen ∈ M[h] is the mirrored generic strategies map map associated to the name R[γ ˙ associated to the name S[γL ]. R and S are defined in Sections 4C (3) and 4E (3). ˙ Looking at DefiniBy assumption t avoids Back I (δ, τ )(Y˙ ) and Back II (δ, τ )(Z). tions 4C.20 and 4E.11 we see that: ˙ ˙ (i) t avoids C(0, δ, ∅, γL ) and D(0, δ, ∅, γL ).
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Starting with condition (i) and using repeated applications of Claim 5C.7 one can easily construct an s which satisfies the requirements of Lemma 5C.9, except that s may not be saturated at τ and obstruction free. We expect saturation to inhibit obstructions using Corollary 5A.12. So the key to the construction is saturation at τ . We intend to secure saturation by making s realize as many e(τ )-iseqs as possible. Let λ denote e(τ ). Let J be the set of all λ-iseqs. Claim 5C.10. The set { σ ∈ J | no extension of t realizes σ } is infinite. Proof. A λ-iseq σ = $ σ (ξ ) | ξ < λ% is constant if σ (ξ ) = σ (0) for all ξ < λ. Let I be the set of constant λ-iseqs σ so that ht( σ (0)) ≤ δ + 1 and t |= σ (0). It is easy to see that I is infinite. The claim follows; if σ belongs to I then it cannot be realized by any extension of t. # J is a subset of M&λ + 1, and hence countable in M[h]. Fix in M[h] some enumeration $ σk | k < ω% of J . λ = e(τ ) is also countable in M[h]. Since τ is a relative limit in W , λ is a limit ordinal. Fix in M[h] some increasing sequence $ζk | k < ω% so that ζk < λ for each k, and supk<ω ζk = λ. We work to construct for each k < ω the objects listed in conditions (A)–(E) below: (A) δk ∈ τ ∩ W and κk ∈ L with κk < δk ; (B) a nice, saturated δk -sequence tk ; (C) a natural number nk which is not used in tk (κk ). Let wk denote tk (κk ) + $nk , δk %. This is a witness for δk + 1 = rdm(tk ). (D) a position Pk of length nk in R[γL , tk , wk ], played according to the generic strategy σgen [tk , wk ]; and (E) a position Qk of length nk in S[γL , tk , wk ], played according to the generic strategy τgen [tk , wk ]. Let κ0 = 0, let δ0 = δ, let n0 = 0, let P0 = Q0 = ∅, and let t0 = t. This assignment trivially satisfies the above condition. We continue the construction by induction on k. Having defined the objects of conditions (A)–(E) for k we fix: (F) a number mk which is not used in tk (κk ), with mk > nk . This is possible by condition (N4) in Definition 5A.1, which states that there are infinitely many numbers not used in tk (κk ). We then define the objects of conditions (A)–(E) for k + 1 according to one of the two cases described below. In both cases we make sure that: (1) tk+1 extends tk strictly; (2) tk+1 (δk + 1) = wk ;
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(3) tk+1 (κk+1 ) is an mk -extension of tk+1 (δk + 1) (in particular κk+1 ≥ δk + 1); (4) Pk+1 extends Pk ; (5) Qk+1 extends Qk ; and ˙ k+1 , δk+1 , Pk+1 , γL ) and D(κ ˙ k+1 , δk+1 , Qk+1 , γL ). (6) tk+1 avoids C(κ Case 1. If there exists an assignment which satisfies conditions (A)–(E) for k + 1, satisfies conditions (1)–(6) above, and satisfies in addition the case condition (I) below. In this case we fix some assignment which satisfies all these conditions. (I) nk+1 = mk ; and tk+1 realizes σk .
# (Case 1)
Case 2. Otherwise. Fix in this second case an assignment which satisfies conditions (A)–(E) for k + 1, satisfies conditions (1)–(6) above, and satisfies in addition: (II) δk+1 > ζk ; nk+1 > mk ; and mk is not used in tk+1 (κk+1 ).
# (Case 2)
The construction described by the two cases above is a matter of folding a bias for realizing λ-iseqs into the natural construction suggested by Claim 5C.7. Note that for each k the objects δk , κk , Pk , Qk , and tk satisfy the assumptions (L1)–(L5) in Section 5C (1). Using Claim 5C.7 it is always possible to find an assignment for k + 1 which satisfies the requirements of case 2. (By routine absoluteness it is possible to pick this assignment in M[h].) But the construction first searches for an assignment which satisfies the conditions of case 1. Only if no such assignment is found does the construction retreat to a use of Claim 5C.7 to obtain the conditions of case 2. Let us develop some properties of the objects constructed above. Some of these properties are trivial and we list them for the record, without proof. Claim 5C.11. The sequence $mk | k < ω% is strictly increasing.
#
Claim 5C.12. For each k < ω, wk+1 is an mk -extension of wk . Proof. wk+1 by definition is equal to tk+1 (κk+1 ) + $nk+1 , δk+1 %. tk+1 (κk+1 ) is an mk extension of wk by conditions (2) and (3). nk+1 ≥ mk by the case conditions (I) and (II). The claim follows. # Define T1 ⊂ ω by setting k ∈ T1 iff the objects of conditions (A)–(E) for k + 1 were constructed according to case 1. The next claim formalizes the bias towards case 1 in the construction. Claim 5C.13 (for k < ω). Suppose that there is a δ¯ ∈ τ ∩ W , a κ¯ < δ¯ in L, a nice, ¯ in (M&τ )<ω , so that the following conditions, ¯ saturated δ-sequence t¯, and P¯ and Q (C1)–(C8), hold: (C1) t¯ extends tk strictly; (C2) t¯(δk + 1) = wk ;
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(C3) t¯(κ) ¯ is an mk -extension of wk ; (C4) t¯(κ) ¯ does not use mk . ¯ This is a witness for δ¯ + 1 = rdm(t¯). Let w¯ = t¯(κ) ¯ + $mk , δ%. (C5) P¯ is a position of length mk in R[γL , t¯, w], ¯ extending Pk and played according to σgen [t¯, w]; ¯ ¯ is a position of length mk in S[γL , t¯, w], (C6) Q ¯ extending Qk and played according ¯ ) to τgen [t¯, w]; ˙ κ, ˙ κ, ¯ P¯ , γL ) and D( ¯ P¯ , γL ); and (C7) t¯ avoids C( ¯ δ, ¯ δ, (C8) t¯ realizes σk . Then k belongs to T1 . Proof. Simply note that the conditions assumed in the claim correspond precisely to the requirements of case 1 of the construction. The use of mk in conditions (C4)–(C6) is not an error; it corresponds to the requirement nk+1 = mk in the case condition (I). # Claim 5C.14 (for k < ω). If k ∈ T1 then tk+1 realizes σk . Proof. If k ∈ T1 then the case condition (I) holds for tk+1 .
#
Define T2 ⊂ ω by setting k ∈ T2 iff the objects of conditions (A)–(E) for k + 1 were constructed according to case 2. (In other words define T2 = ω − T1 .) Claim 5C.15. T2 is infinite. Proof. By Claim 5C.10 there are infinitely many k < ω so that no extension of t realizes σk . Every such k must belong to T2 , since it couldn’t possibly admit an assignment which satisfies the case condition (I). # Claim 5C.16. supk<ω δk = supk<ω κk = λ. Proof. The fact that supk<ω δk = λ follows from Claim 5C.15 and the requirement δk+1 > ζk in the case condition (II). The fact that supk<ω κk = supk<ω δk follows from conditions (A) and (3). # Let t∞ = k<ω tk . The union makes sense by condition (1). t∞ is a nice annotated position of relative domain λ. For each k we have t∞ (δk + 1) = wk by condition (2). Let w∞ = k<ω wk . The union makes sense by Claim 5C.12. w∞ is a witness for λ, amenable to t∞ (see Definition 4A.19). Claim 5C.17. t∞ is M-clear.
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Proof. For each k < ω, tk is a δk -sequence and so in particular M-clear. It is easy to check that the increasing union of M-clear annotated positions is M-clear. The claim follows. # Claim 5C.18 (for k < ω). w∞ is an mk -extension of wk . #
Proof. Immediate from Claims 5C.11 and 5C.12. Claim 5C.19. Suppose k ∈ T2 . Then mk is not used in w∞ .
Proof. By the case condition (II), mk is not used in tk+1 (κk+1 ) and nk+1 > mk . From this it follows that mk is not used in wk+1 = tk+1 (κk+1 ) + $nk+1 , δk+1 %. Since w∞ is an mk+1 -extension of wk+1 , and since mk+1 > mk , it follows that mk is not used in w∞ . # Let x∞ = t∞ , w∞ . Using Claims 5C.18 and 4A.18 we see that x∞ and tk , wk agree to mk . By condition (F), mk is greater than nk . So x∞ and tk , wk certainly agree to nk . From this and condition (D)it follows that Pk is a position in R[γL , x∞ ], played according to σgen [x∞ ]. Let a = k<ω Pk . The union makes sense by condition (4). a is an infinite run of R[γL , x∞ ], played according to σgen [x∞ ]. Using Lemma 1B.2 it ˙ L ][h]. Looking at Definition 4C.18 we conclude that: follows that $ a , x∞ % ∈ R[γ (iv) For every v ∈ (M&λ + 1)ω ∩ M[h], x∞ (ua,γL ⊕ v) is neither: (a) an element of Y˙ [h], nor (b) I-acceptably obstructed. ua,γL here is the I-record attached to a and γL , see Definition 4C.17. Let b = k<ω Qk . By reasoning similar to the above but using conditions (E) and (5), Lemma 1D.2, and Definition 4E.9 we get: ) is neither: (v) For every u ∈ (M&λ + 1)ω ∩ M[h], x∞ (u ⊕ vb,γ L
˙ (a) an element of Z[h], nor (b) II-acceptably obstructed. here is the II-record attached to b and γL , see Definition 4E.8. vb,γ L
. Let s = x∞ y∞ . This is the same as setting s = Let y∞ = ua,γL ⊕ vb,γ L t∞ −−, w∞ , y∞ . Claim 5C.20. s ∈ M[h] is a nice annotated position of relative domain τ + 1, extending t.
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Proof. t∞ is an annotated position of relative domain λ = e(τ ), w∞ is a code for λ, and y∞ belongs to (M&λ + 1)ω . From this it follows that s = t∞ −−, w∞ , y∞ is an annotated position of relative domain τ + 1. s extends t0 by construction, and t0 was set equal to t, so s extends t. s belongs to M[h] since the entire construction takes place inside M[h]. It is easy to verify that s is nice. Let us only make two comments on this matter. From Claims 5C.15 and 5C.19 it follows that there are infinitely many numbers not used in w∞ , and this secures condition (N4) in Definition 5A.1 for κ = λ. Condition (N3) in the case κ = λ is proved using the fact that w∞ is amenable to t∞ , # and the fact that t∞ itself satisfies condition (N3). Claim 5C.21. s is saturated at τ . Proof. This claim is the reason for the inclusion of case 1 in the construction above. Any λ-iseq which is not realized by s corresponds to a stage in the construction where case 2 was taken. In other words it corresponds to a stage where objects satisfying the conditions of case 1 could not be found. We shall use this to argue for saturation. Fix a λ-iseq σ . Let us verify the statement of Definition 5A.7 for this particular iseq. If σ is realized by s then Definition 5A.7 requires nothing. So we may assume: (a) σ is not realized by s. Using the fact that $ σk | k < ω% enumerates J we may find k < ω so that σ is equal to σk . By condition (a), σk is not realized by s and so certainly not realized by any initial segment of s. In particular σk is not realized by tk+1 . Using Claim 5C.14 it follows that: (b) k ∈ T2 . From this and Claim 5C.19 it follows that mk is not used in w∞ . This in turn implies that {mk } ∪ {i < mk | i ∈ dom(w∞ )} is (a finite set) contained in ω − dom(w∞ ) = ω − dom(s(λ)). Let m = mk , let ρ = δk + 1, and let a = {mk } ∪ {i < mk | i ∈ dom(w∞ )}. We prove that this particular assignment witnesses the truth of the existential statement in Definition 5A.7, for the iseq σk . ¯ t¯%. We have to verify that at least one of the conditions in Fix a κ-step $κ, ¯ δ, Definition 5A.7 fails. Suppose otherwise. In other words suppose that: ¯ t¯% is an a-extension of $sρ, s(ρ)% = $tk , wk %; (c) $κ, ¯ δ, ¯ and d(κ, ¯ where c = ua,γ (m) and d = v (m); and (d) t¯ avoids c(κ, ¯ δ) ¯ δ) b,γ L L
(e) t¯ realizes σk . We work to obtain a contradiction from these conditions. In translating from condition (2) in Definition 5A.7 to condition (d) here we already . When we further factor factored in the fact that s(τ ) = y∞ and y∞ = ua,γL ⊕ vb,γ L in Definitions 4C.17 and 4E.8 we get:
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˙ κ, ˙ κ, ¯ bm, ¯ a m, γL ) and D( ¯ δ, γL ). (f) t¯ avoids C( ¯ δ, Recall that a was set equal to {mk } ∪ {i < mk | i ∈ d(w∞ )}. By Claim 5C.18, w∞ is an mk -extension of wk . So an i < mk is used in w∞ iff it is used in wk . Thus: (g) a = {mk } ∪ {i < mk | i ∈ dom(wk )}. ¯ t¯% is an a-extension of $tk , wk % it is easy Using condition (g) and the fact that $κ, ¯ δ, to check that κ, ¯ δ¯ and t¯ satisfy conditions (C1)–(C4) in Claim 5C.13. Following the ¯ w¯ is an mk -extension of t¯(κ), ¯ which notation of Claim 5C.13 let w¯ = t¯(κ) ¯ + $mk , δ%. by condition (C3) is an mk -extension of wk . So w¯ is an mk -extension of wk . Using this, condition (C1), and Claim 4A.18 it follows that t¯, w ¯ agrees with tk , wk to mk . ¯ and x∞ agree to mk . We saw already that tk , wk agrees with x∞ to mk . So t¯, w ¯ κ, It is now easy to verify that conditions (C5) and (C6) of Claim 5C.13 hold for δ, ¯ t¯, ¯ = bm k . The remaining conditions, (C7) and (C8), follow from P¯ = a mk , and Q conditions (e) and (f). Applying Claim 5C.13 we get k ∈ T1 . But this contradicts condition (b) above. # (Claim 5C.21) Claim 5C.22. s is saturated. Proof. s is saturated below τ since each tk is saturated. s is saturated at τ by Claim 5C.21. # So far we proved that s ∈ M[h] is a nice annotated position of relative domain τ +1, extending t, and saturated. Let us now continue to show that s is M-clear, and that s ˙ Conditions (iv) and (v) above provide the starting point. Applying avoids Y˙ and Z. we get: condition (iv) with v = vb,γ L
(vi) s ∈ Y˙ [h]; and (vii) s is not I-acceptably obstructed. Applying condition (v) with u = ua,γL we get: ˙ (viii) s ∈ Z[h]; and (ix) s is not II-acceptably obstructed. Claim 5C.23. s is M-clear. Proof. Use Corollary 5A.12 on the nice annotated position s. The assumptions of Corollary 5A.12 hold for s by Claim 5C.17, conditions (vii) and (ix) above, and Claim 5C.22. # By construction s extends t0 which is equal to t. Claims 5C.20, 5C.22, and 5C.23 combine to state that s is a nice, saturated τ -sequence. Conditions (vi) and (viii) ˙ So s satisfies the combined with the fact that s ∈ M[h] state that s avoids Y˙ and Z. requirements in the conclusion of Lemma 5C.9. # (Lemma 5C.9)
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Remark 5C.24. Note that Claim 5C.7 is used in the proof of Lemma 5C.9. The claim is needed to guarantee the existence of objects satisfying the requirements of case 2, which is used as a fall-back option during the construction. Hypothesis 5C.2 is used through the appeal to Claim 5C.7. But the hypothesis is only needed for χ ∗ < τ , by Remark 5C.8.
5D Woodin limits of Woodin cardinals Our intention next is to prove a parallel of Lemma 5C.9 for the case of a Woodin limit of Woodin cardinals. Definition 5D.1. Suppose δ is a Woodin limit of Woodin cardinals. Let C˙ and D˙ be ˙ δ-names. A δ-sequence t is said to avoid C˙ and D˙ if it belongs to neither C[G(t)] nor ˙ D[G(t)]. Definition 5D.1 is a companion to Definition 5A.4. There is no conflict between the two since Woodin limits of Woodin cardinals are excluded from W . The filter G(t) referred to in Definition 5D.1 is the one given by Definition 4B.23. In our context, the context of an annotated position t of relative domain δ, G(t) is equal to {[σ ]δ | σ is an identity in M&δ and t |= σ }. When t is a δ-sequence (hence in particular M-clear) and δ is a Woodin limit of Woodin cardinals, G(t) is Wδ -generic/M by Corollary 4B.30. Moreover t belongs to M[G(t)] by Claim 4B.41. The requirement in ˙ ˙ Definition 5D.1 that t belongs to neither C[G(t)] nor D[G(t)] is therefore non-trivial; t does belong to the generic extension M[G(t)]. Fix a Woodin limit of Woodin cardinals θ. Let Y˙ and Z˙ be θ -names. We work with these fixed objects for the rest of Section 5D. We shall use Hypothesis 5C.2, but only for χ ∗ < θ . We follow throughout the notation of Sections 4D (1) and 4E (4). In particular α < θ is the least ordinal which seals Y˙ and β < θ is the least ordinal which ˙ C(κ, ˙ ˙ seals Z. δ, n, γ ) and D(κ, δ, n, γ ) are the names determined by Definitions 4D.5 and 4E.13. The former is defined for δ ∈ [α, θ) ∩ W , κ ∈ δ ∩ L, n < ω, and γ ∈ ON. The latter is defined for δ ∈ [β, θ) ∩ W and κ, n, and γ as before. 5D (1) Basic step. Let $γL , γH % be the least pair of local indiscernibles of M relative to θ . ˙ Claim 5D.2. For each δ ∈ [α, θ) ∩ W , each κ ∈ δ ∩ L, and each n < ω, C(κ, δ, n, γL ) ˙ is equal to C(κ, δ, n, γH ). ˙ Proof. Note that C(κ, δ, n, γ ) is uniformly definable in M&γ + ω from the parameters: (1) γ ; (2) c0 coding the canonical equivalent to Y˙ ;
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(3) c1 coding the pullback operation below θ ; and (4) c2 coding the map which assigns to each relative successor δ † < θ and each canonical δ † -name C˙ † , the canonical name in the collapse to the largest Woodin cardinal below δ † for the set of pairs $t, w% so that ϕsuc (t, w, C˙ † ) holds true. ˙ The definition of C(κ, δ, n, γ ) from these parameters is simply the formalization of the definition given in Section 4D(1). With the exception of γ all the parameters listed are elements of M&θ + ω. The claim now follows from the local indiscernibility of γL and γH . # ˙ Claim 5D.3. For each δ ∈ [β, θ) ∩ W , each κ ∈ δ ∩ L, and each n < ω, D(κ, δ, n, γL ) ˙ is equal to D(κ, δ, n, γH ). Proof. Similar to the proof of the previous claim.
#
Fix now some δ ∈ [max{α, β}, θ) ∩ W , some κ ∈ δ ∩ L, some n < ω, and some annotated position s of relative domain δ + 1. Since the relative domain of s is greater ˙ Let us than α and β we can ask whether s is hopeful with respect to each of Y˙ and Z. here work under the following assumptions: ˙ (W1) s is not hopeful, not with respect to Y˙ and not with respect to Z; (W2) s is a nice, saturated δ-sequence; (W3) s(κ) does not use n; and ˙ ˙ δ, n, γL ). (W4) s avoids C(κ, δ, n, γL ) and D(κ, Claim 5D.4 (under the assumptions (W1)–(W4)). Let ζ < θ and m < ω be given. Then there is a δ ∗ ∈ θ ∩ W greater than ζ , an n∗ ∈ ω greater than m, and an annotated position s ∗ so that: (1) s ∗ extends s strictly; (2) s ∗ (δ + 1) = s(κ) + $n, δ%; ˙ (3) s ∗ is not hopeful, not with respect to Y˙ and not with respect to Z; (4) s ∗ is a nice, saturated δ ∗ -sequence; (5) n∗ not used in s ∗ (δ + 1); and ˙ + 1, δ ∗ , n∗ , γL ) and D(δ ˙ + 1, δ ∗ , n∗ , γL ). (6) s ∗ avoids C(δ Condition (3) in the conclusion of Claim 5D.4 is actually redundant; it follows from conditions (W1) and (1) by Claim 4D.3. We include condition (3) in the statement of Claim 5D.4 only to make clearer the connection between the assumptions of the claim and its conclusion. Conditions (3)–(6) precisely match assumptions (W1)–(W4), setting the grounds for an iterated use.
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Proof of Claim 5D.4. Using Claims 5D.2 and 5D.3 we can switch from γL to γH and ˙ ˙ rephrase condition (W4) to say that s avoids C(κ, δ, n, γH ) and D(κ, δ, n, γH ). We may thus fix some g so that: • g is col(ω, δ)-generic/M; • s belongs to M[g]; and ˙ ˙ • s belongs to neither C(κ, δ, n, γH )[g] nor D(κ, δ, n, γH )[g]. ˙ Using the fact that s belongs to M[g] but not C(κ, δ, n, γH )[g], Definition 4D.5, the fact that s is a δ-sequence, and condition (W3), we see that: (i) II wins the game G∗wl (s, κ, δ, n, γH ) of Section 4D (1). Similar reasoning using Definition 4E.13 and the fact that s does not belong to ˙ D(κ, δ, n, γH )[g] shows that: ∗ (s, κ, δ, n, γ ) of Section 4E (4). (ii) I wins the game Hwl H
G∗wl (s, κ, δ, n, γH ) is a finite game consisting of moves ν, δ ∗ , n∗ , and γ ∗ . The rules of the game are such that condition (i) translates to the following: (iii) There exists some ν < θ ; so that for every δ ∗ and n∗ which satisfy rule (2) in Section 4D(1); and for every γ ∗ < γH ; the payoff condition (P) in Section 4D (1) fails. Remark 5D.5. We are assuming here that s is not hopeful with respect to Y˙ . The move γ ∗ in G∗wl (s, κ, δ, n, γH ) therefore corresponds to player I, see Diagram 4.3 and Remark 4D.4. The phrasing of condition (iii) takes account of this fact; γ ∗ , like the other moves for I, is bounded by a universal quantifier in condition (iii). Condition (ii) similarly translates to: (iv) There exists some ν < θ ; so that for every δ ∗ and n∗ which satisfy rule (2) in Section 4E(4); and for every γ ∗ < γH ; the payoff condition (P) in Section 4E (4) fails. Here too γ ∗ is bounded by a universal quantifier, taking account of the fact that s is not ˙ hopeful with respect to Z. Let νa and νb respectively witness the existential statements in conditions (iii) and (iv). Following the notation of Sections 4D(1) and 4E (4) let δ † be the first Woodin cardinal greater than δ, let w = s(κ)+$n, δ%, and let κ ∗ = δ+1. Using the fact that θ is a limit of Woodin cardinals fix some δ ∗ ∈ θ ∩W which is greater than max{νa , νb }, greater than δ † , and greater than ζ . Using condition (N4) in Definition 5A.1 fix some n∗ < ω which is not used in s(κ), larger than m, and distinct from n. Since w = s(κ) + $n, δ% it follows that n∗ is also not used in w. Condition (iii), applied with the objects δ ∗ and n∗ fixed above and with γ ∗ = γL , implies that:
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(v) ϕsuc (s, w, C˙ † ) fails with the assignment C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ), where C˙ ∗ = ˙ ∗ , δ ∗ , n∗ , γL ). C(κ Condition (iv), similarly applied with the objects δ ∗ and n∗ fixed above and with γ ∗ = γL , implies that: (vi) ψsuc (s, w, D˙ † ) fails with the assignment D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ), where D˙ ∗ = ˙ ∗ , δ ∗ , n∗ , γL ). D(κ Condition (v) and (vi) allow for an application of Lemma 5B.1. The lemma produces a nice, saturated δ † -sequence s † which extends s and avoids C˙ † and D˙ † , with s † (δ + 1) = w. An application of Hypothesis 5C.2 then produces a nice, saturated δ ∗ -sequence s ∗ which extends s † and avoids C˙ ∗ and D˙ ∗ . It is easy to check that δ ∗ , n∗ , # and s ∗ satisfy the conditions stated in Claim 5D.4. Remark 5D.6. The proof of Claim 5D.4 uses Hypothesis 5C.2 with χ = δ † , r = s † , and χ ∗ = δ ∗ . Since δ ∗ was chosen smaller than θ , Hypothesis 5C.2 is only needed for χ∗ < θ. 5D (2) Impossibility. Let us again work with δ ∈ [max{α, β}, θ) ∩ W , κ ∈ δ ∩ L, n < ω, and an annotated position s of relative domain δ + 1. But let us now assume that: ˙ (I1) s is hopeful, both with respect to Y˙ and with respect to Z. (I2) s is a nice, saturated δ-sequence; and (I3) s(κ) does not use n. Note how assumption (I1) differs from the assumption (W1) in Section 5D (1). Let γa , γb ∈ ON be given. Assume further that: ˙ ˙ (I4) s avoids C(κ, δ, n, γa ) and D(κ, δ, n, γb ). Claim 5D.7 (under the assumptions (I1)–(I4)). There is a δ ∗ ∈ θ ∩ W , an n∗ ∈ ω, an annotated position s ∗ , and ordinals γa∗ and γb∗ so that: (1) s ∗ extends s strictly; (2) s ∗ (δ + 1) = s(κ) + $n, δ%; ˙ (3) s ∗ is hopeful, both with respect to Y˙ and with respect to Z; (4) s ∗ is a nice, saturated δ ∗ -sequence; (5) n∗ is not used in s ∗ (δ + 1); ˙ + 1, δ ∗ , n∗ , γa∗ ) and D(δ ˙ + 1, δ ∗ , n∗ , γ ∗ ); and (6) s ∗ avoids C(δ b (7) γa∗ < γa and γb∗ < γb , strictly.
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Claim 5D.7 is a parallel of Claim 5D.4 for the case that s is hopeful with respect ˙ ˙ Here too condition (3) is redundant, since it follows from conditions (I1) to Y and Z. and (1) using Claim 4D.3. We include condition (3) in the statement of Claim 5D.7 to make clearer the connection between the conclusion of the claim and its assumptions. The claim is specifically geared for an iterated use. Proof of Claim 5D.7. From assumption (I4) we get: (i) II wins the game G∗wl (s, κ, δ, n, γa ) of Section 4D (1); and ∗ (s, κ, δ, n, γ ) of Section 4E (4). (ii) I wins the game Hwl b
The rules of G∗wl (s, κ, δ, n, γa ) are such that condition (i) translates to: (iii) There exists some ν < θ ; so that for every δ ∗ and n∗ which satisfy rule (2) in Section 4D(1); there exists some γ ∗ < γa ; so that the payoff condition (P) in Section 4D(1) fails. Remark 5D.8. We are assuming in Claim 5D.7 that s is hopeful with respect to Y˙ . It follows from this assumption that the move γ ∗ in G∗wl (s, κ, δ, n, γa ) corresponds to player II, see Diagram 4.3 and Remark 4D.4. In phrasing condition (iii) we therefore bound γ ∗ by an existential quantifier. Condition (ii) similarly translates to: (iv) There exists some ν < θ ; so that for every δ ∗ and n∗ which satisfy rule (2) in Section 4E(4); there exists some γ ∗ < γb ; so that the payoff condition (P) in Section 4E(4) fails. Here too γ ∗ is bounded by an existential quantifier, taking account of the fact that s is ˙ hopeful with respect to Z. Let νa and νb respectively witness the outer existential statements in conditions (iii) and (iv). Following the notation of Sections 4D (1) and 4E (4) let δ † be the first Woodin cardinal greater than δ, let w = s(κ) + $n, δ%, and let κ ∗ = δ + 1. Using the fact that θ is a limit of Woodin cardinals fix some δ ∗ ∈ θ ∩ W which is greater than max{νa , νb }, and greater than δ † . Using condition (N4) in Definition 5A.1 fix some n∗ < ω which is distinct from n and not used in s(κ). Since w = s(κ) + $n, δ% it follows that n∗ is also not used in w. Condition (iii), applied with the objects δ ∗ and n∗ fixed above, states the existence of an ordinal γ ∗ < γa which makes condition (P) in Section 4D (1) fail. Let γa∗ be some such ordinal. Then: (v) ϕsuc (s, w, C˙ † ) fails with the assignment C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ), where C˙ ∗ = ˙ ∗ , δ ∗ , n∗ , γa∗ ). C(κ Condition (iv), similarly applied with the objects δ ∗ and n∗ fixed above, produces some ordinal γb∗ < γb so that:
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(vi) ψsuc (s, w, D˙ † ) fails with the assignment D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ), where D˙ ∗ = ˙ ∗ , δ ∗ , n∗ , γ ∗ ). D(κ b Using conditions (v) and (vi), Lemma 5B.1 produces a δ † -sequence s † which extends s and avoids C˙ † and D˙ † , with s † (δ + 1) = w. Hypothesis 5C.2 then produces a δ ∗ sequence s ∗ which extends s † and avoids C˙ ∗ and D˙ ∗ . It is easy to check that δ ∗ , n∗ , s ∗ , # γa∗ , and γb∗ satisfy the requirements in the conclusion of Claim 5D.7. The reader may wish to carefully compare the proof of Claim 5D.7 to that of Claim 5D.4. In the proof of Claim 5D.7 we have the ordinals γa∗ and γb∗ given to us by conditions (iii) and (iv). In the proof of Claim 5D.4 on the other hand it is up to us to pick γ ∗ when applying conditions (iii) and (iv). This difference between the two cases traces back to the definitions in Sections 4D (1) and 4E (4), specifically to Remarks 4D.4 and 4E.12. Corollary 5D.9. There are no objects which satisfy assumptions (I1)–(I4). Proof. Fix δ, κ, n, s, and γa , γb ∈ ON. Suppose for contradiction that these objects satisfy assumptions (I1)–(I4). Iterated applications of Claim 5D.7 produce, among other things, two sequences of ordinals $γka | k < ω% and $γkb | k < ω%. By condition (7) of the claim these sequences are descending, giving the desired contradiction. # Claim 5D.10. Let δ belong to [max{α, β}, θ) ∩ W . Let s be a nice, saturated, δ˙ ˙ δ, 0, γL ). Then it cannot be sequence. Suppose that s avoids C(0, δ, 0, γL ) and D(0, ˙ ˙ that s is hopeful with respect to both Y and Z. Proof. This is just the special case of Corollary 5D.9 corresponding to κ = 0, n = 0, and γa = γb = γL . # 5D (3) Another impossibility. Again let us work with δ ∈ [max{α, β}, θ) ∩ W , κ ∈ δ ∩ L, n < ω, and an annotated position s of relative domain δ + 1. Conditions (W1) and (I1), which we considered above, covered the cases that s is either hopeful with ˙ or hopeful with respect to both. Let us next consider respect to neither one of Y˙ and Z, the case that s is hopeful with respect to one but not the other. Let us assume that: ˙ (J1) s is hopeful with respect to Y˙ , but not hopeful with respect to Z. (J2) s is a nice, saturated δ-sequence; and (J3) s(κ) does not use n. Let γa ∈ ON be given. Assume further that: ˙ ˙ (J4) s avoids C(κ, δ, n, γa ) and D(κ, δ, n, γL ). ˙ . . ), and the local indisNote how we use the given ordinal γa in our reference to C(. ˙ . . ). cernible γL in our reference to D(.
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Claim 5D.11 (under the assumptions (J1)–(J4)). There is a δ ∗ ∈ θ ∩ W , an n∗ ∈ ω, an annotated position s ∗ , and an ordinal γa∗ so that: (1) s ∗ extends s strictly; (2) s ∗ (δ + 1) = s(κ) + $n, δ%; ˙ (3) s ∗ is hopeful with respect to Y˙ but not hopeful with respect to Z; (4) s ∗ is a nice, saturated δ ∗ -sequence; (5) n∗ is not used in s ∗ (δ + 1); ˙ + 1, δ ∗ , n∗ , γa∗ ) and D(δ ˙ + 1, δ ∗ , n∗ , γL ); and (6) s ∗ avoids C(δ (7) γa∗ < γa strictly.
#
We omit the proof of Claim 5D.11. It is simply a matter of crossing the proof of Claim 5D.4 with the proof of Claim 5D.7. Corollary 5D.12. There are no objects which satisfy assumptions (J1)–(J4). Proof. This is similar to Corollary 5D.9. The proof of Corollary 5D.9 produced two infinite descending sequences of ordinals. Here repeated use of Claim 5D.11 produces only one infinite descending sequence of ordinals. But this is still enough for a contradiction. # Claim 5D.13. Let δ belong to [max{α, β}, θ) ∩ W . Let s be a nice, saturated, δ˙ ˙ δ, 0, γL ). Then it cannot be sequence. Suppose that s avoids C(0, δ, 0, γL ) and D(0, ˙ ˙ that s is hopeful with respect to Y and not hopeful with respect to Z. Proof. This is the special case of Corollary 5D.12 corresponding to κ = 0, n = 0, and # γa = γL . The next claim is a dual to Claim 5D.13. Its proof, which we omit, involves phrasing a dual to Claim 5D.11 to deal with the case that s is hopeful with respect to Z˙ but not Y˙ . Claim 5D.14. Let δ belong to [max{α, β}, θ) ∩ W . Let s be a nice, saturated, δ˙ ˙ δ, 0, γL ). Then it cannot be sequence. Suppose that s avoids C(0, δ, 0, γL ) and D(0, ˙ that s is hopeful with respect to Z and not hopeful with respect to Y˙ . # Remark 5D.15. Claims 5D.10, 5D.13, and 5D.14 all use Hypothesis 5C.2. The hypothesis comes in through Claim 5D.7 and its parallels. Note that the hypothesis is only needed for χ ∗ < θ.
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5D (4) Construction. We now combine the steps of Sections 5D (1) through 5D (3) above to prove Lemma 5D.19, a parallel of Lemma 5C.9 for the case of a Woodin limit of Woodin cardinals. Definition 5D.16. A δ-sequence t is supernice if it is nice and if in addition there exists a witness w for rdm(t) so that: (1) w is amenable to t; and (2) there are infinitely many numbers which are not used in w. Remark 5D.17. If t is supernice then a witness w satisfying the conditions of Definition 5D.16 can be found in any generic extension of M which has t as an element and collapses rdm(t) to be countable. Remark 5D.18. If rdm(t) is a successor ordinal, δ + 1 say, then a witness w satisfying the conditions of Definition 5D.16 exists trivially; simply take w = {$0, δ%}. Nice δ-sequences are therefore trivially supernice when δ ∈ W . The two notions only differ when δ is a Woodin limit of Woodin cardinals. Recall that we are working with a fixed Woodin limit of Woodin cardinals θ, and ˙ Lemma 5D.19 is phrased for these objects. fixed θ -names Y˙ and Z. Lemma 5D.19 (using Hypothesis 5C.2, but only for χ ∗ < θ). Let χ < θ be an element of W . Let r be a nice, saturated χ -sequence. Suppose that r avoids Back I (χ , θ )(Y˙ ) and ˙ Then there exists a supernice, saturated θ-sequence t which extends Back II (χ , θ )(Z). ˙ r and avoids Y˙ and Z. ˙ fix Proof. Using the assumption that r avoids Back I (χ , θ )(Y˙ ) and Back II (χ , θ )(Z) some q so that: • q is col(ω, χ)-generic/M; • r belongs to M[q]; and ˙ • r belongs to neither Back I (χ, θ)(Y˙ )[q] nor Back II (χ , θ )(Z)[q]. Let χ † be the first Woodin cardinal above χ . Recall that α < θ is the least ordinal ˙ Let δ < θ be the first which seals Y˙ and β < θ is the least ordinal which seals Z. element of W which is greater than χ † and greater than max{α, β}. Let w¯ = {$0, χ%}. This is a witness for χ + 1. ˙ We know that r does not ˙ Let C˙ = C(0, δ, 0, γL ) and let A˙ = Back I (χ † , δ)(C). I ˙ belong to Back (χ, θ)(Y )[q]. Looking at Definition 4D.6 we see that this certainly implies: ˙ fails. (i) ϕsuc (r, w, ¯ A) ˙ ˙ Working as we did above Let D˙ = D(0, δ, 0, γL ) and let B˙ = Back I (χ † , δ)(D). ˙ we get: but with Definition 4E.14 and the fact that r ∈ Back II (χ , θ )(Z)[q]
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˙ fails. (ii) ψsuc (r, w, ¯ B) Conditions (i) and (ii) put us in a position to apply Lemma 5B.1. Using that lemma ˙ Now we obtain a nice, saturated χ † -sequence r † which extends r and avoids A˙ and B. I II † † ˙ ˙ ˙ ˙ A = Back (χ , δ)(C) and B = Back (χ , δ)(D). Using Hypothesis 5C.2 on r † we ˙ (The obtain next a nice, saturated δ-sequence s which extends r † and avoids C˙ and D. ∗ hypothesis is used with χ = δ. The use is valid since δ is smaller than θ.) So far we worked under the assumptions of Lemma 5D.19 to construct an s satisfying: (iii) s is a nice, saturated δ-sequence; (iv) s extends r; and ˙ (v) s avoids C˙ and D. s is a δ-sequence, and δ is larger than α and β. We can thus ask whether or not ˙ Using condition (v) we may apply s is hopeful with respect to each of Y˙ and Z. Claims 5D.10, 5D.13, and 5D.14. These claims together rule out the possibility that s ˙ or both. Thus: is hopeful with respect to either Y˙ , or Z, ˙ (vi) s is not hopeful, not with respect to Y˙ and not with respect to Z. Fix, in some generic extension of M, a sequence $ζk | k < ω% of ordinals so that ζk < θ for each k, and supk<ω ζk = θ. We work to construct for each k < ω the following objects: (A) δk ∈ θ ∩ W and κk ∈ L with κk < δk ; (B) a nice, saturated δk -sequence sk ; and (C) a natural number nk which is not used in sk (κk ). We make sure that: ˙ and (1) sk is not hopeful, not with respect to Y˙ and not with respect to Z; ˙ k , δk , nk , γL ) and D(κ ˙ k , δk , nk , γL ). (2) sk avoids C(κ Set δ0 = δ, κ0 = 0, and n0 = 0 to begin with. Conditions (1) and (2) for k = 0 follow from conditions (vi) and (v). We continue by induction on k, using repeated applications of Claim 5D.4. In stage k we use the claim with ζ = ζk and m = 2k so as to ensure: (vii) δk+1 > ζk ; and (viii) nk+1 > 2k. Having used the claim to obtain δk+1 , nk+1 , and sk+1 we set κk+1 = δk + 1. Conditions (A)–(C), (1), and (2) for k+1 follow from the conditions in the conclusion of Claim 5D.4. The conditions in Claim 5D.4 also state that:
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(ix) sk+1 extends sk strictly; and (x) sk+1 (κk+1 ) = sk (κk ) + $nk , δk %. Let t = k<ω sk . t is an annotated position of relative domain supk<ω δk . supk<ω δk is precisely equal to θ by conditions (A) and (vii). Thus t is an annotated position of relative domain θ. t is nice, saturated, and M-clear because it is the increasing union of positions which are nice, saturated, and M-clear. So: Claim 5D.20. t is a nice, saturated θ-sequence.
#
˙ It remains to show that t is supernice, and avoids Y˙ and Z. Claim 5D.21. t is supernice. Proof. Let w = k<ω sk (κk ). We claim that w witnesses the conditions of Definition 5D.16. It is easy to check, directly with Definition 4A.19, that w is amenable to t. Working inductively with condition (x) one can check that w is in fact precisely equal to {$nk , δk % | k < ω}. So the numbers used in w are precisely the numbers nk for k < ω. By condition (viii) the set {nk | k < ω} is thin in the sense that it has at most k + 1 elements below 2k + 1. From this is follows that there are infinitely many numbers in the complement of {nk | k < ω}. In other words there are infinitely many numbers which are not used in w. # Recall that G(t) is the filter given by Definition 4B.23. It is equal to {[σ ] | σ is an identity in M&θ and t |= θ}. We already established that t is M-clear. Since θ is a Woodin limit of Woodin cardinals it follows from Corollary 4B.30 that G(t) is Wθ -generic/M. It follows from Claim 4B.41 that t belongs to M[G(t)]. Claim 5D.22. t does not belong to Y˙ [G(t)]. Proof. Suppose for contradiction that t = t˙[G(t)] belongs to Y˙ [G(t)]. Recall that α is the least ordinal which seals Y˙ . Looking at Definition 4B.43 we see that there must exist some condition [σ ] in G(t) ∩ (M&α) which forces “t˙ ∈ Y˙ .” From this it follows that there must exist some good identity σ in M&α so that t |= σ and [σ ] Wθ “t˙ ∈ Y˙ .” t extends s0 = s which has relative domain δ + 1 > α. Since σ belongs to M&α it has height smaller than α. So t |= σ iff s |= σ . In conclusion we found a good identity σ ∈ M&α so that s |= σ and [σ ] Wθ “t˙ ∈ Y˙ .” But then s is hopeful with respect to Y˙ (see Definition 4D.2), contradicting condition (vi). # ˙ Claim 5D.23. t does not belong to Z[G(t)]. Proof. Similar to the last claim, using this time the fact that s is not hopeful with respect ˙ given again by condition (vi). to Z, # We established already that t is a supernice, saturated θ -sequence. t extends s ˙ This which in turn extends r. The last two claims demonstrate that t avoids Y˙ and Z. completes the proof of Lemma 5D.19. # (Lemma 5D.19)
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5E Relative successors Fix a relative successor δ † ∈ W . Let C˙ † and D˙ † be δ † -names. We work to prove the following Lemma: Lemma 5E.1. Let δ be the largest Woodin cardinal below δ † . Let t be a nice (supernice if δ is a Woodin limit of Woodin cardinals), saturated δ-sequence. Suppose that t avoids Back I (δ, δ † )(C˙ † ) and Back II (δ, δ † )(D˙ † ). Then there exists a nice, saturated δ † -sequence t † which extends t and avoids C˙ † and D˙ † . Proof. We divide the proof of Lemma 5E.1 into two cases, depending on whether δ is an element of W or a Woodin limit of Woodin cardinals. In both cases we follow the notation of Sections 4D(2) and 4E(5). Case 1. Suppose first that δ is an element of W . Since t avoids Back I (δ, δ † )(C˙ † ) and Back II (δ, δ † )(D˙ † ), there exists some g which is col(ω, δ)-generic/M, and such that: (i) t belongs to M[g]; (ii) t ∈ Back I (δ, δ † )(C˙ † )[g]; and (iii) t ∈ Back II (δ, δ † )(D˙ † )[g]. Let w = {$0, δ%}. This is a (rather trivial) witness for δ + 1, the relative domain of t. The definitions in Section 4D(2) are such that from conditions (i) and (ii) it follows that: (iv) ϕsuc (t, w, C˙ † ) fails. (We are using here the negation of condition (A2) in Section 4D (2), applied with the particular witness w listed above.) Working similarly with the definitions in Section 4E (5) it follows from condition (iii) that: (v) ψsuc (t, w, D˙ † ) fails. It is now easy to complete the proof of Lemma 5E.1 using an application of Lemma 5B.1. # (Case 1) Case 2. Suppose next that δ is a Woodin limit of Woodin cardinals. Let G denote the filter associated to t by Definition 4B.23. More precisely let G equal {[σ ]δ | σ is an identity in M&δ and t |= σ }. t is assumed to be a δ-sequence, hence in particular M-clear. δ is a Woodin limit of Woodin cardinals. By Corollary 4B.30, G is Wδ generic/M. Moreover by Claim 4B.41: (i) t belongs to M[G]. By Definition 5D.1 the fact that t avoids Back I (δ, δ † )(C˙ † ) and Back II (δ, δ † )(D˙ † ) says that:
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(ii) t ∈ Back I (δ, δ † )(C˙ † )[G]; and (iii) t ∈ Back II (δ, δ † )(D˙ † )[G]. Let g be col(ω, δ)-generic/M[G]. Using the assumption that t is supernice, and using Remark 5D.17, fix some w so that: (iv) w is a witness for δ; w is amenable to t; there are infinitely many numbers not used in w; and (v) w belongs to M[G][g]. The definitions in Section 4D(2) are such that from conditions (i), (ii), and (v) it follows that: (v) ϕsuc (t, w, C˙ † ) fails. (We are using here the negation of condition (B2) in Section 4D (2), applied with the particular witness w found above.) Working similarly with the definitions in Section 4E (5) it follows from condition (iii) that: (vi) ψsuc (t, w, D˙ † ) fails. It is now easy to complete the proof of Lemma 5E.1 using an application of Lemma 5B.1. We leave the exact details of this to the reader and only note that both Wδ and col(ω, δ) are absorbed into col(ω, δ † ), so t and w can both be absorbed into an # (Case 2) extension M[g † ] where g † is col(ω, δ † )-generic/M. # (Lemma 5E.1)
5F Compositions Recall that the pullback and mirrored pullback operations are defined using five cases. The cases are listed in Section 4D(3) and mirrored in Section 4E (6). Our work so far handles situations which correspond to the three basic cases. Let us now handle situations which correspond to the remaining two cases, listed as cases (IV) and (V) in Section 4D (3). Lemma 5F.1. Let δ † be a relative successor in W . Let C˙ † and D˙ † be δ † -names. Suppose that Hypothesis 5C.2 holds for χ ∗ < δ † . Let δ be the largest Woodin cardinal below δ † . Let χ < δ be an element of W . Let r be a nice, saturated χ-sequence. Suppose that r avoids Back I (χ, δ † )(C˙ † ) and Back II (χ, δ † )(D˙ † ). Then there exists a nice, saturated δ † -sequence t † which extends r and avoids C˙ † and D˙ † . Proof. Let C˙ = Back I (δ, δ † )(C˙ † ) and let D˙ = Back II (δ, δ † )(D˙ † ). The definition in ˙ case (IV) of Section 4D(3) is such that Back I (χ , δ † )(C˙ † ) is equal to Back I (χ , δ)(C). ˙ Thus: Similarly Back II (χ, δ † )(D˙ † ) is equal to Back II (χ , δ)(D).
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˙ and Back II (χ, δ)(D). ˙ (i) r avoids Back I (χ, δ)(C) Applying Hypothesis 5C.2 or Lemma 5D.19, depending on whether δ is an element of W or a Woodin limit of Woodin cardinals, we obtain a nice (supernice if δ is a Woodin limit of Woodin cardinals), saturated δ-sequence t which extends r and avoids C˙ and ˙ Then using Lemma 5E.1 we obtain a nice, saturated δ † -sequence t † which extends D. # t and avoids C˙ † and D˙ † . Remark 5F.2. Let us briefly take note of the use of Hypothesis 5C.2 in the proof of Lemma 5F.1. If δ is an element of W then the proof makes a direct use of the hypothesis, with χ ∗ = δ, to obtain t. If δ is a Woodin limit of Woodin cardinals then the proof makes an indirect use of the hypothesis through Lemma 5D.19. In that case the hypothesis is only needed for χ ∗ < δ, since this is all that Lemma 5D.19 requires. Lemma 5F.3. Let δ be a Woodin limit of Woodin cardinals. Let δ † be the first Woodin cardinal above δ. Let δ ∗ be a Woodin cardinal greater than δ † . Let C˙ ∗ and D˙ ∗ be δ ∗ -names. Suppose that Hypothesis 5C.2 holds for χ ∗ < δ ∗ . Let t be a supernice, saturated δ-sequence. Suppose t avoids Back I (δ, δ ∗ )(C˙ ∗ ) and Back II (δ, δ ∗ )(D˙ ∗ ). Then there exists a nice (supernice if δ ∗ is a Woodin limit of Woodin cardinals), saturated δ ∗ -sequence t ∗ which extends t and avoids C˙ ∗ and D˙ ∗ . Proof. Let C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ) and let D˙ † = Back II (δ † , δ ∗ )(D˙ ∗ ). The definition in case (V) of Section 4D(3) is such that Back I (δ, δ ∗ )(C˙ ∗ ) is equal to Back I (δ, δ † )(C˙ † ). Similarly Back II (δ, δ ∗ )(D˙ ∗ ) is equal to Back II (δ, δ † )(D˙ † ). Thus: (i) t avoids Back I (δ, δ † )(C˙ † ) and Back II (δ, δ † )(D˙ † ). Applying Lemma 5E.1 we obtain a nice, saturated δ † -sequence t † which extends t and avoids C˙ † and D˙ † . Rewriting the last part we see that: (ii) t † avoids Back I (δ † , δ ∗ )(C˙ ∗ ) and Back II (δ † , δ ∗ )(D˙ ∗ ). From this we can obtain a nice (supernice if δ ∗ is a Woodin limit of Woodin cardinal), saturated δ ∗ -sequence which extends t † and avoids C˙ ∗ and D˙ ∗ . We obtain t ∗ through an application of: Lemma 5C.9 if δ ∗ is a relative limit in W ; Lemma 5D.19 if δ ∗ is a Woodin limit of Woodin cardinals; Lemma 5E.1 if δ ∗ is a relative successor and equal to the first Woodin cardinal above δ † ; and Lemma 5F.1 if δ ∗ is a relative successor and # greater than the first Woodin cardinal above δ † . Remark 5F.4. Let us take note of the use of Hypothesis 5C.2 in the proof of Lemma 5F.3. The use of the hypothesis is always indirect. If δ ∗ is a relative limit in W then the hypothesis comes in through its use in Lemma 5C.9. If δ ∗ is a Woodin limit of Woodin cardinals then the hypothesis comes in through its use in Lemma 5D.19. If δ ∗ is a relative successor and greater than the first Woodin cardinal above δ † then the hypothesis comes in through its use in Lemma 5F.1. In all cases the hypothesis is only needed for χ ∗ < δ ∗ .
5G Conclusion
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5G Conclusion The lemmas of the previous sections combine to yield the following theorem: Theorem 5G.1. Let δ ∗ be a Woodin cardinal. Let C˙ ∗ and D˙ ∗ be δ ∗ -names. Let δ be a Woodin cardinal smaller than δ ∗ . Let t be a supernice, saturated δ-sequence. Suppose t avoids Back I (δ, δ ∗ )(C˙ ∗ ) and Back II (δ, δ ∗ )(D˙ ∗ ). Then there exists a supernice, saturated δ ∗ -sequence t ∗ which extends t and avoids C˙ ∗ and D˙ ∗ . Proof. We work by induction on δ ∗ . Note that Theorem 5G.1 subsumes Hypothesis 5C.2. Indeed the hypothesis is simply the special case of the theorem where both δ and δ ∗ belong to W . (By Remark 5D.18, nice and supernice are the same in this special case.) Thus when proving Theorem 5G.1 for a particular δ ∗ we may inductively assume that: (H) Hypothesis 5C.2 holds for χ ∗ < δ ∗ . Let us now fix δ ∗ , C˙ ∗ , D˙ ∗ , and t satisfying the assumptions of the theorem, and work to obtain t ∗ . Case (I). If δ ∗ is a relative limit in W and δ belongs to W . In this case we obtain t ∗ through an application of Lemma 5C.9, with τ = δ ∗ . By condition (H) we have enough of Hypothesis 5C.2 to allow this use of Lemma 5C.9. The lemma produces a nice, saturated δ ∗ -sequence t ∗ which extends t and avoids C˙ ∗ and D˙ ∗ . The reader may be slightly worried that Theorem 5G.1 asks for a supernice δ ∗ -sequence, and Lemma 5C.9 only provides a nice t ∗ . But for δ ∗ ∈ W the two notions are the same, by Remark 5D.18. (Similar reasoning on the move from nice to supernice applies also in cases (III)–(V) below, but we shall not comment on it anymore.) # Case (II). If δ ∗ is a Woodin limit of Woodin cardinals and δ belongs to W . In this case we obtain t ∗ through an application of Lemma 5D.19. Condition (H) provides enough of Hypothesis 5C.2 to allow this use of the lemma. # Case (III). If δ ∗ is the first Woodin cardinal above δ. In this case we obtain t ∗ through an application of Lemma 5E.1. # Case (IV). If δ ∗ is a relative successor in W , δ is an element of W , and δ ∗ is strictly larger than the first Woodin cardinal above δ. In this case we obtain t ∗ through an application of Lemma 5F.1. (Our current δ plays the role of χ in that lemma.) As usual condition (H) provides enough of Hypothesis 5C.2 to allow this use of the lemma. # Case (V). If δ is a Woodin limit of Woodin cardinals, and δ ∗ is strictly larger than the first Woodin cardinal above δ. In this case we obtain t ∗ through an application of Lemma 5F.3. Again condition (H) provides enough of Hypothesis 5C.2 to allow this use of the lemma. #
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5 When both players lose
The cases listed above correspond to the cases listed in Section 4D (3) and mirrored in Section 4E (6). It is easy to check that they are mutually exclusive, and exhaust all # (Theorem 5G.1) possible layouts for δ and δ ∗ . We have been working in M and in generic extensions of M throughout this chapter. It should be noted that Theorem 5G.1 only asserts the existence of t ∗ in some generic extension of M. No assertion is made about its existence in V. A similar comment applies to Corollary 5G.3 below. ˙ to Definition 5G.2. Let δ be a Woodin cardinal. Let C˙ be a δ-name. Define ϕini (δ, C) ˙ be the formula “ϕsuc (t0 , w0 , C0 ) holds true with: • t0 equal to the empty annotated position of relative domain 0; • w0 equal to the empty witness for 0; and ˙ where δ0 is the first Woodin cardinal of M.” • C˙ 0 equal to Back(δ0 , δ)(C) ˙ be defined similarly, but using ψsuc and the mirrored For a δ-name D˙ let ψini (δ, D) pullback. Corollary 5G.3. Let δ be a Woodin cardinal. Let C˙ and D˙ be δ-names. Suppose ˙ and ψini (δ, D) ˙ both fail. Then there exists a supernice, that the formulae ϕini (δ, C) ˙ saturated δ-sequence t which avoids C˙ and D. ˙ and let Proof. Let δ0 , t0 , and w0 be as in Definition 5G.2. Let C˙ 0 = Back I (δ0 , δ)(C) II ˙ ˙ ˙ ˙ D0 = Back (δ0 , δ)(D). The assumption that ϕini (δ, C) and ψini (δ, D) both fail tells us that: (i) ϕsuc (t0 , w0 , C˙ 0 ) and ψsuc (t0 , w0 , D˙ 0 ) both fail. Using this condition, an application of Lemma 5B.1 produces a real y0 so that: (ii) t0 −−, w0 , y0 is a nice (supernice by Remark 5D.18) δ0 -sequence; and ˙ and D˙ 0 = Back II (δ0 , δ)(D). ˙ (iii) t0 −−, w0 , y0 avoids C˙ 0 = Back I (δ0 , δ)(C) t0 −−, w0 , y0 is trivially saturated, since there are no relative limits below δ0 + 1. If δ is equal to δ0 we are done; we can simply take t = t0 −−, w0 , y0 . Otherwise an application of Theorem 5G.1 produces a supernice, saturated δ-sequence t which extends ˙ # t0 −−, w0 , y0 and avoids C˙ and D. Corollary 5G.3 at last achieves the goal stated at the start of this chapter. We will apply it later on with C˙ and D˙ naming sets whose union includes all δ-sequences in the appropriate generic extension of M. Using the theorem we will be able to argue that at ˙ and ψini (δ, D) ˙ holds. least one of ϕini (δ, C)
Chapter 6
Along a single branch
In Chapter 4 we defined a pullback operation on names. We worked in a transitive model M of ZFC∗ . Given Woodin cardinals δ < δ ∗ in M, and given a δ ∗ -name C˙ ∗ , we defined a δ-name Back I (δ, δ ∗ )(C˙ ∗ ), called the (δ, δ ∗ )-pullback of C˙ ∗ . Intuitively we thought of Back I (δ, δ ∗ )(C˙ ∗ ) as naming the set of δ-sequences from which player I can win to either enter an interpretation of a shift of C˙ ∗ , or reach a I-acceptable obstruction along the way. We now work to make this intuition precise. In Section 6A we define precisely a game through which players I and II collaborate to create an embedding j ∗ and a j ∗ (δ ∗ )-sequence t ∗ . I’s goal in the game, roughly speaking, is to either make t ∗ enter an interpretation of j ∗ (C˙ ∗ ), or reach a I-acceptable obstruction during the construction of t ∗ . For the rest of the chapter we work on constructing a winning strategy for I in this game, starting from t in an interpretation of the (δ, δ ∗ )-pullback of C˙ ∗ . The end products of our work are the results in Section 6G. Theorem 6G.1 in particular is the precise meaning of our intuitive expectations from the pullback operation.
6A The game Let M be a transitive model of ZFC∗ . Let δ < δ ∗ be Woodin cardinals of M. Let C˙ ∗ be a δ ∗ -name in M. Let t be a δ-sequence over M. We take this section to describe a branch (M, t, δ ∗ )(C˙ ∗ ). The game, which is played in V, branch = G game we denote G sets rules for the construction of the following objects through a collaboration between players I and II: • a wellfounded model M ∗ ; • an elementary embedding j ∗ : M → M ∗ ; and • a j ∗ (δ ∗ )-sequence t ∗ over M ∗ , extending t. The format of the collaboration is for the most part standard: I plays iteration trees of length ω; II picks branches through those trees; and the two players work together to produce the real part of t ∗ . But every once in a while we allow exceptions to this standard format. The exceptions come in through rules (L5) and (L6) below. These rules let player II insert intervals of her choice into t ∗ , and insert extenders of her choice into the iteration which creates M ∗ and j ∗ . The construction of M ∗ , j ∗ , and t ∗ may hit some snags along the way. For example it may reach an illfounded model, or it may reach an obstruction it cannot remove,
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6 Along a single branch
or it may reach a limit which is not countable. Conditions (I1)–(I7) below spell out the possible snags and decide the winner in the case of a construction which is left incomplete because of a snag. A complete construction is one which ends through the payoff condition (P1) or the payoff condition (P2) below. It produces a model M ∗ = Mβ+1 , an embedding j ∗ = j0,β+1 : M → Mβ+1 , and a j0,β+1 (δ ∗ )-sequence t ∗ = tβ+1 over Mβ+1 . Player I wins a complete construction just in case that t ∗ belongs to an interpretation of j ∗ (C˙ ∗ ). branch (M, t, δ ∗ )(C˙ ∗ ). We Let us now begin the precise description of the rules of G set M0 = M and t0 = t to begin with. The game starts with mega-round 0, which is played according to the rules of the successor case below. Successor case. At the start of a (zero or) successor mega-round β we have, either through the initial settings above or through the work of the players in mega-rounds prior to β, the following objects: • a wellfounded model Mβ ; • an elementary embedding j0,β : M0 → Mβ ; and • an annotated Mβ -position tβ . These objects are such that: (i) tβ is Mβ -clear; and (ii) rdm(tβ ) < j0,β (δ ∗ ). It follows from condition (ii) that there are Woodin cardinals of Mβ above rdm(tβ ). Let δβ† denote the least such. Then δβ† ≤ j0,β (δ ∗ ). Mega-round β proceeds according to the following rules: (S1) I picks a witness wβ for rdm(tβ ) over Mβ ; (S2) I and II collaborate in the usual fashion to produce a real yβ ; (S3) I plays a length ω iteration tree Tβ on Mβ , with critical points above rdm(tβ ) and using only extenders which are countable in V (see Appendix A); and (S4) II plays a cofinal branch bβ through Tβ . Let Qβ be the direct limit along the branch bβ of Tβ and let kβ : Mβ → Qβ be the direct limit embedding. branch ends and player I wins. (I1) If Qβ is illfounded then G From now on assume that Qβ is wellfounded. The restriction in rule (S3) is such that crit(kβ ) > rdm(tβ ). From this it follows that tβ may be regarded as an annotated position over Qβ . Similarly wβ may be regarded as a witness over Qβ . Working over Qβ set tβ† = tβ −−, wβ , yβ . This is an annotated position, of relative domain kβ (δβ† ) + 1, over Qβ .
6A The game
/
Mβ
kkk Sk k Sk SSSS k / β
Qβ
=
211
/
Mβ+1
Tβ
Diagram 6.1. The successor case.
branch ends. Player I wins just in case that there (I2) If tβ† is obstructed over Qβ then G exists a I-acceptable obstruction for tβ† over Qβ . From now on assume that tβ† is obstruction free over Qβ . Set Mβ+1 = Qβ , jβ,β+1 = kβ , and tβ+1 = tβ† . Combining condition (i) above with the fact that tβ† is obstruction free over Qβ we see that tβ+1 is Mβ+1 -clear. So condition (i) above holds for β +1. Let δβ+1 = kβ (δβ† ). The relative domain of tβ+1 is δβ+1 + 1. Note that δβ+1 ≤ j0,β+1 (δ ∗ )
since δβ† ≤ j0,β (δ ∗ ).
branch = G branch (M, t, δ ∗ )(C˙ ∗ ) ends. Player I wins (P1) If δβ+1 = j0,β+1 (δ ∗ ) then G just in case that there exists some g so that: (1) g is col(ω, j0,β+1 (δ ∗ ))-generic/Mβ+1 ; and (2) tβ+1 ∈ j0,β+1 (C˙ ∗ )[g]. If δβ+1 < j0,β+1 (δ ∗ ) then we pass to mega-round β + 1.
# (Successor case)
At the start of a limit mega-round β we have, through the work of the players in previous mega-rounds, the following objects: • wellfounded models Mξ for ξ < β; • elementary embeddings jζ,ξ : Mζ → Mξ for ζ < ξ < β; and • annotated Mξ -positions tξ for ξ < β. The objects constructed in mega-rounds prior to β are such that: (a) tξ is Mξ -clear for each ξ < β; (b) rdm(tξ ) < j0,ξ (δ ∗ ) for each ξ < β; (c) the sequence $tξ | ξ < β% is strictly increasing; and (d) crit(jξ,ξ ∗ ) ≥ rdm(tξ ) for each ξ < β and all ξ ∗ ∈ (ξ, β). Let Mβ be the direct limit of the system $Mξ , jζ,ξ | ζ ≤ ξ < β%, and let jξ,β for ξ < β be the direct limit embeddings. branch ends and player I wins. (I3) If Mβ is illfounded then G
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6 Along a single branch
branch ends and player I loses. (I4) If β = ω1V then G There is no need to specify priority between conditions (I3) and (I4), since they cannot both hold: if β = ω1V then Mβ is the direct limit of ω1V wellfounded models, and so by necessity wellfounded itself. Suppose from now on that Mβ is wellfounded, and that β is smaller than ω1V . From condition (d) it follows that: (e) crit(jξ,β ) ≥ rdm(tξ ) for each ξ < β. From this it follows that tξ may be regarded as an annotated position over Mβ . Let tβ = ξ <β tξ . The union makes sense by condition (c). tβ is an annotated Mβ -position. Using conditions (a) and (b) we see that: (f) tβ is Mβ -clear; and (g) rdm(tβ ) ≤ j0,β (δ ∗ ). Using condition (c) and the fact that β is a limit ordinal we get: (h) rdm(tβ ) is a limit of Woodin cardinals in Mβ . The rules for mega-round β divide into two cases, depending on whether or not rdm(tβ ) itself is Woodin. Phantom limit case. If rdm(tβ ) is a Woodin cardinal in Mβ . There are no moves in mega-round β in this case. We simply set Mβ+1 = Mβ , jβ,β+1 = id, and tβ+1 = tβ . Let δβ and δβ+1 equal rdm(tβ ). δβ+1 is a Woodin limit of Woodin cardinals in Mβ+1 . We have δβ+1 ≤ j0,β+1 (δ ∗ ) by condition (g). (Note that j0,β+1 = j0,β , since jβ,β+1 = id.) branch = G branch (M, t, δ ∗ )(C˙ ∗ ) ends. Player I wins (P2) If δβ+1 = j0,β+1 (δ ∗ ) then G just in case that there exists some G so that: (1) G is j0,β+1 (Wδ ∗ )-generic/Mβ+1 ; and (2) tβ+1 ∈ j0,β+1 (C˙ ∗ )[G]. If δβ+1 < j0,β+1 (δ ∗ ) then we pass to mega-round β + 1.
# (Phantom limit case)
Standard limit case. If rdm(tβ ) is not a Woodin cardinal in Mβ . Let λβ = rdm(tβ ). We have λβ ≤ j0,β (δ ∗ ) by condition (g). Equality is impossible since j0,β (δ ∗ ) is Woodin in Mβ , and λβ is not. So λβ < j0,β (δ ∗ ) strictly. It follows that there are Woodin cardinals of Mβ above λβ . Let τβ denote the least such. We have τβ ≤ j0,β (δ ∗ ). Mega-round β, in the case that rdm(tβ ) is not Woodin in Mβ , starts according to the following rules: (L1) I picks a witness wβ for λβ over Mβ ; (L2) I plays a length ω iteration tree Tβ on Mβ , with critical points larger than λβ and using only extenders which are countable in V; and
6A The game
213
(L3) II plays a cofinal branch bβ through Tβ . Let Qβ be the direct limit along the branch bβ of Tβ and let kβ : Mβ → Qβ be the direct limit embedding. branch ends and player I wins. (I5) If Qβ is illfounded then G Assume from now on that Qβ is wellfounded. Mega-round β continues according to the following rule: (L4) I picks yβ in (Qβ &λβ + 1)ω . The restriction in rule (L2) is such that crit(kβ ) > λβ = rdm(tβ ). tβ may thus be regarded as an annotated position over Qβ . Working over Qβ set sβ = tβ −−, wβ , yβ . sβ is an annotated position, of relative domain kβ (τβ ) + 1, over Qβ . branch ends (I6) If sβ is obstructed, but not I-acceptably obstructed over Qβ , then G and player I loses. Suppose from now on that sβ is either obstruction free or I-acceptably obstructed over Qβ . Mega-round β proceeds in one of two ways. If sβ is obstruction free over Qβ then player II may elect an early end to megaround β. In this case we set Mβ+1 = Qβ , jβ,β+1 = kβ , and tβ+1 = sβ . We let δβ+1 = kβ (τβ ). The relative domain of tβ+1 is δβ+1 + 1. We have δβ+1 ≤ j0,β+1 (δ ∗ ) since τβ ≤ j0,β (δ ∗ ). If δβ+1 = j0,β+1 (δ ∗ ) then the game ends with the payoff condition (P1) stated above. If δβ+1 < j0,β+1 (δ ∗ ) then we pass to mega-round β + 1. If sβ is obstructed, or if player II does not elect an early end, then mega-round β continues with what we call a leap. The leap consists of moves by player II alone, subject to rules (L5) and (L6) below: (L5) II plays Eβ∗ subject to the following conditions: (1) Eβ∗ is a kβ (τβ ) + 1-strong extender in some model Q∗β which agrees with Qβ to kβ (τβ ) + 1; (2) Eβ∗ is countable in V; and (3) crit(Eβ∗ ) = λβ .
Mβ and Qβ agree past λβ by rule (L2). Folding this into the conditions in rule (L5) it follows that Mβ and Q∗β agree past the critical point of Eβ∗ . So Eβ∗ can be applied to Mβ . Let Mβ+1 = Ult(Mβ , Eβ∗ ) and let jβ,β+1 be the ultrapower embedding. The situation is illustrated in Diagram 6.2. branch ends and player I wins. (I7) If Mβ+1 is illfounded then G Suppose from now on that Mβ+1 is wellfounded. Note that the agreement between Mβ+1 and Mβ is sufficient that sβ may be regarded as an annotated position over Mβ+1 . Let Wβ+1 denote the class W of Section 4A, computed in Mβ+1 . Let uβ be the “even half” of yβ . More precisely let uβ = $yβ (2i) | i < ω%.
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6 Along a single branch jβ,β+1
/
Mβ
kkk ∗ Sk k Sk SSSS k / Qβ _ _ _ _ _ _ Qβ β
Eβ∗
&
Mβ+1
/
Tβ
Diagram 6.2. Standard limit case with a leap according to rules (L5) and (L6).
(L6) II plays δβ+1 ∈ jβ,β+1 (λβ ) ∩ Wβ+1 , and an annotated position tβ+1 of relative domain δβ+1 + 1 over Mβ+1 , subject to the following conditions: (1) tβ+1 extends sβ (perhaps not strictly); (2) tβ+1 is Mβ+1 -clear; and (3) there exists some n∗β < ω so that uβ (n∗β ) is a λβ -functor in the sense of Mβ , and so that tβ+1 ∈ cβ+1 (λβ , δβ+1 )[∗] where cβ+1 = jβ,β+1 (uβ (n∗β )). Remark 6A.1. Rule (L6) lets player II choose tβ+1 . But player I gets to regulate this choice using the even half of her move yβ in rule (L4). The even half of yβ affects the possible choices of tβ+1 through condition (3) in rule (L6). Note that the condition makes sense: uβ (n∗β ) is a λβ -functor in the sense of Mβ . It ¯ ∈ (λβ ∩ Lβ ) × (λβ ∩ Wβ ) | is thus a function with domain equal to the set I = ${κ, ¯ δ% ¯ κ¯ < δ}. Using the fact that δβ+1 belongs to jβ,β+1 (λβ ) ∩ Wβ one can check that $λβ , δβ+1 % belongs to jβ,β+1 (I ), namely to the domain of cβ+1 . cβ+1 is a functor over ¯ ¯ in its domain. So the to each pair $κ, ¯ δ% Mβ+1 . It therefore assigns a col(ω, δ)-name reference to cβ+1 (λβ , δβ+1 )[∗] in condition (3) makes sense. Rule (L6) ends mega-round β in the case of a leap. We pass to mega-round β + 1 equipped with the annotated Mβ+1 -position tβ+1 chosen by II subject to this rule. Note that rdm(tβ+1 ) must be smaller than j0,β+1 (δ ∗ ). This follows from the requirement δβ+1 ∈ jβ,β+1 (λβ ) ∩ Wβ+1 in rule (L6) and the fact that λβ is smaller than j0,β (δ ∗ ). # (Standard limit case) The successor case and two limit cases complete the description of the game branch (M, t, δ ∗ )(C˙ ∗ ) should be branch (M, t, δ ∗ )(C˙ ∗ ). A winning strategy for I in G G viewed as a construction mechanism. Our goal in this chapter is to create the mechanism. We intend to prove: Theorem (6G.1). 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 branch (M, t, δ ∗ )(C˙ ∗ ). player I has a winning strategy in G We will use the construction mechanism given by this theorem as part of our work in the next chapter.
6A The game
215
branch (M, t, δ ∗ )(C˙ ∗ ) as being given by seRemark 6A.2. We think of positions in G quences of the form P = $Tξ , bξ , Eξ∗ , tξ +1 | ξ < β%. Strictly speaking not all the objects in this sequence are defined for each ξ . For example Eξ∗ is only defined if ξ is a standard limit, and only if II elected a leap in mega-round ξ . Tξ and bξ are only defined if ξ is a successor or a standard limit. Let us for uniformity adopt the convention that Tξ is the tree which consists entirely of “padding” if ξ is a phantom limit, and bξ in this case is the unique branch through Tξ . That way Eξ∗ is the only object which need not be defined for all ξ . Note that the convention fits with the settings in phantom limit cases. jξ,ξ +1 is the identity in such cases, so we may think of it as the direct limit embedding through a tree which consists entirely of “padding.” There are moves in mega-round ξ which are not listed among the objects Tξ , bξ , Eξ∗ , and tξ +1 . But all the moves in mega-round ξ can be recovered from (tξ and) these objects. Thus a sequence P = $Tξ , bξ , Eξ∗ , tξ +1 | ξ < β% gives a complete account of branch (M, t, δ ∗ )(C˙ ∗ ). a position of length β in G branch (M, t, δ ∗ )(C˙ ∗ ) gives rise to: Remark 6A.3. A position P of length β in G • models Mξ for ξ ≤ β; • embeddings jζ,ξ : Mζ → Mξ for ζ < ξ ≤ β; and • annotated positions tξ over Mξ . We refer to the sequence $Mξ , jζ,ξ , tξ | ζ < ξ ≤ β%, which consists of the entire array of models, embeddings, and annotated positions, as the history of P . We refer to the triple $Mβ , j0,β , tβ %, which consists only of the final model, embedding, and annotated position, as the outcome of P . Remark 6A.4. Let $Mβ , j0,β , tβ % be the outcome of a position of length β in branch (M, t, δ ∗ )(C˙ ∗ ). Then: G (1) tβ is Mβ -clear; and (2) if β < ω1V then j0,β preserves countability (see the section on extenders in Appendix A). branch and, in the case of condiBoth conditions are easily implied by the structure of G tion (2), Fact 3 and Claim 4 in Appendix A. Note that a position of length β = ω1V is lost by I subject to the snag (I4). So condition (2) holds for all outcomes of positions which are not lost by I. 6A (1) How to leap. We describe here a scenario through which player II may obtain legal moves for rules (L5) and (L6) in the standard limit case above. The point of the description is to record the scenario for future reference, and also to shed some light on these two rules. We shall see that they fit tightly with the notion of I-acceptable obstructions.
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6 Along a single branch
branch (M, t, δ ∗ )(C˙ ∗ ). Let Let P be a non-terminal position of limit length β in G $Mβ , j0,β , tβ % be the outcome of P . Let λβ = rdm(tβ ) and let τβ be the first Woodin cardinal of Mβ above λβ . branch (M, t, δ ∗ )(C˙ ∗ ) Suppose λβ is not Woodin in Mβ , so that mega-round β of G following P is played subject to the rules of the standard limit case. Let wβ , Tβ , bβ , and yβ be legal moves corresponding to rules (L1)–(L4). Suppose that these moves are branch non-terminal. More precisely suppose that they do not bring about the end of G through one of the snags (I5) and (I6). Following the notation of the standard limit case let Qβ be the direct limit along the branch bβ of Tβ , and let kβ : Mβ → Qβ be the direct limit embedding. Let sβ = tβ −−, wβ , yβ . This is an annotated position of relative domain kβ (τβ )+1 over Qβ . Lemma 6A.5. Let Q∗β be a transitive model which agrees with Qβ to kβ (τβ ) + 1. Let Wβ∗ denote the class W of Section 4A, computed in Q∗β . Let δβ∗ be an element of Wβ∗ , and let tβ∗ be an annotated position of relative domain ∗ δβ + 1 over Q∗β . Suppose that tβ∗ is I-acceptably obstructed over Q∗β . Let $Eβ∗ , σβ∗ % be an obstruction witnessing this. Suppose that: (1) Q∗β &δβ∗ + ω is countable in V; (2) tβ∗ extends sβ (perhaps not strictly); (3) every strict initial segment of tβ∗ is Q∗β -clear (tβ∗ itself of course is obstructed by the initial assumptions); (4) crit(Eβ∗ ) is equal to λβ ; and (5) Ult(Mβ , Eβ∗ ) is wellfounded. Let δβ+1 = δβ∗ and let tβ+1 = tβ∗ . Then Eβ∗ , δβ+1 , and tβ+1 satisfy the demands of rules (L5) and (L6). Proof. We work through a series of claims to establish that the conditions in rules (L5) and (L6) hold for the objects of the lemma. We repeatedly use the fact that $Eβ∗ , σβ∗ % is a I-acceptable obstruction for tβ∗ over Q∗β . Claim 6A.6. δβ∗ is greater than or equal to kβ (τβ ). Proof. This follows from the fact that tβ∗ extends sβ .
#
Claim 6A.7. Eβ∗ is a δβ∗ + 1-strong extender in Q∗β , and belongs to Q∗β &δβ∗ + ω. Proof. $Eβ∗ , σβ∗ % is by assumption an obstruction for tβ∗ over Q∗β . It therefore satisfies the conditions of Definition 4B.26. The current claim follows from condition (1) in that Definition. # Corollary 6A.8. Eβ∗ is countable in V, and at least kβ (τβ ) + 1-strong in Q∗β .
6A The game
217
Proof. This follows immediately from the last two claims, and the assumption in # Lemma 6A.5 that Q∗β &δβ∗ + ω is countable in V. Claim 6A.9. Q∗β and Mβ agree past λβ . Proof. Mβ agrees with Qβ past λβ , since all extenders in Tβ have critical points above λβ by the conditions in rule (L2). By assumption Q∗β agrees with Qβ up to kβ (τβ ) + 1, which is greater than λβ . So Q∗β and Mβ agree past λβ . # Let Mβ+1 = Ult(Mβ , Eβ∗ ). The ultrapower makes sense by the previous claim. Let jβ,β+1 : Mβ → Mβ+1 be the ultrapower embedding. Let Wβ+1 denote the class W of Section 4A, computed in Mβ+1 . Claim 6A.10. δβ∗ belongs to jβ,β+1 (λβ ) ∩ Wβ+1 . Proof. The strength of Eβ∗ given by Claim 6A.7 is such that Mβ+1 and Q∗β agree to δβ∗ + 1. It follows from this that Wβ∗ and Wβ+1 are the same up to δβ∗ + 1. So δβ∗ belongs to Wβ+1 . The ultrapower embedding by Eβ∗ sends its critical point past the strength of Eβ∗ (see Fact 2 in Appendix A). So jβ,β+1 (λβ ) > δβ∗ + 1. It follows from this and from the # conclusion of the previous paragraph that δβ∗ belongs to jβ,β+1 (λβ ) ∩ Wβ+1 . Claim 6A.11. tβ∗ is an annotated position over Mβ+1 , and all strict initial segments of tβ∗ are Mβ+1 -clear. Proof. This follows from the initial assumptions about tβ∗ over Q∗β , and the fact that # Mβ+1 agrees with Q∗β to δβ∗ + 1 = rdm(tβ∗ ). Claim 6A.12. tβ∗ is obstruction free over Mβ+1 . Proof. $Eβ∗ , σβ∗ % is a I-acceptable obstruction for tβ∗ over Q∗β . It therefore satisfies the conditions in Definition 4C.6. In particular it is a minimal obstruction for tβ∗ over Q∗β . So tβ∗ is obstruction free over Ult(Q∗β , Eβ∗ ). The claim follows from this and from the # standard agreement between Ult(Q∗β , Eβ∗ ) and Mβ+1 = Ult(Mβ , Eβ∗ ). Corollary 6A.13. tβ∗ is Mβ+1 -clear. Proof. This is simply the conjunction of the last two claims.
#
Let uβ be the even half of yβ . Precisely, let uβ = $yβ (2i) | i < ω%. Claim 6A.14. For each n < ω, uβ (n) is a λβ -functor in the sense of Mβ . Proof. Since Q∗β and Mβ agree past λβ it is enough to check that each uβ (n) is a λβ -functor in the sense of Q∗β . This property of uβ follows from the fact that $Eβ∗ , σβ∗ % is a I-acceptable obstruction for tβ∗ over Q∗β , using the suitability requirement in condition (2) of Definition 4C.6. #
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6 Along a single branch
Let π : Q∗β → Ult(Q∗β , Eβ∗ ) be the ultrapower embedding of Q∗β by Eβ∗ . Recall that jβ,β+1 is the ultrapower embedding of Mβ by Eβ∗ . Since both π and jβ,β+1 are created through an ultrapower by Eβ∗ , they agree on subsets of crit(Eβ∗ ) = λβ . In particular they agree on λβ -functors. Claim 6A.15. There exists n∗β < ω so that tβ∗ belongs to jβ,β+1 (cβ∗ )(λβ , δβ∗ )[∗], where cβ∗ stands for uβ (n∗β ). Proof. One last time we use the fact that $Eβ∗ , σβ∗ % is a I-acceptable obstruction for tβ∗ over Q∗β , and therefore satisfies the conditions in Definition 4C.6. The current claim follows directly from condition (3) in that definition, using the observation above that # π and jβ,β+1 agree on λβ -functors. Equipped with the claims and corollaries above it is easy to go over the conditions in rules (L5) and (L6), and verify that they hold true for Eβ∗ , δβ+1 = δβ∗ , and tβ+1 = tβ∗ . # (Lemma 6A.5) Lemma 6A.5 provides a scenario through which player II may obtain legal moves branch . The lemma shows a connection between leaps and I-acceptable for a leap in G obstructions. It was the crucial assumption that Eβ∗ is part of a I-acceptable obstruction for tβ∗ that carried us through the proof. 6A (2) The skipping game. We noted earlier that the main goal of this chapter is to branch . As a main construct winning strategies for player I in various instances of G step toward this goal we will first construct winning strategies in a related game which branch . We now define this skipping game. allows player I to “skip” through rounds of G Work as before with a transitive model M; Woodin cardinals δ < δ ∗ in M; a δ skip (M, t, δ ∗ )(C˙ ∗ ) to be the sequence t over M; and a δ ∗ -name C˙ ∗ in M. Define G branch (M, t, δ ∗ )(C˙ ∗ ), except that the game which is played according to the rules of G successor mega-round is played as follows: Successor case for skipping games. Let β be a successor (or zero). Mega-round β branch . This produces begins with rules (S1)–(S4), listed above in the successor case of G wβ , yβ , Tβ , and bβ . Following the definitions in the successor case let Qβ be the direct limit along the branch bβ of Tβ and let kβ be the direct limit embedding. If Qβ is illfounded then following condition (I1) in the successor case the game ends and I wins. Assuming that Qβ is wellfounded, let tβ† = tβ −−, wβ , yβ . If tβ† is obstructed over Qβ then the game ends in the manner of condition (I2) in the successor case. Suppose now that tβ† is obstruction free over Qβ . Player I has two options at this point. She can declare an early end to megaround β. In this case we simply continue to follow the successor case in the definition branch . We set Mβ+1 = Qβ , jβ,β+1 = kβ , tβ+1 = t † , and δβ+1 = kβ (δ † ). If of G β β δβ+1 = j0,β+1 (δ ∗ ) then the game ends subject to condition (P1) in the successor case. If δβ+1 < j0,β+1 (δ ∗ ) then the game continues and we pass to mega-round β + 1 (of the skipping game).
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6A The game
Alternatively player I can call for a skip. In this case mega-round β continues with branch . Let Wβ denote the class rules additional to the ones of the successor case of G W of Section 4A, computed in Mβ . The additional rules are: (S5) I plays δβ∗ and C˙ β∗ subject to the following conditions: (1) δβ∗ belongs to Wβ ; (2) δβ∗ is greater than δβ† and smaller than j0,β (δ ∗ ); (3) C˙ β∗ is a δβ∗ -name in Mβ . Let C˙ β† = Back I (δβ† , δβ∗ )(C˙ β∗ ), where the pullback is computed inside Mβ . I must play C˙ β∗ in such a way that: (4) tβ† belongs to an interpretation of kβ (C˙ β† ) over Qβ . (S6) II plays hβ , Mβ+1 , and tβ+1 so that: (1) Mβ+1 is a wellfounded model of ZFC∗ , hβ : Qβ → Mβ+1 (displayed in Diagram 6.3) is elementary, crit(hβ ) > kβ (δβ† ), and hβ is preserves countability; (2) tβ+1 is an (hβ ◦ kβ )(δβ∗ )-sequence over Mβ+1 ; (3) tβ+1 extends tβ† ; and (4) tβ+1 belongs to an interpretation of (hβ ◦ kβ )(C˙ β∗ ) over Mβ+1 . These two rules should be viewed as an “allow and choose” arrangement. Player I through her choice of δβ∗ and C˙ β∗ determines which extensions of tβ† are allowed—these are the ones which belong to interpretations of shifts of C˙ β∗ —and player II chooses among them. hβ
/
Mβ
kkk Sk k Sk SSSS k / Qβ β '
Mβ+1
/
Tβ
Diagram 6.3. Successor case with a skip.
Rules (S5) and (S6) complete mega-round β in the case of a skip. We set jβ,β+1 = hβ ◦ kβ , set δβ+1 = jβ,β+1 (δβ∗ ) and pass to mega-round β + 1 using the objects Mβ+1 and tβ+1 chosen by player II subject to rule (S6). # (Skip case) branch (M, t, δ ∗ )(C˙ ∗ ) sets rules for the construction of: Recall that G
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6 Along a single branch
• a wellfounded model M ∗ ; • an elementary embedding j ∗ : M → M ∗ ; and • a j ∗ (δ ∗ )-sequence t ∗ over M ∗ , extending t. skip (M, t, δ ∗ )(C˙ ∗ ) sets similar rules for the construction of the same objects. The G skip allows player I to skip along the only difference between the two games is that G ∗ way to δ . I can in mega-round β choose some δβ∗ , possibly much larger than δβ† , and catapult the game directly to the level of a δβ∗ -sequence without having to actually play through all the Woodin cardinals between δβ† and δβ∗ .
skip . Then Remark 6A.16. Let j0,β+1 be an embedding created through a position in G j0,β+1 preserves countability. This follows from the restrictions to extenders which are countable in V in rules (S3), (L2), and (L5), the restriction to an hβ which preserves countability in rule (S6), and the restriction in condition (I4) which implies that β is countable. Our goal next is to construct winning strategies for I in skipping games. This is done in Sections 6B through 6E, with the central cases being Section 6C which handles pullbacks from relative limits, and Section 6D which handles pullbacks from Woodin branch and construct winning limits of Woodin cardinals. In Section 6F we return to G strategies for I in that game by sewing together winning strategies in various skipping games. The end results are listed in Section 6G. Chapter 7 will appeal to these end skip . branch , but not to any of the intermediary results on G results about G
6B Successors, basic step Let M be a model of ZFC∗ . Let δ † be a relative successor in W (computed in M) and let C˙ † ∈ M be a δ † -name. Let µ = e(δ † ) and let t be an annotated M-position of relative domain µ. Let w be a witness for µ over M. We work to isolate the successor case of branch , and see how I can play in that case. G suc (M, t, w, C˙ † ) to be the following game: Define G • players I and II collaborate as usual to produce a real y; • I plays a length ω iteration tree T on M, with critical points above rdm(t) and using only extenders which are countable in V; and • II plays a cofinal branch b through T . This completes the game. We let Q be the direct limit along the branch b of T and let k : M → Q be the direct limit embedding. We let t † = t−−, w, y. This is an annotated position of relative domain k(δ † ) + 1 over Q. Player I wins the run consisting of y, T , and b just in case that one of the following conditions holds:
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(B1) Q is illfounded; (B2) t † is I-acceptably obstructed over Q; or (B3) Q is wellfounded, t † is obstruction free over Q, and there exists some g so that: (1) g is col(ω, k(δ)† )-generic/Q, and (2) t † ∈ k(C˙ † )[g]. Remark 6B.1. It t † belongs to k(C˙ † )[g] then it has to be obstruction free over Q. (Even more than that: t † has to be a k(δ † )-sequence over Q and therefore Q-clear.) So the statement that t † is obstruction free in condition (B3) is redundant. We include it to emphasize the contrast with condition (B2). branch (. . . ). The rules suc (M, t, w, C˙ † ) is closely related to the successor case of G G † ˙ of Gsuc (M, t, w, C ) correspond precisely to rules (S2)–(S4) in Section 6A. Moreover the payoff conditions (B1)–(B3) correspond very closely to conditions (I1), (I2), and (P1) in Section 6A. Lemma 6B.2. Suppose that: (1) M&δ † + 1 is countable in V; (2) t and w exist in a generic extension of M by a forcing of size µ or less (in M if µ = 0); and (3) ϕsuc (t, w, C˙ † ) (defined in Section 4C(2)) holds in that generic extension. suc (M, t, w, C˙ † ). Then I has a winning strategy in G Proof. This is a straightforward application of the techniques of Chapter 1. The main point is the correspondence between the payoff conditions (B2) and (B3), and the definition of A† in Section 4C(2). We leave the precise details to the reader. Let us only note that the assumption on the size of the forcing in condition (2) is used in conjunction with Remark 4C.9 and Lemma 1C.7, to see that the relevant iteration embeddings extend to act on the generic extension which contains {t, w}. #
6C Relative limits Let M be a transitive model of ZFC∗ . Let τ be a relative limit in W (where W is computed in M). Let Y˙ be a τ -name in M. We work with these fixed objects throughout Section 6C. Later in the section we will work with some t which belongs to an interpretation of a pullback of Y˙ , and aim to construct a winning strategy for player I skip (M, t, τ )(Y˙ ). But first we work to establish some auxiliary claims. in G
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6 Along a single branch
We follow the notation of Section 4C(3). We use C˙ to denote the function which assigns, to each ordinal γ < jumpM (τ ), each P ∈ (M&τ )<ω , each δ ∈ τ ∩ W , and ˙ each κ ∈ δ ∩ L, the name C(κ, δ, P , γ ) given by Definition 4C.14. The function C˙ belongs to M. It can be shifted via elementary embeddings which act on M. We use R˙ to denote the function which assigns to each ordinal γ < jumpM (τ ) the ˙ ] of Definition 4C.18. This function too belongs to M, and can be shifted via name R[γ ˙ ] is a name in elementary embeddings which act on M. Let λ = e(τ ). Recall that R[γ col(ω, τ ) for a subset of (M&τ )ω × (M&λ)ω . Remark 6C.1. The restriction to γ < jumpM (τ ) is in line with the definitions of Section 1F. Its point is to make C˙ and R˙ set functions which literally belong to M, rather than class functions definable over M. It is easy to check that all the ordinals γ which come up in the constructions later in the section are smaller than the appropriate shift of jumpM (τ ). (Note especially that γL in Definition 6C.3 is smaller than jumpM (τ ), by Claim 1A.16.) So the restriction to γ < jumpM (τ ) in the domain of C˙ does not pose any problem. ˙ τ , and X = M&λ. Let Rmix denote the mixed pivot games map associated to R, This is a Lipschitz continuous map which assigns to each x ∈ (M&λ)ω the mixed pivot game Rmix [x]. Round n of this game is displayed in Diagram 6.4, which we copy from Chapter 1. Precise details can be found in Section 1F. f (n), T f (n) + 1 I ······ ······ II
γn
···
ln , pn , un Ef (n) , Ef (n)+1 , wn
Diagram 6.4. Round n of Rmix [x].
We use the letter P to range over both finite positions in Rmix [x] and infinite runs of Rmix [x]. We say that P has length n if it covers rounds 0 through n − 1. We say that P has length n + 0.2 if in addition it covers the moves f (n) and T f (n) + 1 in round n. This is illustrated by the vertical dotted line in Diagram 6.4. A position of length n + 0.2 covers moves precisely up to this dotted line. We refer to the part of round n left of the dotted line as the first fifth of round n. 6C (1) Finite. Let x¯ be some element of (M&λ)ω . Let n be a natural number. Let P ¯ P gives rise to: be a position of length n + 0.2 in Rmix [x]. • a list of ordinals γ0 , . . . , γn−1 ; • an increasing list of natural numbers f (0), . . . , f (n); • an iteration tree T f (n) + 1 on M, with a final model Mf (n) and an embedding j0,f (n) : M → Mf (n) ; and • a sequence P = $a0 , . . . , an−1 % in (Mf (n) &j0,f (n) (τ ))n .
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We follow the notation of Section 1F(1) when discussing these objects. We work to develop some terminology concerning P, leading to Claim 6C.7 which we will need later on. Claim 6C.2. The critical point of j0,f (n) is greater than λ. Proof. The rules of the mixed pivot game, stated in Section 1F (1), force all extenders used in j0,f (n) to have critical points above rank(X). X in our case is M&λ, see Remark 4C.12. So all the extenders used in T f (n) + 1 must have critical points above rank(M&λ), and the claim follows. # Definition 6C.3. If n = 0 define γ (P) to be j0,f (0) (γL ) where $γL , γH % is the least pair of local indiscernibles of M relative to τ . If n > 0 define γ (P) to be equal to jf (m),f (n) (γm ) where m < n is largest so that f (m) belongs to [0, f (n))T . If 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, #
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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.
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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].
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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.
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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 $Mβ , jβ,β ¯ | β¯ ≤ β < α%. Let jβ,α be the skip (M, t, τ )(Y˙ ) ends and player I direct limit embeddings. If Mα is illfounded then G wins through condition (I3) in Section 6A. So suppose that Mα is wellfounded. Let tα = β<α tβ . This is an annotated Mα -position. Let Sα be the set consisting of zero and all successor ordinals smaller than α. The relative domain of tα is equal to sup{δβ + 1 | β ∈ Sα }. From condition (vi) it follows that: (a) rdm(tα ) ≤ j0,α (λ). Using conditions (4) and (iii) it is easy to see that the numbers nβ for β ∈ Sα are distinct. Since α is a limit it follows that nβ → ∞ as β → α. For each n < ω let βn < α be a successor ordinal large enough that: (b) nξ ≥ n for all successor ξ ∈ [βn , α). Let wα = {$nβ , δβ % | β ∈ Sα }. This is a witness for rdm(tα ). Let xα = tα , wα . Using condition (4) we see that wα extends tβn (κβn ) for each n < ω. Adding condition (b) we see that wα is in fact an n-extension of tβn (κβn ). By Claim 4A.18 it follows that: (c) x¯βn and xα agree to n. Let φα = β∈Sα (jβ,α ◦ φβ ). The union makes sense and yields a function by condition (5). Let aα = {nβ | β ∈ Sα }. This is the set of numbers used in wα . Define α : ω → Mα &j0,α (τ ) + 1 by: φα (r(i)) if r(i) ∈ aα × ω; and α (i) = ∅ otherwise. Using our definitions, condition (b), and condition (ii) it is easy to see that: (d) jβn ,α ◦ ¯ βn and α agree to n. By condition (xi), the range of φα contains the range of jβ,α ◦ ψβ for each successor β < α. Recall that ψβ was picked a surjection on Mβ &j0,β (τ ) + 1. So the range of φα contains the image under jβ,α of Mβ &jβ,α (τ ) + 1 for each successor β < α. It follows that φα : aα × ω → Mα &j0,α (τ ) + 1 is a surjection. This in turn implies that: (e) α : ω → Mα &j0,α (τ ) + 1 is a surjection. Given condition (c) we may regard jβn ,α (Pβn ) as a position in j0,α (Rmix )[xα ]. The position is useful and played according to j0,α (σmix )[α , xα ]. Using conditions (6) and (b)it is easy to see that jβn+1 ,α (Pβn+1 ) extends jβn ,α (Pβn ) for each n < ω. Set Pα = n<ω jβn ,α (Pβn ). Then: (f) Pα is an infinite run of j0,α (Rmix )[xα ]. Pα is useful and played according to j0,α (σmix )[α , xα ].
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Let Tα be the iteration tree given by Pα . Pα and Tα will be the driving forces behind our moves for player I in mega-round α. Remark 6C.15. The reader should compare the limit settings here with the limit construction in Section 3B(3). In both constructions we use the agreement secured in mega-rounds prior to α to argue for the convergence which produces the mixed pivot needed at α. In both constructions the argument hinges on the fact that the numbers nβ are distinct, and therefore converge to ∞. We already saw that the numbers nβ for β ∈ Sα are distinct. It follows from this that skip (M, t, τ )(Y˙ ) does not end with a loss for I through condition (I4) α is countable. So G in Section 6A. We now describe how to play in mega-round α. We divide into three cases, but one of them is degenerate. Degenerate limit. If all cofinal branches through Tα lead to illfounded direct limits. skip (M, t, τ )(Y˙ ) by playing Tα at the first opportunity. In this case player I can win G The opportunity comes already at mega-round α if α falls under the conditions of the standard limit case in Section 6A. Otherwise the opportunity comes at mega-round α + 1. Once Tα is played, II is forced into a loss through one of the conditions (I1) and (I3) in Section 6A. # (Degenerate limit) Suppose now that some cofinal branch through Tα leads to a wellfounded direct limit. Claim 6C.16. rdm(tα ) is equal to j0,α (λ). Proof. Immediate from Claim 6C.11, applied over Mα . To apply the claim we must work with a cofinal wellfounded branch through Tα . Which particular wellfounded branch we take does not matter here. # It follows from the last claim that rdm(tα ) is a limit of Woodin cardinals in Mα , but not itself a Woodin cardinal. Mega-round α therefore falls under the standard limit case in Section 6A, played subject to rules (L1)–(L6). The assignments of wα and Tα above cover the moves corresponding to rules (L1) and (L2). Let the imaginary opponent play bα subject to rule (L3). Let Qα be the direct limit along the branch bα of Tα . Let kα : Mα → Qα be the skip (M, t, τ )(Y˙ ) ends and player I direct limit embedding. If Qα is illfounded then G wins through condition (I5). So suppose that Qα is wellfounded. Let uα = uPα ,bα , where uPα ,bα is given by Definition 6C.10 applied over Mα . Let vα be given by Claim 6C.11, applied with Pα and bα over Mα . Let yα = uα ⊕ vα . This last assignment covers the move corresponding to rule (L4) in Section 6A. Set sα = tα −−, wα , yα . Translating the conclusion of Claim 6C.11 to the current context we get: (g) sα is either I-acceptably obstructed over Qα ; or else it is obstruction free and belongs to an interpretation of (kα ◦ j0,α )(Y˙ ).
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skip (M, t, τ )(Y˙ ) does not end with a loss for I through In particular it follows that G condition (I6) in Section 6A. Our work so far covers the moves corresponding to rules (L1)–(L4) in Section 6A. The imaginary opponent may now elect an early end to mega-round α, or elect to continue with a leap subject to rules (L5) and (L6). Early end. If the imaginary opponent elects an early end. Following the notation of Section 6A set Mα+1 = Qα , jα,α+1 = kα , and tα+1 = sα . skip do not allow sα must be obstruction free over Qα , since otherwise the rules of G II to elect an early end in mega-round α. Using condition (g) it follows that sα belongs to an interpretation of (kα ◦ j0,α )(Y˙ ). In other words: • tα+1 belongs to an interpretation of j0,α+1 (Y˙ ). skip (M, t, δ)(Y˙ ) ends, and player I wins through the payoff condiThis means that G tion (P1) in Section 6A. # (Early end) Leap. If the imaginary opponent elects a leap in mega-round α. Let Eα∗ , δα+1 , and tα+1 be the moves made by the imaginary opponent subject to rules (L5) and (L6). Following the notation of Section 6A let Mα+1 = Ult(Mα , Eα∗ ) and let jα,α+1 be the skip (M, t, τ )(Y˙ ) ends and player I ultrapower embedding. If Mα+1 is illfounded then G wins through condition (I7) in Section 6A. So suppose that Mα+1 is wellfounded. The moves δα+1 and tα+1 played by the imaginary opponent must satisfy the conditions of rule (L6) in Section 6A. Let n∗α witness that they satisfy condition (3) in that rule. Let cα∗ = uα (n∗α ) and let cα+1 = jα,α+1 (cα∗ ). Let λα = j0,α (λ). Condition (3) in rule (L6) tells us that tα+1 belongs to an interpretation of cα+1 (λα , δα+1 ). Let gα+1 witness this. Then: (xii) gα+1 is col(ω, δα+1 )-generic/Mα+1 and tα+1 ∈ cα+1 (λα , δα+1 )[gα+1 ]. Let P∗α be the position trunc(Pα , bα , n∗α ) given by Definition 6C.12 applied over Mα . Using Claim 6C.13 over Mα we see that: (xiii) For every pair $κ, δ% ∈ j0,α (I ), and every g which is col(ω, δ)-generic/Mα , ˙ δ, P∗α )[g]. cα∗ (κ, δ)[g] is contained in j0,α (K)(κ, Let Pα+1 = jα,α+1 (P∗α ). Shifting the last condition to Mα+1 and applying it with the pair $λα , δα+1 % we see that: ˙ α , δα+1 , Pα+1 )[gα+1 ]. (xiv) cα+1 (λα , δα+1 )[gα+1 ] ⊂ j0,α+1 (K)(λ Combining condition (xiv) and the final clause in condition (xii) we get: ˙ α , δα+1 , Pα+1 )[gα+1 ]. (xv) tα+1 belongs to j0,α+1 (K)(λ Let κα+1 = λα and let nα+1 = n∗α . Let φα+1 equal jα,α+1 ◦ φα . We have defined all the objects of conditions (A)–(G), listed above in the successor case, for α + 1. It is easy to check that conditions (1)–(6) hold for α + 1 with our definitions. Let us only
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make the following comments: Condition (3) for α + 1 follows from condition (xv) above. Condition (4) for α + 1 follows from the definition of wα and the fact that tα+1 (λα ) = wα . We may now proceed to mega-round α + 1 of the construction. # (Leap) The various cases above provide a complete inductive description of a construction skip (M, t, τ )(Y˙ ), joint with an imaginary opponent who plays for II. In of a run of G the “early end” case the construction terminates with a victory for player I through the payoff condition (P1) of Section 6A. In the other cases the construction may terminate through one of the conditions (I1)–(I7), but it only terminates with victory for player I. The construction can therefore be formalized into a winning strategy for player I in skip (M, t, τ )(Y˙ ). # (Lemma 6C.14) G
6D Woodin limits of Woodin cardinals Let M be a transitive model of ZFC∗ . Let θ be a Woodin limit of Woodin cardinals in M. Let Y˙ be a θ-name in M. We work with these fixed objects throughout Section 6D. Later in the section we will fix some r which belongs to an interpretation of a pullback skip (M, r, θ)(Y˙ ). of Y˙ , and aim to construct a winning strategy for I in G We follow the notation of Section 4D(1). α < θ is the least ordinal which seals Y˙ . We use C˙ to denote the function which associates to each δ ∈ [α, θ ) ∩ W , each κ ∈ ˙ δ ∩ L, each n ∈ ω, and each ordinal γ < jumpM (θ ), the name C(κ, δ, n, γ ) given by ˙ Definition 4D.5. There is a dependence on the name Y in that definition. But Y˙ is fixed and we suppress it in the notation. The function C˙ belongs to M. It can (and will) be shifted via elementary embeddings which act on M. Remark 6D.1. Again the point of the restriction to γ < jumpM (θ ) is to make C˙ a set function which literally belongs to M, rather than a class function definable over M. The ordinals γ which come up in the constructions later in the section are all smaller than jumpM (θ ), so this restriction does not pose any problems. 6D (1) Hopeful. Fix in addition to the objects listed above: • δ ∈ [α, θ ) ∩ W ; and • a δ-sequence s over M. Let $γL , γH % be the least pair of local indiscernibles of M relative to θ . By Claim 1A.16, ˙ . . , γL ) and C(. ˙ . . , γH ). γL and γH are smaller than jumpM (θ). So we may talk about C(. Suppose that: (H1) θ is countable in V; (H2) s is hopeful with respect to Y˙ over M; and
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˙ (H3) s belongs to an interpretation of C(0, δ, 0, γL ). Note that assumption (H2) makes sense, since s is a δ-sequence and δ belongs to the interval [α, θ ). Lemma 6D.2 (under the assumptions (H1)–(H3) listed above). Player I has a winning skip (M, s, θ)(Y˙ ). strategy in G skip (M, s, θ)(Y˙ ). We work Proof. Fix an imaginary opponent willing to play for II in G skip (M, s, θ)(Y˙ ). with the imaginary opponent to construct a run of ω mega-rounds in G We shall verify after ω mega-rounds that the run is won by player I. At the start of mega-round β, for β < ω, we will have: (A) a wellfounded model Mβ ; (B) an elementary embedding j0,β : M → Mβ ; (C) δβ ∈ Wβ and a δβ -sequence tβ over Mβ ; (D) a generic gβ for col(ω, δβ ) over Mβ ; (E) a function β : ω → j0,β (θ); (F) κβ ∈ Lβ , and nβ < ω. Wβ and Lβ above are the classes W and L of Section 4A, computed in Mβ . We intend to secure the following inductive conditions: (1) β is cofinal in j0,β (θ); (2) tβ is hopeful with respect to j0,β (Y˙ ) over Mβ ; and ˙ β , δβ , nβ , j0,β (γL ))[gβ ]. (3) tβ belongs to j0,β (C)(κ We set to begin with M0 = M, δ0 = δ, t0 = s, κ0 = 0, and n0 = 0. Condition (2) follows from assumption (H2). Assumption (H3) allows picking g0 so as to satisfy condition (3). Assumption (H1) allows picking 0 so as to satisfy condition (1). Let us start mega-round β of the construction. Using Claim 5D.2 inside Mβ we may switch from j0,β (γL ) in condition (3) to the ordinal j0,β (γH ). We get: ˙ β , δβ , nβ , j0,β (γH ))[gβ ]. (i) tβ belongs to j0,β (C)(κ By Definition 4D.5 this means in particular that: (ii) nβ is not used in tβ (κβ ); and (iii) player I wins the game j0,β (G∗wl )(tβ , κβ , δβ , nβ , j0,β (γH )).
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Remark 6D.3. The game j0,β (G∗wl )(tβ , . . . ) consists of moves ν, δ ∗ , n∗ , and γ ∗ subject to the rules listed in Section 4D(1), shifted to Mβ . The precise format of this game is displayed in Diagram 4.3. Note particularly that the move γ ∗ is the responsibility of player II. This is because tβ is hopeful with respect to j0,β (Y˙ ) over Mβ , by condition (2). Keeping the last remark in mind we see that condition (iii) expands to: (iv) For every ν < j0,β (θ), there is a δ ∗ and an n∗ satisfying the shift to Mβ of rule (2) in Section 4D(1), so that for every γ ∗ < j0,β (γH ), the shift to Mβ of the payoff condition (P) in Section 4D(1) holds. Following notation similar to that of Section 4D (1) let wβ = tβ (κβ ) + $nβ , δβ %, let κβ∗ = δβ + 1, and let δβ† be the first Woodin cardinal of Mβ above δβ . Let νβ = β (β). Let δβ∗ and n∗β be given by condition (iv), applied with ν = νβ . These objects satisfy the shift to Mβ of rule (2) in Section 4D (1). In particular this means that δβ∗ > νβ and δβ∗ < j0,β (θ). Folding into this the value of νβ we have: (v) δβ∗ > 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<ω tβ . tω is an annotated position over Mω . Translating the limit case condition (f) in Section 6A to our context we get: (xii) tω is Mω -clear. Claim 6D.4. rdm(tω ) = j0,ω (θ). Proof. Condition (v) shifted to Mβ+1 states that δβ+1 > (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<ω δβ+1 = j0,ω (θ). The claim follows. Corollary 6D.5. rdm(tω ) is Woodin in Mω . Proof. This follows from the previous claim, since θ is Woodin in M.
#
Let W ∈ M denote the forcing Wθ of Section 4B, defined in M. Let t˙ ∈ M be the name of Definition 4B.39. Let G(tω ) be the filter associated to tω by Definition 4B.23, executed over Mω . Claim 6D.6. G(tω ) is j0,ω (W)-generic/Mω and j0,ω (t˙)[G(tω )] = tω . Proof. Claim 6D.4 and condition (xii) combine to state that tω is an Mω -clear annotated position of relative domain j0,ω (θ) over Mω . The current claim follows through an application of Corollary 4B.30 and an application of Claim 4B.41. # Claim 6D.7. tω belongs to j0,ω (Y˙ )[G(tω )]. Proof. Recall that one of the initial assumptions in Lemma 6D.2, assumption (H2) to be exact, states that s is hopeful with respect to Y˙ over M. By Definition 4D.2 this means that there exists some good identity σ ∈ M&α so that s |= σ and [σ ] W “t˙ ∈ Y˙ .” (α here is the least ordinal which seals Y˙ .) Condition (xi) implies that j0,ω : M → Mω has critical point greater than δ0 . δ0 was set equal to δ at the start of the construction. δ, which was fixed at the start of Section 6D (1), is greater than α. So σ , which belongs to M&α, is not moved by j0,ω . We know that [σ ] forces “t˙ ∈ Y˙ ” over M. Applying j0,ω , and using the fact that σ is not moved, we see that [σ ] j0,ω (W) “j0,ω (t˙) ∈ j0,ω (Y˙ )” over Mω . tω extends t0 , which was set equal to s at the start of the construction. Using the fact that s |= σ over M it follows that tω |= σ over Mω . So [σ ] belongs to G(tω ). Since [σ ] forces “j0,ω (t˙) ∈ j0,ω (Y˙ )” over Mω , it follows that j0,ω (t˙)[G(tω )] ∈ j0,ω (Y˙ )[G(tω )]. But j0,ω (t˙)[G(tω )] is equal to tω by Claim 6D.6. So tω ∈ j0,ω (Y˙ )[G(tω )]. #
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skip (M, s, θ)(Y˙ ). CorolRemember that we are now working on mega-round ω of G lary 6D.5 tells us that this mega-round follows the rules of the phantom limit case in Section 6A. Following the notation of the phantom limit case let δω+1 = δω = rdm(tω ), and skip (M, s, θ)(Y˙ ) let jω,ω+1 = id. By Claim 6D.4, δω+1 = j0,ω+1 (θ ). This means that G ends with the payoff condition (P2) in Section 6A. By Claim 6D.7 player I wins. Working with an imaginary opponent who plays for II we described how to construct skip (M, s, θ)(Y˙ ). The mechanism we described always leads to runs which a run of G are won by player I. It can therefore be formalized into a winning strategy for I in skip (M, s, θ )(Y˙ ). # (Lemma 6D.2) G 6D (2) Not hopeful. Recall that we are working with a transitive model M of ZFC∗ , a Woodin limit of Woodin cardinals θ in M, and a θ-name Y˙ over M. Recall that α < θ is the least ordinal which seals Y˙ . Recall that C˙ ∈ M is the function which associates to each δ ∈ [α, θ) ∩ W , each κ ∈ δ ∩ L, each n < ω, and each ordinal γ < jumpM (θ ), ˙ the name C(κ, δ, n, γ ) given by Definition 4D.5. As in Section 6D(1) let us fix: • δ ∈ [α, θ ) ∩ W ; and • a δ-sequence s over M. Let $γL , γH % be the least pair of local indiscernibles of M relative to θ. Suppose that: (N1) θ is countable in V; (N2) s is not hopeful with respect to Y˙ over M; and ˙ (N3) s belongs to an interpretation of C(0, δ, 0, γL ). These assumptions are similar to the assumptions (H1)–(H3) in Section 6D (1), except that here we assume that s is not hopeful. Lemma 6D.8 (under the assumptions (N1)–(N3) listed above). Player I has a winning skip (M, s, θ)(Y˙ ). strategy in G skip (M, s, θ)(Y˙ ). We work Proof. Fix an imaginary opponent willing to play for II in G skip (M, s, θ)(Y˙ ). with the imaginary opponent to construct a run of ω mega-rounds in G At the start of mega-round β, for β < ω, we will have: (A) a wellfounded model Mβ ; (B) an elementary embedding j0,β : M → Mβ ; (C) δβ ∈ Wβ and a δβ -sequence tβ over Mβ ; (D) a generic gβ for col(ω, δβ ) over Mβ ; (E) a function β : ω → j0,β (θ);
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(F) κβ ∈ Lβ , and nβ < ω; and (G) an ordinal γβ in Mβ . This list of objects is similar to the one at the start of the proof of Lemma 6D.2. But note the addition here of the ordinal γβ listed in condition (G). We intend to secure the following inductive conditions: (1) β is cofinal in j0,β (θ); (2) tβ is not hopeful with respect to j0,β (Y˙ ) over Mβ ; and ˙ β , δβ , nβ , γβ )[gβ ]. (3) tβ belongs to j0,β (C)(κ This list too is similar to the one in the proof of Lemma 6D.2. But there are a couple of differences. In condition (2) we state that tβ is not hopeful. In condition (3) we use the ordinal γβ of condition (G). Set to begin with M0 = M, δ0 = δ, t0 = s, κ0 = 0, n0 = 0, and γ0 = γL . Condition (2) follows from assumption (N2). Assumption (N3) allows picking g0 so as to satisfy condition (3). Assumption (N1) allows picking 0 so as to satisfy condition (1). Let us start mega-round β of the construction. By Definition 4D.5, condition (3) implies that: (i) nβ is not used in tβ (κβ ); and (ii) player I wins the game j0,β (G∗wl )(tβ , κβ , δβ , nβ , γβ ). The game j0,β (G∗wl )(tβ , . . . ) consists of moves ν, δ ∗ , n∗ , and γ ∗ subject to the rules listed in Section 4D(1), shifted to Mβ . We refer the reader to Diagram 4.3 for the precise format of the game. Note that the move γ ∗ is the responsibility of player I here. This is because tβ is not hopeful with respect to j0,β (Y˙ ) over Mβ , by condition (2). Remark 6D.9. The fact that the move γ ∗ here falls on player I should be contrasted with the situation in Lemma 6D.2. There γ ∗ was the responsibility of player II, see Remark 6D.3. Keeping in mind that γ ∗ is the responsibility of player I in j0,β (G∗wl )(tβ , . . . ) we see that condition (ii) expands to: (iii) For every ν < j0,β (θ), there is a δ ∗ and an n∗ satisfying the shift to Mβ of rule (2) in Section 4D(1), and there is an ordinal γ ∗ < γβ , so that the shift to Mβ of the payoff condition (P) in Section 4D(1) holds. Note that γ ∗ is bounded by an existential quantifier here. This is in line with the fact that γ ∗ is the responsibility of player I in j0,β (G∗wl )(tβ , . . . ). Let wβ = tβ (κβ ) + $nβ , δβ %, let κβ∗ = δβ + 1, and let δβ† be the first Woodin cardinal of Mβ above δβ .
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Let νβ = β (β). Let δβ∗ , n∗β , and γβ∗ be given by condition (iii), applied with ν = νβ . Note that the ordinal γβ∗ is given to us by condition (iii). The constraints placed by that condition are such that: (iv) γβ∗ < γβ . ˙ ∗ , δ ∗ , n∗ , γ ∗ ) and let C˙ † = Back I (δ † , δ ∗ )(C˙ ∗ ). Let C˙ β∗ = j0,β (C)(κ β β β β β β β β Construct Mβ+1 , jβ,β+1 , δβ+1 , tβ+1 , β+1 , κβ+1 , and nβ+1 by copying precisely the construction of these objects in the proof of Lemma 6D.2, only using the names C˙ β∗ and C˙ β† defined above. We omit the actual copying, but note that among other things we get: (v) tβ+1 belongs to an interpretation of jβ,β+1 (C˙ β∗ ). ˙ ∗ , δ ∗ , n∗ , γ ∗ ), conLet γβ+1 = jβ,β+1 (γβ∗ ). Since C˙ β∗ was set equal to j0,β (C)(κ β β β β dition (v) allows picking gβ+1 so as to satisfy the inductive condition (3) of the current construction. The inductive condition (2) carries over to β +1 since tβ+1 extends tβ , and since crit(jβ,β+1 ) > 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 accord skip (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.
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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<ρ Pι . Suppose that each ∗ ∗ branch (M, t, δ ∗ )(C˙ ∗ ). ˙ Pι is secure in Gbranch (M, t, δ )(C ). Then P∞ too is secure in G Proof. For each ι < ρ let fι be the unique route to Pι . By Corollary 6F.12, $fι | ι < ρ% is increasing. Let g = ι<ρ fι . Let γ∞ be the domain of g. It is easy to see that γ∞ is a limit ordinal. Let η∞ be the length of P∞ . It is easy to see that g is cofinal in η∞ . Define f∞ : γ∞ + 1 → η∞ + 1 by: g(β) if β < γ∞ ; and f∞ (β) = if β = γ∞ . η∞ It is easy to see that f∞ is a route to P∞ .
#
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branch (M, t, δ ∗ )(C˙ ∗ ), and secure. Then P is won Claim 6F.16. Let P be terminal in G by player I. Proof. Let f be the unique route to P . Let R = red(P , f ). Using the fact that P is ter skip (M, t, δ ∗ )(C˙ ∗ ). branch (M, t, δ ∗ )(C˙ ∗ ) it is easy to see that R is terminal in G minal in G R is consistent with skip (M, t, δ ∗ , C˙ ∗ ) by Definition 6F.9, and skip (M, t, δ ∗ , C˙ ∗ ) is skip (M, t, δ ∗ )(C˙ ∗ ). a winning strategy for player I. So R = red(P , f ) is won by I in G ∗ branch (M, t, δ )(C˙ ∗ ). It follows from this that P is won by player I in G # 6F (2) Extensions. Let $M, t, δ ∗ , C˙ ∗ % be promising. Let P be secure in the game branch (M, t, δ ∗ )(C˙ ∗ ), and non-terminal. We work on ways to extend P by one megaG round, in such a way that the extension is either secure, or else it translates to a position skip in which player I calls for a skip. The former case of course sets the grounds in G for an iteration. In the latter case we show that I’s moves give rise to a new promising tuple. Let f be the unique route to P . Let R = red(P , f ). Observe that R is non-terminal skip (M, t, δ ∗ )(C˙ ∗ ). This can be seen easily using the assumption that P is nonin G branch (M, t, δ ∗ )(C˙ ∗ ). Let η = lh(P ) and let γ = lh(R). We follow terminal in G the notation established in Section 6F(1). P is given by a sequence $Tξ , bξ , Eξ∗ , tξ +1 | ξ < η%. The sequence gives rise to models Mξ and embeddings jζ,ξ for ζ < ξ ≤ η. branch (M, t, δ ∗ )(C˙ ∗ ) branch (M, t, δ ∗ , C˙ ∗ )[P ] to denote mega-round η of G We use G following the position P . This is a game of just one mega-round, played according to the rules of either the successor case or the limit cases in Section 6A. skip (M, t, δ ∗ )(C˙ ∗ ) fol skip (M, t, δ ∗ , C˙ ∗ )[R] to denote mega-round γ of G We use G lowing the position given by R. This too is a game of just one mega-round, played according to the rules of either the successor case for skipping games or the limit cases in Section 6A. skip (M, t, δ ∗ , C˙ ∗ )[R] are precisely branch (M, t, δ ∗ , C˙ ∗ )[P ] and G If η is a limit then G the same game. They both start from Mη and tη , and they are both played according to the rules of the limit cases in Section 6A. branch (M, t, δ ∗ , C˙ ∗ )[P ] is an initial segment of If η is a successor or zero then G ∗ ∗ skip (M, t, δ , C˙ )[R]. Both games start from Mη and tη . G branch (M, t, δ ∗ , C˙ ∗ )[P ] G skip (M, t, δ ∗ , C˙ ∗ )[R] may is played according to rules (S1)–(S4) in Section 6A. G continue with rules (S5) and (S6). skip (M, t, δ ∗ , C˙ ∗ ) to skip (M, t, δ ∗ , C˙ ∗ )[R] to denote the restriction of We use skip (M, t, δ ∗ , C˙ ∗ )[R] is thus a mega-round γ following the position given by R. skip (M, t, δ ∗ , C˙ ∗ )[R]. strategy for player I in G branch (M, t, δ ∗ , C˙ ∗ )[P ] to be the restriction of the strategy Definition 6F.17. Define ∗ ∗ branch (M, t, δ ∗ , C˙ ∗ )[P ]. skip (M, t, δ , C˙ )[R] to the game G Definition 6F.17 is made under the assumption that P is non-terminal and secure in branch (M, t, δ ∗ )(C˙ ∗ ). R stands for red(P , f ) where f is the unique route to P . Note G branch (. . . )[P ] that the restriction mentioned in Definition 6F.17 makes sense, because G skip (. . . )[R] or else it is an initial segment of that game. is either exactly equal to G
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branch (M, t, δ ∗ )(C˙ ∗ ), extending P . Claim 6F.18. Let P + be a position of length η+1 in G Suppose that η is a limit ordinal. Suppose further that mega-round η in P + is played branch (M, t, δ ∗ )(C˙ ∗ ). branch (M, t, δ ∗ , C˙ ∗ )[P ]. Then P + is secure in G according to Proof. This is a simple matter of checking that the natural extension of the route which leads to P gives a route which leads to P + . Recall that f is the unique route to P and γ = lh(f ). Let f + : γ + 2 → η + 2 be the function defined by: f (β) if β ≤ γ ; and + f (β) = η + 1 if β = γ + 1. We claim that f + is a route to P + . To prove this we must check that R + = red(P + , f + ) skip (M, t, δ ∗ , C˙ ∗ ). is consistent with + skip (M, t, δ ∗ , C˙ ∗ ) since f R γ is equal to red(R, f ) which is consistent with is a route to P . So it is enough to check that mega-round γ in R + is consistent with skip (M, t, δ ∗ , C˙ ∗ )[R]. η is assumed to be a limit ordinal. It follows from this that the two games skip (M, t, δ ∗ , C˙ ∗ )[R] are precisely the same, and that branch (M, t, δ ∗ , C˙ ∗ )[P ] and G G ∗ skip (M, t, δ ∗ , C˙ ∗ )[R] are precisely the the strategies branch (M, t, δ , C˙ ∗ )[P ] and + skip (M, t, δ ∗ , C˙ ∗ )[R] same. The fact that mega-round γ in R is consistent with + thus follows from the assumption that mega-round η in P is played according to branch (M, t, δ ∗ , C˙ ∗ )[P ]. # branch (M, t, δ ∗ )(C˙ ∗ ), extending P . Claim 6F.19. Let P + be a position of length η+1 in G Suppose that η is a successor or zero. Suppose further that mega-round η in P + is played branch (M, t, δ ∗ , C˙ ∗ )[P ]. Then one of the following conditions holds: according to branch (M, t, δ ∗ )(C˙ ∗ ); or (1) P + is secure in G skip (M, t, δ ∗ , C˙ ∗ )[R] calls for a skip following the moves made in mega-round (2) η of P + . Proof. Define f + and R + as in the proof of Claim 6F.18. Mega-round γ in R + follows skip (M, t, δ ∗ , C˙ ∗ )[R] for the moves corresponding to rules (S1)–(S4) in Section 6A, skip (M, t, δ ∗ , C˙ ∗ )[R] and then calls for an early end. If this call is consistent with then an argument similar to the one in Claim 6F.18 shows that P + is secure and we skip (M, t, δ ∗ , C˙ ∗ )[R] calls obtain condition (1) in Claim 6F.19. If on the other hand for a skip then we obtain condition (2). # branch (M, t, δ ∗ )(C˙ ∗ ) if it falls under Definition 6F.20. We say that P + opens a skip in G the situation of condition (2) in Claim 6F.19. Claims 6F.18 and 6F.19 together show that an extension P + of P obtained branch (M, t, δ ∗ , C˙ ∗ )[P ]—or in other words through a use of through a use of G ∗ ∗ skip (M, t, δ , C˙ )[R]—is either secure or else it opens a skip. In the former case G skip (M, t, δ ∗ , C˙ ∗ )[R + ]. we can iterate our work, moving onward to extend P + using
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skip (M, t, δ ∗ , C˙ ∗ )[R] calls for a skip following the moves in But in the latter case mega-round η of P + , and there are no moves in P + to answer that call. It is therefore skip (M, t, δ ∗ , C˙ ∗ ) to continue and extend P + . impossible in the latter case to use We work for the rest of Section 6F(2) in the situation of this latter case, the case that P + opens a skip. We shall see that the moves in this case give rise to a new promising skip which can serve as a temporary tuple, allowing access to a new instance of skip (M, t, δ ∗ , C˙ ∗ ) above. replacement for the instance Let wη , yη , Tη , and bη be the moves made in mega-round η of P + . These moves correspond to rules (S1)–(S4) in Section 6A. Let Qη be the direct limit along the branch bη of Tη . Let kη : Mη → Qη be the direct limit embedding. The settings in the branch are such that: successor case of G (i) Mη+1 = Qη ; (ii) jη,η+1 = kη ; (iii) δη+1 = kη (δη† ); and (iv) tη+1 = tη −− wη , yη . branch (M, t, δ ∗ , C˙ ∗ )[P ], The moves wη , yη , Tη , and bη are played according to ∗ ∗ skip (M, t, δ , C˙ )[R] to rules (S1)–(S4). We which by definition is the restriction of skip (M, t, δ ∗ , C˙ ∗ )[R] thereare working under the assumption that P + opens a skip. fore calls for a skip after the moves wη , yη , Tη , and bη . skip (M, t, δ ∗ , C˙ ∗ )[R] subject to rule (S5), Let δγ∗ and C˙ γ∗ be the moves played by in response to wη , yη , Tη , and bη . Following the notation in Section 6A let δη† be the first Woodin cardinal of Mη above rdm(tη ), and let Wη be the class W computed in Mη . Translating the conditions in rule (S5) to the current context we see that: (v) δγ∗ belongs to Wη ; (vi) δγ∗ is greater than δη† and smaller than j0,η (δ ∗ ); (vii) C˙ γ∗ is a δγ∗ -name in Mη ; and (viii) tη −−, wη , yη belongs to an interpretation of kη (C˙ γ† ) where C˙ γ† is equal to Back I (δη† , δγ∗ )(C˙ γ∗ ) computed in Mη . (The indexing is a bit awkward here. Let us remind the reader that mega-round η in branch corresponds to mega-round γ in G skip .) G + ∗ + Let δ = kη (δγ ) and let C˙ = kη (C˙ γ∗ ). With this notation the conditions above combine to yield the following claims: Claim 6F.21. δ + is smaller than j0,η+1 (δ ∗ ). Proof. Immediate from the definition of δ + and conditions (ii) and (vi). Claim 6F.22. $Mη+1 , tη+1 , δ + , C˙ + % is promising.
#
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Proof. The fact that M&δ ∗ + 1 is countable in V, the fact that j0,η+1 preserves countability (see Remark 6A.4), and the last claim combine to imply that Mη+1 &δ + + 1 is countable in V. This gives condition (1) in Definition 6F.1. Condition (2) in Definition 6F.1 translates in our context to the statement that tη+1 belongs to an interpretation of Back I (δη+1 , δ + )(C˙ + ). This statement follows from condition (viii) above, using the # definitions of δ + and C˙ + , and conditions (iii) and (iv). Now that we know that $Mη+1 , tη+1 , δ + , C˙ + % is promising we may attempt to skip (Mη+1 , tη+1 , δ + , C˙ + ), instead of skip (M, t, δ ∗ , C˙ ∗ ), to create extenuse + sions of P . Section 6F(3) below develops the terminology needed for such a use. skip (Mη+1 , tη+1 , δ + , C˙ + ) can be used until finally it produces a position which is ter skip (and of course won by player I). We shall see minal in the respective instance of G skip (M, t, δ ∗ , C˙ ∗ ). that at that point it is possible to return to using 6F (3) Closing skips. Let $M, t, δ ∗ , C˙ ∗ % be promising. Let θ be an ordinal. Let branch (M)(t, δ ∗ , C˙ ∗ ). Q Q = $Tξ , bξ , Eξ∗ , tξ +1 | ξ < θ% be a position of length θ in G gives rise to: • models Mξ for ξ ≤ θ; and • embeddings jζ,ξ : Mζ → Mξ for ζ < ξ ≤ θ . Let η be an ordinal smaller than θ. Let P = Qη and let P + = Qη + 1. Suppose that: branch (M, t, δ ∗ )(C˙ ∗ ); (C1) P = Qη is secure in G branch (M, t, δ ∗ , C˙ ∗ )[P ]; and (C2) mega-round η in Q is played according to branch (M, t, δ ∗ , C˙ ∗ ). (C3) P + = Qη + 1 opens a skip in G Let δ + and C˙ + be defined as in Section 6F(2). By Claims 6F.21 and 6F.22, δ + is smaller than j0,η+1 (δ ∗ ) and, most importantly, $Mη+1 , tη+1 , δ + , C˙ + % is promising. Definition 6F.23. tail(Q, η) denotes $Tξ , bξ , Eξ∗ , tξ +1 | ξ ∈ [η + 1, θ)%. tail(Q, η) is the sequence consisting of mega-rounds η + 1 and above in Q. It branch (Mη+1 , tη+1 , j0,η+1 (δ ∗ ))(j0,η+1 (C˙ ∗ )). If rdm(tξ +1 ) ≤ is literally a position in G + jη+1,ξ +1 (δ ) + 1 for each ξ ∈ [η + 1, θ) then tail(Q, η) may also be regarded as a branch (Mη+1 , tη+1 , δ + )(C˙ + ). position in G Let us suppose that: (C4) rdm(tξ +1 ) ≤ jη+1,ξ +1 (δ + ) + 1 for each ξ ∈ [η + 1, θ), so that tail(Q, η) is a branch (Mη+1 , tη+1 , δ + )(C˙ + ); position in G branch (Mη+1 , tη+1 , δ + )(C˙ + ); and (C5) tail(Q, η) is terminal in G branch (Mη+1 , tη+1 , δ + )(C˙ + ). (C6) tail(Q, η) is won by player I in G
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Lemma 6F.24 (under assumptions (C1)–(C6) above). Exactly one of the following conditions holds: branch (M, t, δ ∗ )(C˙ ∗ ) and won by player I; or (1) Q is terminal in G branch (M, t, δ ∗ )(C˙ ∗ ). (2) Q is non-terminal and secure in G Proof. By assumptions (C5) and (C6) the position tail(Q, η) is terminal and won by branch (Mη+1 , tη+1 , δ + )(C˙ + ). A victory in G branch may come either through player I in G one of the snags described in conditions (I1)–(I7) in Section 6A, or through one of the payoff conditions (P1) and (P2) in that section. We divide the proof into two cases depending on which of these two possibilities holds for tail(Q, η). Case 1. If tail(Q, η) is won by player I through one of the snags described in conditions (I1)–(I7) of Section 6A. It is easy to see that in this case Q is terminal in branch (M, t, δ ∗ )(C˙ ∗ ) and won by player I through exactly the same snag. So we G obtain condition (1) in Lemma 6F.24. # (Case 1) Case 2. If tail(Q, η) is won by player I through one of the payoff conditions (P1) and branch (Mη+1 , tη+1 , δ + )(C˙ + ). Translating these conditions to the (P2), in the game G current context we see that: (i) θ is a successor ordinal; (ii) tθ is a jη+1,θ (δ + )-sequence over Mθ ; and (iii) tθ belongs to an interpretation of jη+1,θ (C˙ + ). By Claim 6F.21, δ + < j0,η+1 (δ ∗ ). So condition (ii) implies that the relative domain of tθ is strictly smaller than j0,θ (δ ∗ ). It follows from this that Q is non-terminal in branch (M, t, δ ∗ )(C˙ ∗ ). To obtain condition (2) in Lemma 6F.24 we must check that Q G branch (M, t, δ ∗ )(C˙ ∗ ). is secure in G Following the notation in Section 6F(2) let f be the unique route to P , let γ = lh(f ), and let R = red(P , f ). Define f ∗ : γ + 2 → θ + 1 by: f (β) if β < γ ; ∗ f (β) = η if β = γ ; and θ if β = γ + 1. The top two cases could be merged into one, since f (γ ) = η by condition (2) in Definition 6F.5. But we wish to emphasize the fact that f ∗ sends γ to η and γ + 1 to θ . Using the fact that θ is a successor ordinal it is easy to check that f ∗ is a skip frame for Q. f ∗ extends f . Let R ∗ = red(Q, f ∗ ). R ∗ then has the format of a reduced skip (M, t, δ ∗ )(C˙ ∗ ), extending R. Mega-round γ of R ∗ is position of length γ + 1 in G given by Tη , bη , hγ , and tθ , where: (iv) hγ = jη+1,θ .
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(We are simply copying from Definition 6F.8, using the fact that f ∗ (γ ) = η and f ∗ (γ + 1) = θ .) skip (M, t, δ ∗ )(C˙ ∗ ), consistent with Claim 6F.25. R ∗ is a reduced position in G ∗ ∗ ˙ skip (M, t, δ , C ). skip (M, t, δ ∗ , C˙ ∗ ) we only have Proof. Since R is a reduced position consistent with ∗ to worry about mega-round γ in R . Let wη , yη , Tη , and bη be the moves in megaround η of Q. These are the moves corresponding to rules (S1)–(S4) in mega-round skip (M, t, δ ∗ , C˙ ∗ )[R] by condition (C2). γ of R ∗ . These moves are consistent with branch (M, t, δ ∗ )(C˙ ∗ ). By condition (C3), P + = Qη + 1 opens a skip in the game G ∗ ∗ ˙ skip (M, t, δ , C )[R] therefore calls for a skip in response wη , yη , Tη , and bη . Let δγ∗ and C˙ γ∗ be the moves, for rule (S5) of the skip case in Section 6A, played by skip (M, t, δ ∗ , C˙ ∗ )[R] in response to wη , yη , Tη , and bη . the strategy Remember that our goal is to verify that mega-round γ in R ∗ is legal in skip (M, t, δ ∗ , C˙ ∗ )[R]. The moves correGskip (M, t, δ ∗ )(C˙ ∗ ) and consistent with sponding to rule (S6) in mega-round γ of R ∗ are hγ , Mθ , and tθ . We have to check that they can be regarded as a legal response to the moves δγ∗ and C˙ γ∗ made by skip (M, t, δ ∗ , C˙ ∗ )[R]. In other words we have to check that they satisfy the con ditions of rule (S6) in Section 6A, translated to the current context. Most of the conditions in rule (S6) are easy to verify. Let us just comment on condition (4). Translated to the current context it says that tθ must belong to an interpretation of (hγ ◦ kη )(C˙ γ∗ ), or in other words to an interpretation of jη+1,θ (C˙ + ). (The move from (hγ ◦ kη )(C˙ γ∗ ) to jη+1,θ (C˙ + ) uses condition (iv) above and the definition of C˙ + in Section 6F (2).) The fact that tθ does indeed belong to an interpretation of jη+1,θ (C˙ + ) follows from condition (iii) above. It is through the appeal to condition (iii) that we are using the crucial assumption in condition (C6), that tail(Q, η) is won by player I. # (Claim 6F.25) It follows from the last claim that f ∗ is a route to Q. Q is therefore secure in branch (M, t, δ ∗ )(C˙ ∗ ). Gbranch (M, t, δ ∗ )(C˙ ∗ ). We already saw that Q is non-terminal in G So we get condition (2) in Lemma 6F.24. # (Case 2) # (Lemma 6F.24) 6F (4) Discussion. We have now the necessary tools for sewing together instances of branch . The actual argument which does skip into winning strategies for player I in G this is given in Sections 6F(5) and 6F(6). Let us here try to view the argument as an induction. Fix a promising tuple $M, t, δ ∗ , C˙ ∗ %. We aim to show that player I has a winning branch (M, t, δ ∗ )(C˙ ∗ ). Suppose “inductively” that: strategy in G branch (N, s, δ + )(C˙ + ) whenever (∗) Player I has a winning strategy in the game G $N, s, δ + , C˙ + % is promising and δ + < j (δ) for some elementary j : M → N .
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branch (M, t, δ ∗ )(C˙ ∗ ), and win. We now describe how to play for I in G skip (M, t, δ ∗ , C˙ ∗ ). This can be done until, if ever, a Start playing by following branch is the same position is reached where that strategy calls for a skip. (Notice that G skip so long as skips are not called for.) as G skip (M, t, δ ∗ , C˙ ∗ ) eventually leads to a victory If no skip is ever called for then for player I. So suppose that at some point a position, P + of length η + 1 say, is reached skip (M, t, δ ∗ , C˙ ∗ ) calls for a skip. where Notice that P + opens a skip in the sense of Definition 6F.20. Let δ + and C˙ + be defined as in Section 6F(2). By Claims 6F.22 and 6F.21, $Mη+1 , tη+1 , δ + , C˙ + % is promising, and δ + < j0,η+1 (δ ∗ ). Using the “inductive” condition (∗) above it follows branch (Mη+1 , tη+1 , δ + )(C˙ + ). Let + be some such that I has a winning strategy in G strategy. branch (M, t, δ ∗ )(C˙ ∗ ), from the position P + onward, Continue now playing for I in G + . More precisely, continue by creating extensions Q of P + subject to by following + . This will lead to a position Q the condition that tail(Q, η) is played according to branch (Mη+1 , tη+1 , δ + )(C˙ + ). so that tail(Q, η) is won by I in G If Q is terminal in Gbranch (M, t, δ ∗ )(C˙ ∗ ) then by case 1 in Lemma 6F.24 it is won by player I in this game, as required. So suppose that Q is not terminal in branch (M, t, δ ∗ )(C˙ ∗ ). Again by Lemma 6F.24, this time case 2, it must be that Q G allows closing the skip that was opened by P + . Thus, from Q onward it is possible to skip (M, t, δ ∗ , C˙ ∗ ). return to using skip (M, t, δ ∗ , C˙ ∗ ) until, if ever, Proceed by iterating this process. From Q follow it calls for a skip. At that point make another use of the “inductive” condition (∗) to play for I until reaching a position where the skip can be closed, etc. skip (M, t, δ ∗ , C˙ ∗ ) can be combined with The sketch above shows how a use of branch (M, t, δ ∗ )(C˙ ∗ ). The key uses of the “inductive” condition (∗) to play for I in G points are: (a) the way rule (S5) in Section 6A (2)—the rule that opens a skip—gives branch rise to a new promising tuple; and (b) the way a victory for I in the instance of G corresponding to this new promising tuple gives precisely the moves needed for rule (S6) in Section 6A (2)—the rule that closes a skip. These two points are given precisely in Claim 6F.22 and Lemma 6F.24. There is though a problem with the sketch. The condition (∗) is not, strictly speaking, inductive. It has an inductive nature, in that it provides for “lower” promising tuples $N, s, δ + , C˙ + % the property which we are trying to prove for the tuple $M, t, δ ∗ , C˙ ∗ %. If “lower” were to mean a tuple with δ + < δ ∗ then the condition would actually be inductive. But “lower,” as stated and applied above, means a tuple with δ + < j (δ ∗ ) for some elementary j : N → M. Condition (∗) may thus be viewed as inductive not over ordinals in M, but over ordinals in some directed system of iterates of M. Unfortunately that system need not branch is automatically be wellfounded. Still, any illfoundedness reached in a play of G won by player I through one of the snags (I1), (I3), (I5), or (I7) in Section 6A. So some hope remains that an argument whose spirit follows the sketch given above may branch (M, t, δ ∗ )(C˙ ∗ ). produce a winning strategy for I in G
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Sections 6F(5) and 6F(6) formulate such an argument. Induction is replaced by nesting. When reaching a position P + which opens a skip we do not appeal to branch (Mη+1 , tη+1 , δ + )(C˙ + ), but condition (∗) to obtain a winning strategy for I in G rather nest a construction of such a strategy into the main construction of a strategy branch (M, t, δ ∗ )(C˙ ∗ ). The nested construction may itself reach a skip, in which in G case we nest another construction inside the nested construction, etc. We can proceed this way, and eventually reach a terminal position won by player I, for as long as the nesting levels are finite. We shall see that infinite nesting, if it occurs, leads to a position which gives rise to an illfounded iterate of M—this is related to the nature of “lower” explained above and follows precisely from Claim 6F.21—and such positions too are won by player I. 6F (5) Safe positions. Let $M, t, δ ∗ , C˙ ∗ % be promising. Let θ be an ordinal. Let Q = branch (M, t, δ ∗ )(C˙ ∗ ). $Tξ , bξ , Eξ∗ , tξ +1 | ξ < θ% be a position of length θ in the game G Q gives rise to: • models Mξ for ξ ≤ θ; and • embeddings jζ,ξ : Mζ → Mξ for ζ < ξ ≤ θ . The definitions below are made with reference to these models and embeddings. branch (M, t, δ ∗ )(C˙ ∗ ) just in case that it is secure in Definition 6F.26. Q is level 0 in G ∗ ∗ branch (M, t, δ )(C˙ ). G branch (M, t, δ ∗ )(C˙ ∗ ). Let Suppose for the next definition that Q is not secure in G η < θ be largest so that Qη is secure in that game. The existence of a largest such η follows from Corollary 6F.15. Let P = Qη, and let P + = Qη + 1. branch (M, t, δ ∗ )(C˙ ∗ ) just in case that: Definition 6F.27. Q is level n + 1 in G branch (M, t, δ ∗ , C˙ ∗ )[P ]; (1) mega-round η in Q is played according to branch (M, t, δ ∗ )(C˙ ∗ ); and (2) P + opens a skip in G branch (Mη+1 , tη+1 , δ + )(C˙ + ). (3) tail(Q, η) is a level n position in G Definition 6F.27 is made by induction on n, simultaneously for all Q and all promising tuples, where the base case is given by Definition 6F.26. Notice that condition (3) in Definition 6F.27 makes sense inductively, since the tuple $Mη+1 , tη+1 , δ + , C˙ + % is promising by Claim 6F.22. branch (M, t, δ ∗ )(C˙ ∗ ) if it is level n for some n < ω. Definition 6F.28. Q is safe in G If Q is safe then one can check that there is a unique n < ω witnessing this. We branch (M, t, δ ∗ )(C˙ ∗ ). refer to this unique n as the level of Q in G branch (M, t, δ ∗ )(C˙ ∗ ). If Q is terminal in Claim 6F.29. Suppose that Q is safe in G branch (M, t, δ ∗ )(C˙ ∗ ) then it is won by player I. G
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Proof. Work by induction on the level of Q in G(M, t, δ ∗ )(C˙ ∗ ), simultaneously for all Q and all promising $M, t, δ ∗ , C˙ ∗ %. The base case is given by Claim 6F.16. We leave the inductive case to the reader, and only note that it uses case 1 in Lemma 6F.24. # branch (M, t, δ ∗ )(C˙ ∗ ) in a way that always Eventually we intend to play for I in G sticks to safe positions. We will formulate a strategy that does this later on. Let us here check that this tactic does not fail at limits. Claim 6F.30. Suppose that θ is a limit ordinal. Suppose that every strict initial segment branch (M, t, δ ∗ )(C˙ ∗ ), but Q itself is not. (In particular Q is not secure.) of Q is safe in G branch (M, t, δ ∗ )(C˙ ∗ ). Let P = Qη Let η < θ be largest so that Qη is secure in G + and let P = Qη + 1. Then: branch (M, t, δ ∗ , C˙ ∗ )[P ]; and (1) mega-round η in Q is played according to branch (M, t, δ ∗ )(C˙ ∗ ). (2) P + opens a skip in G Proof. Qη + 1 is a strict initial segment of Q since θ is a limit. By assumption it must branch (M, t, δ ∗ )(C˙ ∗ ). It cannot be secure, because of the maximality of η. be safe in G So it must be level n + 1 for some n. The conditions in Definition 6F.27, applied on Qη + 1, yield the conclusion of the current claim. # Claim 6F.31. Suppose that θ is a limit ordinal. Suppose that every strict initial segment branch (M, t, δ ∗ )(C˙ ∗ ), but Q itself is not. of Q is safe in G branch (M, t, δ ∗ )(C˙ ∗ ). Let P = Qη, Let η < θ be largest so that Qη is secure in G + + + let P = Qη + 1, and let δ and C˙ be defined as in Section 6F (2). Then every strict branch (Mη+1 , tη+1 , δ + )(C˙ + ), but tail(Q, η) itself initial segment of tail(Q, η) is safe in G is not. Proof. Let θ¯ be larger than η and smaller than or equal to θ. Using the previous claim and working directly with Definition 6F.27 it’s clear that Qθ¯ is level n + 1 in branch (Mη+1 , tη+1 , δ + )(C˙ + ). The branch (M, t, δ ∗ )(C˙ ∗ ) iff tail(Qθ, ¯ η) is level n in G G current claim follows immediately from this. # Corollary 6F.32. Suppose that θ is a limit ordinal. Suppose that every strict initial branch (M, t, δ ∗ )(C˙ ∗ ), but Q itself is not. Then Mθ is illfounded. segment of Q is safe in G Proof. Iterated applications of the previous claim produce sequences $ηk | k < ω%, $δk+ | k < ω%, and $C˙ k+ | k < ω% so that: (i) $ηk | k < ω% is an increasing sequence of ordinals smaller than θ ; (ii) for each k < ω, $Mηk +1 , tηk +1 , δk+ , C˙ k+ % is promising; (iii) for each k < ω, every strict initial segment of tail(Q, ηk ) is safe in the game branch (Mη +1 , tη +1 , δ + )(C˙ + ) but tail(Q, ηk ) itself is not; and G k
k
k
k
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6 Along a single branch
+ (iv) for each k < ω, δk+1 is smaller than jηk +1,ηk+1 +1 (δk+ ).
In obtaining conditions (ii) and (iv) we make use of Claims 6F.22 and 6F.21 respectively. It is the use of Claim 6F.21 that is most important here. Condition (iv), which traces back to this claim, implies that Mθ is illfounded. # Corollary 6F.33. Suppose that Q is a position of limit length. Suppose that every strict branch (M, t, δ ∗ )(C˙ ∗ ). Then either: initial segment of Q is safe in G branch (M, t, δ ∗ )(C˙ ∗ ); or (1) Q itself is also safe in G branch (M, t, δ ∗ )(C˙ ∗ ) and won by player I. (2) Q is a terminal position in G Proof. If Mθ is illfounded then condition (I3) in Section 6A states that Q is terminal branch and won by player I. If Mθ is wellfounded then by the previous corollary Q in G must be safe. # Corollary 6F.33 formalizes the method of dealing with infinite nesting indicated at the end of Section 6F(4). The crucial ingredient leading to the corollary is Claim 6F.21, which states that δ + is smaller than j0,η+1 (δ ∗ ) in situations which open a skip. This same claim was the key to the “induction” discussed in Section 6F (4). 6F (6) The strategy. For each promising $M, t, δ ∗ , C˙ ∗ % and for each position Q which branch (M, t, δ ∗ )(C˙ ∗ ) define a strategy (M, t, δ ∗ , C˙ ∗ )[Q] is non-terminal and safe in G branch (M, t, δ ∗ , C˙ ∗ )[Q] subject to the appropriate condition for player I in the game G below: branch (M, t, δ ∗ )(C˙ ∗ ) then define (M, (T1) If Q is secure in G t, δ ∗ , C˙ ∗ )[Q] to be branch (M, t, δ ∗ , C˙ ∗ )[Q] of Definition 6F.17. equal to the strategy η+1 , tη+1 , δ + , C˙ + )[tail(Q, η)] (T2) Otherwise set (M, t, δ ∗ , C˙ ∗ )[Q] equal to (M branch (M, t, δ ∗ )(C˙ ∗ ), and δ + and C˙ + where η is largest so that Qη is secure in G are defined as in Section 6F(2). The definition of (M, t, δ ∗ , C˙ ∗ )[Q] is made by induction on the level of Q in ∗ ∗ branch (M, t, δ )(C˙ ). The base case of level 0 is handled in condition (T1). The G inductive case is given by condition (T2). To make sense of this inductive condition branch (Mη+1 , tη+1 , δ + )(C˙ + ); that one has to know that tail(Q, η) is non-terminal in G it is safe in this game; and that its level in this game is smaller than the level of Q branch (M, t, δ ∗ )(C˙ ∗ ). The last two properties are immediate from the definitions in G in Section 6F (5). The first property follows from the fact that Q is non-terminal in branch (M, t, δ ∗ )(C˙ ∗ ) using Claim 6F.29 and Lemma 6F.24: If tail(Q, η) were terminal G branch (Mη+1 , tη+1 , δ + )(C˙ + ) then by Claim 6F.29 it would be won by player I in this in G branch (M, t, δ ∗ )(C˙ ∗ ). game. But then by Lemma 6F.24, Q would have to be terminal in G (Q cannot be secure in this game because of the case assumption of condition (T2).)
6F Skips
265
Claim 6F.34. Let $M, t, δ ∗ , C˙ ∗ % be promising. Let P be non-terminal and secure in branch (M, t, δ ∗ )(C˙ ∗ ). Let P + extend P by one mega-round, and suppose that this G branch (M, t, δ ∗ , C˙ ∗ )[P ]. Then P + is either level 0 or mega-round is consistent with ∗ ∗ ˙ level 1 in Gbranch (M, t, δ )(C ). branch (M, t, δ ∗ )(C˙ ∗ ) Proof. Let η = lh(P ). If η is a limit then P + is secure in G by Claim 6F.18 and therefore has level 0 in this game. If η is a successor then by branch (M, t, δ ∗ )(C˙ ∗ ) or else it opens a skip in this Claim 6F.19 P + is either secure in G game. In the former case P + is level 0, and in the latter case it is easy to check directly # from the definitions in Section 6F(5) that P + is level 1. Claim 6F.35. Let $M, t, δ ∗ , C˙ ∗ % be promising. Let Q be non-terminal and safe branch (M, t, δ ∗ )(C˙ ∗ ). Let Q+ extend Q by one mega-round, and suppose that in G this mega-round is consistent with (M, t, δ ∗ , C˙ ∗ )[Q]. Then Q+ is safe in ∗ ∗ branch (M, t, δ )(C˙ ). G branch (M, t, δ ∗ )(C˙ ∗ ). The base case follows Proof. By induction on the level of Q in G from Claim 6F.34. The inductive case is straightforward using the nature of condition (T2) and the definitions in Section 6F(5). # Corollary 6F.36. Let $M, t, δ ∗ , C˙ ∗ % be promising. Then player I has a winning strategy branch (M, t, δ ∗ )(C˙ ∗ ). in G branch (M, t, δ ∗ )(C˙ ∗ ) as follows: a strategy for player I in G Proof. Define , branch (M, t, δ ∗ )(C˙ ∗ ) then let copy • if Q is (non-terminal and) safe in G ∗ ∗ ˙ (M, t, δ , C )[Q] for moves in mega-round lh(Q) of Gbranch (M, t, δ ∗ )(C˙ ∗ ) following Q; and play these moves in your favorite fashion. • otherwise let is just the natural join of the strategies (M, t, δ ∗ , C˙ ∗ )[Q] on safe positions Q. We does on positions which are not safe. do not care what ∗ branch (M, t, δ ∗ )(C˙ ∗ ), and suppose that Q∗ is Let Q be a terminal position in G consistent with . Using Claims 6F.35 and 6F.33 it is easy to prove by induction on θ branch (M, t, δ ∗ )(C˙ ∗ ), or terminal and that for every θ ≤ lh(Q∗ ), Q∗ θ is either safe in G won by player I (or possibly both). The successor case of the induction uses Claim 6F.35, and the limit case uses Corollary 6F.33. (It is the use of Corollary 6F.33 that forces us to include the possibility that Q∗ θ is terminal and won by player I.) Now Claim 6F.29 states that a safe position, if terminal, is won by player I. So taking θ = lh(Q∗ ) above we see that Q∗ itself is in either case won by player I. The argument of the preceding paragraph shows that terminal positions played branch (M, t, δ ∗ )(C˙ ∗ ). So is a winning are won by player I in G according to strategy for I in this game. #
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6 Along a single branch
6G Conclusion The work of the previous section culminates in the following result: Theorem 6G.1. Let M be a 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. Suppose that: (1) M&δ ∗ + 1 is countable in V; and (2) t belongs to an interpretation of Back I (δ, δ ∗ )(C˙ ∗ ), where the pullback is computed in M. branch (M, t, δ ∗ )(C˙ ∗ ). Then player I has a winning strategy in G Proof. This is simply Corollary 6F.36.
#
The theorem should be viewed as formalizing our intuition in Chapter 4 that Back I (δ, δ ∗ )(C˙ ∗ ) names the set of positions from which player I can win to either enter an interpretation of a shift of C˙ ∗ , or reach a I-acceptable obstruction along the way. We proved Theorem 6G.1 through iterated appeals to Lemma 6F.2 which dealt with skipping games. Lemma 6F.2 in turn was proved by cases, depending on the nature of δ and δ ∗ . The two main cases were the one corresponding to relative limits in W , handled in Section 6C, and the one corresponding to Woodin limits of Woodin cardinals, handled in Section 6D. For the former we used the methods of Chapters 1 and 3 on the names defined in Chapter 4. For the latter the driving force was our adherence to clear annotated positions, which accumulate to form a generic for Woodin’s extender algebra. Corollary 6G.2. Let M be a model of ZFC∗ , let δ be a Woodin cardinal of M, and let C˙ be a δ-name in M. Suppose that: (1) M&δ + 1 is countable in V; and ˙ holds in M. (2) ϕini (δ, C) branch (M, ∅, δ)(C). ˙ Then player I has a winning strategy in G The formula ϕini in Corollary 6G.2 is the one given by Definition 5G.2. ∅ in branch (M, ∅, δ)(C) ˙ is the empty annotated M-position of relative domain 0. G ˙ δ0 being the first Woodin cardinal Proof of Corollary 6G.2. Let C˙ 0 = Back I (δ0 , δ)(C), ˙ holds it can be seen that in M. Using Lemma 6B.2 and the assumption that ϕini (δ, C) 0 be such a strategy. ˙ player I has a winning strategy in Gbranch (M, ∅, δ0 )(C0 ). Let The corollary can be proved by composing 0 with an application of Theorem 6G.1. 0 leads into an interpretation of a shift of C˙ 0 , and from there Theorem 6G.1 provides ˙ We leave the exact a strategy for I which leads into an interpretation of a shift of C. details to the reader. #
6G Conclusion
267
Remark 6G.3. If δ is a Woodin limit of Woodin cardinals in M then the assumption of condition (1) in Corollary 6G.2 can be weakened to the assumption that δ is countable in V. Similarly in Theorem 6G.1, if δ ∗ is a Woodin limit of Woodin cardinals in M then the assumption of condition (1) can be weakened to the assumption that δ ∗ is countable in V. This can be seen by noticing that Lemma 6D.11 assumes that θ, not M&θ + 1, is countable in V, and tracing this fact through subsequent lemmas.
Chapter 7
Games which reach local cardinals
We work here to assemble the results of Chapters 4 through 6 into a powerful theorem on determinacy. The strength of the theorem stems from its reference to Woodin’s extender algebra of Section 4B. It makes runs of long games generic, not over a collapse algebra, but over Woodin’s extender algebra. Genericity over Woodin’s extender algebra is reached through a construction which involves a non-linear composition of length ω iteration trees. Along each branch of this branch , and it is a non-linear composition the construction develops a run of the game G strategy given by the results of the previous chapter that perpetuates the construction. (This provides some late justification for the title of the previous chapter. The work of that chapter handles each single branch of the construction here.) It is too early to go into details, but let us indicate that most of the work which ties the current construction to the previous chapter is done in Sections 7B and 7C. The main theorem itself is described in Section 7A, and proved in Sections 7D and 7E. We end the chapter with applications of the main theorem. First we prove determinacy for games which run up to the first γ which is not collapsed to ω by a function constructible from the play. Then we consider similar games with “constructible from” replaced by “ in” where is a pointclass. Finally we present an application due to Woodin, demonstrating the consistency of the statement that all ordinal definable games of length ω1 on natural numbers are determined.
7A Shifted payoff Let M be a transitive model of ZFC∗ , let be an iteration strategy for M, and let θ be a Woodin limit of Woodin cardinals in M. We work with these fixed objects for the rest of the section. Our discussion refers to Woodin’s algebra Wθ , defined in Section 4B. Let A˙ ∈ M be a Wθ -name for a set of sequences of reals of length θ. (Of course we local (M, , θ, A) ˙ mean that the sequences have length θ , not the reals.) Define the game G to be played as follows: Players I and II alternate natural numbers as usual creating reals zξ for ξ < ω1V . If ever an α < ω1V is reached so that condition (P) below holds (in V) then the game ends and player I wins. If no such α is reached prior to ω1V then player I loses. (P) There exists a countable length iteration tree U on M, leading to a final model M ∗ and to an iteration embedding j ∗ : M → M ∗ , and there exists some H , so that: (1) U is consistent with the iteration strategy ;
7A Shifted payoff
269
(2) H is j ∗ (Wθ )-generic/M ∗ ; and ˙ ]. (3) the sequence $zξ | ξ < α% belongs to j ∗ (A)[H local (M, , θ, A) ˙ which is won by Claim 7A.1. Let $zξ | ξ < α% be a position in G ∗ player I. Let U and H witness this. Then α is precisely equal to j (θ ), where j ∗ is the iteration embedding given by U. Proof. This follows from clause (3) in condition (P) using the fact that A˙ names a set of sequences of length θ. # local (M, , θ, A) ˙ which is won by Claim 7A.2. Let $zξ | ξ < α% be a position in G player I. Let U and H witness this. Let M ∗ be the final model of U. Then: (1) $zξ | ξ < α% belongs to M ∗ [H ]; and (2) α is a cardinal in M ∗ [H ]. Proof. The fact that $zξ | ξ < α% belongs to M ∗ [H ] is a direct consequence of clause (3) in condition (P). Let us check that α is a cardinal of M ∗ [H ]. Lemma 4B.19 states that Wθ has the θ chain condition in M. So j ∗ (Wθ ) has the ∗ j (θ ) chain condition in M ∗ . It follows that j ∗ (θ ) is a cardinal in M ∗ [H ]. By the # previous claim α is precisely equal to j ∗ (θ). So α is a cardinal in M ∗ [H ]. local . It provides a lower bound Claim 7A.2 is the source of strength of the game G on the length of any run of Glocal (M, . . . ) which is won by I. Specifically the claim says that such a run must at the very least reach ω1 in the sense of M ∗ [H ], where M ∗ [H ] is some generic extension of an iterate of M and contains the run in question. Let B˙ ∈ M be another Wθ -name for a set of sequences of reals of length θ . Define local (M, , θ, B) ˙ to be played as follows: Players I and II again alternate the game H natural numbers as usual to create reals zξ for ξ < ω1V . If ever an α < ω1V is reached so that condition (Q) below holds (in V) then the game ends and player II wins. If no such α is reached prior to ω1V then player II loses. (Q) There exists a countable length iteration tree U on M, leading to a final model M ∗ and to an iteration embedding j ∗ : M → M ∗ , and there exists some H , so that: (1) U is consistent with the iteration strategy ; (2) H is j ∗ (Wθ )-generic/M ∗ ; and ˙ ]. (3) the sequence $zξ | ξ < α% belongs to j ∗ (B)[H local . Note particularly that in the local precisely mirrors the game G The game H local the payoff is phrased for player II, rather than I. Claims similar to 7A.1 case of H local , but we do not phrase them explicitly. and 7A.2 hold also for H
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7 Games which reach local cardinals
The main result of this chapter is the following theorem. It is similar in format to the local and earlier Theorems 2A.2 and 3E.1. But the use of Woodin’s extender algebra in G Hlocal makes the current theorem apply to much longer games than those considered in Chapters 2 and 3, as we shall see in Section 7F. Theorem (7E.1). Suppose that θ is countable in V. Then at least one of the following three cases holds: local (M, , θ, A); ˙ (1) player I has a winning strategy in G local (M, , θ, B); ˙ or (2) player II has a winning strategy in H (3) there exists some G ∈ V which is Wθ -generic/M, and there exists some sequence ˙ nor of reals $zξ | ξ < θ% ∈ M[G], so that $zξ | ξ < θ% belongs to neither A[G] ˙ B[G]. Moreover M can distinguish this. More precisely, there are formulae ϕ and ψ so that: if ˙ then case (1) holds; if M |= ψ[θ, B] ˙ then case (2) holds; and otherwise M |= ϕ[θ, A] case (3) holds. Remark 7A.3. Both conditions (P) and (Q) above require the length of the iteration tree U to be countable. So only countable length iteration trees come up through the statement of Theorem 7E.1. Nonetheless for the proof of the theorem it is essential that acts not only on countable length iteration trees but also on iteration trees of length ω1V . The reason for this is explained in Remark 7D.6.
7B Layout Let α be an ordinal. By a tree order on α we mean an order U so that: (1) U is a sub-order of <α; (2) for each ordinal η < α, the set {ζ | ζ U η} is linearly ordered by U ; (3) for each ξ so that ξ + 1 < α, the ordinal ξ + 1 is a successor in U ; and (4) for each limit ordinal γ < α, the set {ζ | ζ U γ } is cofinal in γ . A tree order is precisely the kind of order given by an iteration tree. Condition (2) allows talking about successors in U . An ordinal η < α is a successor in U just in case that {ζ | ζ U η} has a maximal element. (Note that maximal in U is the same as maximal in <, by condition (1).) The following claim relates successors in U to successors in <. Claim 7B.1. Let U be a tree order on α. Let η be smaller than α. Then η is a successor in U iff it is a successor ordinal.
7B Layout
271
Proof. Suppose first that η is a successor in U . {ζ | ζ U 0} is empty by condition (1), so η cannot be equal to 0. η cannot be a limit ordinal by condition (4). So it must be a successor ordinal. Suppose next that η is a successor ordinal, ξ + 1 say. Then by condition (3) it is also a successor in U . # Remark 7B.2. We use UEQ to denote the non-strict order associated to U . More precisely we set ζ UEQ ξ just in case that (ζ U ξ ) ∨ (ζ = ξ ). Definition 7B.3. A branch through U is a subset of α which is linearly ordered by U and downward closed under U . A branch is cofinal if it cofinal in α. Let M be a model of ZFC∗ . A tree of trees (henceforth referred to as a tot) of length α on M is a structure U which consists of the following objects, and which satisfies conditions (S), (U), and (L) below: • a tree order U on α; • models Mξ for ξ < α; • embeddings jζ,ξ : Mζ → Mξ for ζ U ξ , commuting in the natural way; • for each ξ < α, a length ω iteration tree Tξ on Mξ ; • for each ξ < α, a cofinal branch bξ through Tξ ; and • for some (possibly not all) ξ so that ξ + 1 < α, an extender Eξ in Qξ . Qξ in the last item is the direct limit along the branch bξ of Tξ . We use kξ : Mξ → Qξ to denote the direct limit embedding. Note that Eξ need not be defined for all ξ so that ξ + 1 < α. Given ξ so that ξ + 1 < α we use the phrase Eξ = “undefined” to indicate that Eξ is not defined. We allow padding in iteration trees. In fact Tξ may be a tree which consists entirely of padding. In this case Qξ is equal to Mξ and kξ is equal to the identity. We demand the following conditions from the objects of a tot: (S) If Eξ = “undefined” then Mξ +1 = Qξ , the U -predecessor of ξ + 1 is equal to ξ , and jξ,ξ +1 is equal to kξ . (U) If Eξ is defined then Mξ +1 = Ult(Mζ , Eξ ) where ζ is the U -predecessor of ξ +1, and jζ,ξ +1 is equal to the ultrapower embedding. (L) Mγ for limit γ is the direct limit of the models $Mξ | ξ U γ % under the embeddings $jζ,ξ | ζ U ξ U γ %. jξ,γ : Mξ → Mγ for ξ U γ are the direct limit embeddings.
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7 Games which reach local cardinals
The labels (S), (U), and (L) stand for “simple,” “ultrapower,” and “limit” respectively. The situations of conditions (S) and (U) are illustrated in Diagrams 7.1 and 7.2. (Note that ζ may be equal to ξ in the case of condition (U). Diagram 7.2 is not meant to imply that it is always strictly smaller.) Condition (L) is the natural limit condition for any kind of iteration. Note that the set of ξ so that ξ U γ is linearly ordered by U , so the condition makes sense. kkk / / Mξ S kk Sk SSSS k / Qξ = Mξ +1 ξ Tξ
Diagram 7.1. When Eξ = “undefined.”
jζ,ξ +1
/
Mζ
kkk / Sk _ _ _ _ _ _ _ Mξ k SkSSS S kξ
Eξ Qξ
'
Mξ +1
/
Tξ
Diagram 7.2. When Eξ is defined.
A tot U of length α on M may be regarded as one big iteration tree on M, consisting of all the models Mξ , all the models along the trees Tξ , and all the models Qξ . We use merge(U) to denote this iteration tree, which we call the merge of U. If α is a successor then merge(U) is an iteration tree of length ω · α + 1 on M, leading to the model Qα−1 . If α is a limit then merge(U) is an iteration tree of length ω · α on M, with the models Mξ for ξ < α forming its backbone. Definition 7B.4. Let be an iteration strategy for M. A tot U is consistent with just in case that the iteration tree merge(U) is consistent with . We say that a tot U of length α on M is regular just in case that it satisfies conditions (R1)–(R4) below: (R1) For each ξ < α, Mξ has (in order type) at least ξ + 1 Woodin cardinals. Let δξ† be the ξ -th Woodin cardinal of Mξ . By this we mean that δξ† is the first Woodin cardinal of Mξ so that there are (in order type) ξ Woodin cardinals of Mξ below it. The existence of such δξ† is given by condition (R1). Let δξ +1 = kξ (δξ† ). (R2) For each η < α, Tη only uses extenders with critical points greater than sup{δξ +1 | ξ < η}. (R3) All extenders used in Tη are countable in V.
7B Layout
273
(R4) For each η so that η + 1 < α, Eη (if defined) is δη+1 + 1-strong in Qη . We think of conditions (R2) and (R3) as being satisfied trivially if Tη is the tree which consists entirely of “padding.” For the rest of the section let us work with a regular tot U of length α on M. The claims below establish facts about this regular tot which will later allow associating to it a sequence of annotated positions. Claim 7B.5. Let η be smaller than α. Then: (1) for each ξ < η, δξ +1 is precisely the ξ -th Woodin cardinal of Mη ; (2) Mη and all subsequent models in U agree past sup{δξ +1 | ξ < η}; (3) δη+1 is precisely the η-th Woodin cardinal of Qη ; (4) Qη and all subsequent models in U agree past δη+1 . Proof. Prove all conditions simultaneously by induction on η. The main driving force in the proof is the agreement between models of U given by conditions (R2) and (R4). We leave the exact details to the reader. # Conditions (3) and (4) in the last claim can informally be viewed as saying that U leaves a “trail of Woodin cardinals” in its progress. For each limit ordinal γ < α let λγ = sup{δξ +1 | ξ < γ }. λγ is a limit of Woodin cardinals in Mγ . We say that γ is a phantom limit (in U) if λγ is itself Woodin in Mγ . Otherwise we say that γ is a standard limit (in U). This distinction corresponds to the branch in Section 6A. division of limit cases in the definition of G Claim 7B.6. Let η be an ordinal smaller than α. Then the sequence $λγ | γ is a standard limit and γ ≤ η% precisely enumerates, in increasing order, the ordinals smaller than δη+1 which are limits of Woodin cardinals in Qη+1 but not themselves Woodin. Proof. From conditions (1), (2), and (3) in the previous claim it follows that $δξ +1 | ξ < η% precisely enumerates the Woodin cardinals of Qη below δη+1 . The current claim follows from this and from the definition of a standard limit. # Let K U denote the set {η < α | η is a successor, or zero, or a standard limit}. This is simply the set of ordinals smaller than α, except those which are phantom limits. Claim 7B.7. Let η be an ordinal smaller than α. Let Wη denote the class W of Section 4A, computed in Mη . Then the sequence $δξ +1 | ξ ∈ η ∩ K U % precisely enumerates, in increasing order, the set δη† ∩ Wη . Proof. Immediate from condition (1) in Claim 7B.5 and from the definition of K U . Let us just note that for each γ < η, δγ +1 is a Woodin limit of Woodin cardinals in Mη just # in case that γ is a phantom limit (and in fact δγ +1 is equal to λγ in this case).
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Claim 7B.8. Let γ < α be a limit ordinal. Then sup{τ + 1 | τ ∈ δγ† ∩ Wγ } is precisely equal to λγ . Proof. The set γ ∩ K U is cofinal in γ , since it contains all successors below γ . Using this observation and Claim 7B.7 it follows that sup{τ + 1 | τ ∈ δγ† ∩ Wγ } is equal to # sup{δξ +1 + 1 | ξ < γ }, which in turn is easily seen to be equal to λγ . Define λ0 = 0. For each successor ordinal ξ + 1 < α define λξ +1 by: if ξ is a phantom limit; and δξ +1 λξ +1 = δξ +1 + 1 otherwise. Claim 7B.9. Let ξ + 1 be a successor ordinal smaller than α. Then sup{τ + 1 | τ ∈ δξ†+1 ∩ Wξ +1 } is precisely equal to λξ +1 . Proof. From condition (1) in Claim 7B.5 (for η = ξ + 1) and from the definition of δξ†+1 it follows that δξ +1 is the largest Woodin cardinal of Mξ +1 below δξ†+1 . If ξ is not a phantom limit then δξ +1 belongs to Wξ +1 by Claim 7B.7, and so sup{τ + 1 | τ ∈ δξ†+1 ∩ Wξ +1 } = δξ +1 + 1. If ξ is a phantom limit then δξ +1 does not belong to Wξ +1 , but arbitrarily large Woodin cardinals of Mξ +1 below it do, so sup{τ + 1 | τ ∈ # δξ†+1 ∩ Wξ +1 } = δξ +1 . Claim 7B.10. Let η be smaller than α. Then the sequence $δξ +1 | ξ ∈ η ∩ K U % precisely enumerates, in increasing order, the set λη ∩ Wη . Proof. This follows from Claim 7B.7. Simply note that there are no elements of Wη in # the interval [λη , δη† ). For an ordinal η < α and for δ ∈ Wη let eη (δ) = sup{τ + 1 | τ ∈ δ ∩ Wη }. This definition of eη (. . . ) is simply the shift of the definition of e(. . . ) in Section 4A to Mη . Claim 7B.11. Let η belong to K U . Then eη (δη† ) is precisely equal to λη . Proof. The limit case follows from Claim 7B.8, the successor case follows from Claim 7B.9, and the case of η = 0 can be verified directly. # Claim 7B.12. Let ξ belong to K U and let η < α be greater than ξ . Then eη (δξ +1 ) is equal to λξ . Proof. Recall that δξ +1 = kξ (δξ† ). The previous claim tells us that eξ (δξ† ) = λξ . kξ has critical point greater than λξ by condition (R2). So shifting the fact that eξ (δξ† ) = λξ via kξ we see that e(δξ +1 ), in the sense of Qξ , is equal to λξ . The current claim follows # from this and the fact that Qξ and Mη agree past δξ +1 . Claim 7B.13. Let η be an ordinal smaller than α. Let Lη denote the class L of Section 4A, computed in Mη . Then the sequence $λξ | ξ ∈ η ∩ K U % precisely enumerates, in increasing order, the set λη ∩ Lη .
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Proof. The set λη ∩ Lη is precisely equal to {eη (τ ) | τ ∈ λη ∩ Wη }, which by Claim 7B.10 is equal to {eη (δξ +1 ) | ξ ∈ η ∩ K U }. The current claim follows from this and Claim 7B.12. # Equipped with the above claims and definitions we can comfortably associate annotated positions to the tot U. By a U-sequence let us mean a sequence $wξ , yξ | ξ ∈ K U % so that: (C1) for each ξ ∈ K U , wξ is a witness for λξ over Mξ ; (C2) if ξ ∈ K U is a successor ordinal or zero then yξ is a real; and (C3) if ξ ∈ K U is a standard limit then yξ belongs to (Qξ &λξ + 1)ω . Let us for the rest of the section work with a fixed sequence of this kind. Definition 7B.14. For each η < α define tη to be the function given by the following clauses: (1) the domain of tη is equal to {δξ +1 | ξ ∈ η ∩ K U } ∪ {λξ | ξ ∈ η ∩ K U }; (2) for each ξ ∈ η ∩ K U , tη (λξ ) is equal to wξ ; and (3) for each ξ ∈ η ∩ K U , tη (δξ +1 ) is equal to yξ . Claim 7B.15. tη is an annotated position of relative domain λη over Mη . Proof. This follows easily from the definition of tη , conditions (C1)–(C3), and claims 7B.10, 7B.12, and 7B.13 above. We leave the precise details to the reader. Let us only note that by Claims 7B.10 and 7B.13 the domain of tη is equal to λη ∩ (Wη ∪ Lη ), as required. # Claim 7B.16. z(tη ) is precisely equal to $y−1+ξ +1 | ξ + 1 < 1 + η%. Proof. Recall that z(tη ) denotes the real part of tη , given by Definition 4A.21. It is equal to the sequence $tη (δ) | δ ∈ dom(tη ) and δ is a relative successor%. Using Claim 7B.10 it is easy to check that the relative successors in the domain of tη are precisely the ordinals δζ +1 where ζ is smaller than η and is either zero or a successor. Since tη (δζ +1 ) is by definition equal to yζ for such ζ we see that: z(tη ) = $yζ | ζ is smaller than η and is either zero or a successor%. The sequence on the right-hand-side can be presented more uniformly as the sequence # $y−1+ξ +1 | ξ + 1 < 1 + η%. We continue to work with a regular tot U of length α on M, and with a U-sequence $wξ , yξ | ξ ∈ K U %. The definitions below are made with reference to these objects, and also with reference to the annotated positions of Definition 7B.14. Fix an ordinal η < α. Let r denote the set {ζ | ζ U η}. This is simply the branch of U leading to Mη . r is linearly ordered by U , and ζ U ξ ⇐⇒ ζ < ξ for ζ, ξ ∈ r.
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(Both properties follow from the definition of a tree order at the start of this section.) Let β be the order type of r. Let f : β + 1 → r ∪ {η} be the unique order preserving isomorphism of β ∪ {β} and r ∪ {η}. Then ζ < ξ ⇐⇒ f (ζ ) U f (ξ ) for ζ, ξ ≤ β. For each ξ < β, f (ξ + 1) is a successor ordinal. (This follows from Claim 7B.1.) Let Eξ∗ equal Ef (ξ +1)−1 . By this we also mean that Eξ∗ =“undefined” if Ef (ξ +1)−1 = “undefined.” Eξ∗ is thus the object (extender, or “undefined”) used to determine Mf (ξ +1) and the embedding jf (ξ ),f (ξ +1) : Mf (ξ ) → Mf (ξ +1) via conditions (S) and (U) above. Definition 7B.17. Let Pη be the sequence $Tf (ξ ) , bf (ξ ) , Eξ∗ , tf (ξ +1) | ξ < β%, where β, f , and Eξ∗ for ξ < β are the objects defined above. We refer to Pη as the strand of U and $wξ , yξ | ξ ∈ K U % leading to η. Definition 7B.17 breaks U and $wξ , yξ | ξ ∈ K U % into strands, taken along the branches of U . To each η < lh(U) the definition associates a sequence Pη . Pη is generated through the part of U and $wξ , yξ | ξ ∈ K U % which corresponds to the branch of U leading to η. Note that this part has the format of a position of length β in branch . The notation for positions in that game is explained in Remark 6A.2, the game G and it is easy to check that Pη fits the same format. Indeed, conditions (R2)–(R4) and branch , as (C1)–(C3) above are almost enough to literally make Pη a legal position in G they fit exactly with the rules (S1)–(S4) and (L1)–(L4) of Section 6A. (We will see this precisely in Claims 7C.3 and 7C.4 below.) But two elements are still missing. The first is a demand of triviality for the part of U which corresponds to phantom limits, in line with the triviality of the phantom limit case in Section 6A. The second element missing is a connection between the moves made in Pη and the rules for leaps in Section 6A. This connection will be made in Section 7C, with early ends used in Section 7C (1) and leaps handled in Section 7C(2). Claim 7B.18. lh(Pη ) is: a successor if η is a successor; a limit if η is a limit; and zero if η = 0. Proof. The length of Pη is equal to the order type of r = {ζ | ζ U η}. By Claim 7B.1 it is a successor iff η is a successor. The current claim follows from this, the fact that {ζ | ζ U 0} is empty, and the fact that {ζ | ζ U γ } is not empty (in fact it is cofinal in γ ) for limit γ . # 7B (1) Extensions. Let U be a regular tot of length α. By an extension of U to a tot of length α + 0.2 we mean a structure U+ which consists of the following objects: • a tree order U + on α + 1, extending U ; • the models, iteration trees, extenders, and embeddings of U; • (if α is a successor) an additional extender Eα−1 in Qα−1 , or the additional assignment Eα−1 =“undefined”; • an additional model Mα , determined subject to the appropriate condition (S), (U), or (L) in the definition of a tot; and
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• additional embeddings jζ,α : Mζ → Mα for ζ U α, again determined subject to the appropriate condition (S), (U), or (L) in the definition of a tot. If α is a successor and Eα−1 =“undefined” then the U + -predecessor of α must be equal to α − 1, in line with condition (S) in the definition of a tot. U+ is regular if U is regular and if in addition, when α is a successor, Eα−1 (if defined) is δα + 1-strong in Qα−1 . The last clause simply places on the additional extender Eα−1 the same requirements that are placed on prior extenders through condition (R4) in the definition of regularity for a tot of length α. Remark 7B.19. If α is a limit then an extension of U to a tot of length α + 0.2 is fully determined by the set {ζ | ζ U + α}. This set is a cofinal branch through U , and any such branch gives rise to an extension of U. If α is a successor then an extension of U to a tot of length α +0.2 is fully determined by the value of Eα−1 and by the U + -predecessor of α. (If Eα−1 =“undefined” then even this information already involves a redundancy, since the U + -predecessor of α must equal to α − 1.) An extension of U to a regular tot of length α + 0.2 allows extending many of the claims and definitions made above for η < α, and applying them also to η = α. We leave it to the reader to check precisely which claims and definitions extend in this way. But let us note that Definitions 7B.14 and 7B.17 are among them. So an extension of U to a regular tot U+ of length α + 0.2 allows talking about tα and Pα . tα is in fact independent of the particular extension used. But Pα depends on U+ . We say that Pα is defined with reference to U+ to emphasize this dependence.
7C Basic step Let M be a transitive model of ZFC∗ . Let be an iteration strategy for M. Let η be an ordinal and let U be a regular tot of length η + 1 on M, consistent with the iteration strategy . Let $w, y% = $wξ , yξ | ξ ∈ K U % be a U-sequence. We work with these fixed objects, referring freely the terminology, notation, and definitions of Section 7B. Recall most importantly that Definition 7B.17 breaks U and $wξ , yξ | ξ ∈ K U % into strands along the branches of U . Each of these strands has the format of a position branch . We work here under the assumption that each of these strands is in fact a in G branch , and played according to a fixed strategy for I. We show how legal position in G to extend U so as to obtain the same properties for the additional strand generated by the extension. Let us be more precise. Let θ be a Woodin limit of Woodin cardinals in M, and suppose that θ is countable in V. Let Y˙ be a θ -name in M. Suppose that: branch (M, ∅, θ)(Y˙ ). (A1) For each η¯ ≤ η, Pη¯ is a legal position in G
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We work under this assumption for the rest of the section. The assumption states that branch . We will each of the strands of U and $wξ , yξ | ξ ∈ K U % is a legal position in G add more assumptions later, to the effect that each strand is played according to a fixed strategy for I. Claim 7C.1. For each η¯ ≤ η: (1) the outcome of Pη¯ is precisely equal to $Mη¯ , j0,η¯ , tη¯ %; and ¯ (2) the history of Pη¯ is precisely equal to $Mξ , jζ,ξ , tξ | ζ U ξ UEQ η%. We refer the reader to Remark 6A.3 for the definitions of “outcome” and “history.” Proof. This is easy to prove directly from the definition of Pη¯ , using the similarities between: (a) the definitions in Section 6A of the models and embeddings created branch ; and (b) the way conditions (S), (U), and (L) in Section 7B through a position in G ¯ # determine the models and embeddings along the branch {ξ | ξ UEQ η}. branch (M, ∅, θ)(Y˙ ). Let β¯ = lh(Pη¯ ). Claim 7C.2. Suppose that Pη¯ is non-terminal in G ˙ ¯ Then mega-round β of Gbranch (M, ∅, θ)(Y ) following Pη¯ is played subject to the rules of: (1) the phantom limit case in Section 6A if η¯ is a phantom limit in U; (2) the standard limit case in Section 6A if η¯ is a standard limit in U; and (3) the successor case in Section 6A if η¯ is zero or a successor. Proof. By Claim 7B.18, β¯ is a limit iff η¯ is a limit. Condition (3) follows immediately from this. Conditions (1) and (2) follow using the fact that the model and annotated position in the outcome of Pη¯ are precisely Mη¯ and tη¯ , and using the similarities between: (a) the distinction between the phantom limit case and the standard limit case in the branch in Section 6A; and (b) the distinction between phantom limits and Definition of G standard limits in Section 7B. # Recall that the phantom limit case in Section 6A contains no moves at all. So the situation of phantom limits in U is rather trivial. We neglect it in the discussion below. Claim 7C.3. Let η¯ ≤ η be a successor or zero. Let β¯ = lh(Pη¯ ). Suppose that Pη¯ branch (M, ∅, θ)(Y˙ ). Then the objects wη¯ , yη¯ , Tη¯ , and bη¯ (given is non-terminal in G branch (M, ∅, θ)(Y˙ ) by the fixed U, w, and y) are legal moves in mega-round β¯ of G following Pη¯ . Proof. We have to show that wη¯ , yη¯ , Tη¯ , and bη¯ satisfy the requirements of rules (S1)–(S4) in Section 6A, for mega-round β¯ following Pη¯ . The starting point for this mega-round is the outcome of Pη¯ , which we know is equal to $Mη¯ , j0,η¯ , tη¯ %. So the requirements of rules (S1)–(S4) translate in our context to the following conditions: (1) wη¯ is a witness for rdm(tη¯ );
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(2) yη¯ is a real; (3) Tη¯ is a length ω iteration tree on Mη¯ , with critical points above rdm(tη¯ ) and using only extenders which are countable in V; and (4) bη¯ is a cofinal branch through Tη¯ . Checking that these conditions hold true is a simple matter of matching them with the properties of U, w, and y listed in Section 7B: condition (C1) in that section implies that wη¯ is a witness for rdm(tη¯ ); condition (C2) states that yη¯ is a real; condition (R2) implies that Tη¯ uses only extenders with critical points above rdm(tη¯ ); and condition (R3) states # that Tη¯ uses only extenders which are countable in V. Claim 7C.4. Let η¯ ≤ η be a standard limit in U. Let β¯ = lh(Pη¯ ). Suppose that Pη¯ branch (M, ∅, θ)(Y˙ ). Then the objects wη¯ , Tη¯ , bη¯ , and yη¯ (given is non-terminal in G by the fixed U, w, and y) are legal moves for rules (L1)–(L4) in mega-round β¯ of branch (M, ∅, θ )(Y˙ ) following Pη¯ . G Proof. This is similar to the previous claim. Let us only note that condition (C3) in Section 7B matches the requirement of rule (L4) in Section 6A. # branch (M, ∅, θ)(Y˙ ). Fix Suppose now that player I has a winning strategy in G such a strategy, branch , for the rest of the section. Given a non-terminal position P branch (M, ∅, θ)(Y˙ ) let branch [P ] denote the restriction of branch to mega-round in G branch (M, ∅, θ)(Y˙ ) following P . β = lh(P ) in G branch we add the following conditions to the standing assumptions Equipped with of the section: branch (M, ∅, θ)(Y˙ ) and played according (A2) For each η¯ ≤ η, Pη¯ is non-terminal in G to branch . (A3) Let η¯ ≤ η be a successor or zero. Then wη¯ , yη¯ , Tη¯ , and bη¯ are consistent with branch [Pη¯ ]. (A4) Let η¯ ≤ η be a standard limit in U. Then wη¯ , Tη¯ , bη¯ , and yη¯ are consistent with branch [Pη¯ ]. (A5) η itself is not a phantom limit in U. Assumption (A5) is simply part of the neglect of the trivial phantom limits. Assumptions (A2)–(A4) are more serious. They say that each strand of U and $wξ , yξ | ξ ∈ K U % branch (M, ∅, θ)(Y˙ ) which is played according to I’s winning strategy is a position of G branch , and moreover the moves at the endpoint of the strand are also consistent with branch . We now work to extend U to a tot U+ of length η + 1.2, in a way that secures legality branch also for the resulting new strand, namely the strand leading and consistency with to η + 1. We do it in such a way that the new strand is not terminal through any of the “snags” in Section 6A. Since θ is a Woodin limit of Woodin cardinals, a position
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branch (M, ∅, θ)(Y˙ ) which is terminal but not through a snag can only occur after a of G phantom limit. η is not a phantom limit, so altogether the extension we aim for is such branch , and non-terminal. In other words that the new strand is legal, consistent with + we aim to construct an extension U of U which secures condition (A2) for η + 1. The final model in U is the model Qη . Working over that model let tη† = tη −−, wη , yη . tη† is an annotated position of relative domain kη (δη† ) + 1 over Qη . Claim 7C.5. Every strict initial segment of tη† is Qη -clear. Proof. tη is the largest strict initial segment of tη† . So it suffices to show that tη is Qη -clear. branch (M, ∅, θ)(Y˙ ). By Claim 7C.1, By assumption (A1), Pη is a legal position in G the outcome of Pη is equal to $Mη , j0,η , tη %. Using condition (1) in Remark 6A.4 it follows that tη is Mη -clear. Using now the agreement between Mη and Qη implied by # condition (R2) in Section 7B it follows that tη is Qη -clear. Claim 7C.6. tη† is either obstruction free over Qη , or else it is I-acceptably obstructed over Qη . Proof. The claim hinges on the fact that annotated positions which are obstructed but branch . Let us be more not I-acceptably obstructed always result in a loss for I in G precise. Suppose first that η is a successor or zero. Assumptions (A2) and (A3) imply that branch (M, ∅, θ)(Y˙ ) which consists of Pη followed by the moves wη , the position in G branch . So this position cannot be yη , Tη , and bη , is according to I’s winning strategy lost by I. In particular it cannot be lost by I through condition (I2) in Section 6A. The claim follows from this. The proof when η is a standard limit is similar, using assumptions (A2) and (A4) above, and condition (I6) in Section 6A. # Remember that our goal is to extend U to a regular tot U+ of length η + 1.2, in a way branch (M, ∅, θ)(Y˙ ), non-terminal, and consistent that makes the strand Pη+1 legal in G with branch . We divide the construction into two cases, depending on whether tη† is obstruction free over Qη , or I-acceptably obstructed. By the last claim these two cases exhaust all possibilities. 7C (1) Obstruction free. Suppose first that tη† is obstruction free over Qη . Let U+ be the extension of U, to a regular tot of length η + 1.2, determined by the assignment Eη =“undefined.” This assignment determines the extension completely, see Remark 7B.19. The extension allows talking about tη+1 and Pη+1 . Lemma 7C.7 (assuming that tη† is obstruction free over Qη ). Let Pη+1 be defined with reference to the extension U+ of U determined by the assignment Eη =“undefined.” branch (M, ∅, θ)(Y˙ ), non-terminal, and played according to Then Pη+1 is legal in G branch .
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Proof. Suppose first that η is a successor or zero. Let β = lh(Pη ). Using the fact that η is the U + -predecessor of η + 1 it follows, directly from Definition 7B.17, that Pη+1 = Pη −−, Tη , bη , Eη , tη+1 . Eη is equal to “undefined” and tη+1 is simply equal to tη −−, wη , yη . So Pη+1 is simply the position given by Pη followed by the moves wη , yη , Tη , and bη for mega-round β. Assumption (A1) and Claim 7C.3 imply that Pη+1 is a branch (M, ∅, θ)(Y˙ ). Assumptions (A2) and (A3) for η¯ = η combine to legal position in G branch (M, ∅, θ)(Y˙ ) branch . Pη+1 cannot be terminal in G state that Pη+1 is according to through the payoff condition (P1) in Section 6A. This is because θ is a Woodin limit of Woodin cardinals in M and δη+1 is a relative successor in Mη+1 , so δη+1 couldn’t possibly be equal to j0,η+1 (θ). Pη+1 cannot be terminal through condition (I2) in Section 6A since tη+1 = tη† is not obstructed over Mη+1 = Qη . Finally Pη+1 cannot be terminal through condition (I1) in Section 6A since Qη , being a model on a tot consistent with the iteration strategy , is wellfounded. The proof when η is a standard limit is similar. Pη+1 is the position given by Pη followed by the moves wη , Tη , bη , and yη for rules (L1)–(L4) in Section 6A, followed by an early end rather than a leap. (Formally this early end is indicated by the fact branch (M, ∅, θ)(Y˙ ) that Eη =“undefined.”) The fact that Pη+1 is a legal position in G and played according to branch follows as before, only using Claim 7C.4 instead of Claim 7C.3, and assumption (A4) instead of assumption (A3). Pη+1 cannot be terminal branch (M, ∅, θ)(Y˙ ) through the payoff condition (P1) in Section 6A. This is because in G θ is a Woodin limit of Woodin cardinals in M and δη+1 is a relative limit in Mη+1 , so δη+1 couldn’t possibly be equal to j0,η+1 (θ). Pη+1 cannot be terminal through condition (I6) in Section 6A since tη+1 = tη† is not obstructed over Mη+1 = Qη , and it cannot be terminal through condition (I5) since Qη is wellfounded. # 7C (2) I-acceptably obstructed. Suppose now that tη† is I-acceptably obstructed over Qη . Let $E, σ % be a I-acceptable obstruction for tη† over Qη . Following the notation in Section 7B let δη+1 = kη (δη† ). Using the fact that $E, σ % is an obstruction for tη† , and copying from the conditions in Definition 4B.26, we see that: (i) E is δη+1 + 1-strong in Qη ; and (ii) crit(E) is smaller than δη+1 , and is not Woodin in Qη . It follows from conditions (i) and (ii) that crit(E) is a limit of Woodin cardinals in Qη , but not itself Woodin. By Claim 7B.6 there must therefore exist some γ so that: (iii) γ ≤ η is a standard limit; and (iv) crit(E) is equal to λγ . Qη and Mγ agree past λγ by Claim 7B.5. So the extender E can be applied to the model Mγ . Let U+ be the extension of U which makes this application. In other words let U+ be the extension of U, to a tot of length η + 1.2, determined by the assignments:
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(a) Eη = E; and (b) the U + -predecessor of η + 1 is γ . These assignments completely determine U+ , see Remark 7B.19. Claim 7C.8. U+ is regular. Proof. Immediate from the strength of E given by condition (i).
#
Claim 7C.9. Mη+1 = Ult(Mγ , Eη ) is wellfounded. Proof. Immediate from the fact that U is consistent with .
#
Claim 7C.10. Let sγ be equal to tγ −− wγ , yγ over Qγ . Then tη† extends sγ (perhaps not strictly). Proof. If γ < η then sγ is simply equal to tγ +1 , and tη† extends it strictly. If γ = η then sγ and tη† are equal by their definitions. # branch (M, ∅, θ)(Y˙ ) following Let β = lh(Pγ ). By Claim 7C.2 mega-round β of G Pγ is played subject to the rules of the standard limit case in Section 6A. By Claim 7C.4, wγ , Tγ , bγ , and yγ are legal moves for rules (L1)–(L4) in that mega-round. branch (M, ∅, θ)(Y˙ ) which consists of Pγ Claim 7C.11. Let Pγ∗ denote the position in G followed by the moves wγ , Tγ , bγ , and yγ for rules (L1)–(L4) in mega-round β. Then Pγ∗ is: branch ; and (1) played according to branch (M, ∅, θ)(Y˙ ). (2) non-terminal in G Proof. Assumptions (A2) and (A4) for η¯ = γ combine to state that Pγ∗ is played branch is a branch . This establishes condition (1) of the claim. Since according to ∗ winning strategy for I it also establishes that Pγ is not lost by player I. So Pγ∗ cannot be terminal through condition (I6) in Section 6A. Pγ∗ cannot be terminal through condition (I5) in Section 6A since Qγ , being a model along a tot consistent with the iteration strategy , is wellfounded. # Claim 7C.12. Qη &δη+1 + ω is countable in V. Proof. Assumption (A2) states that Pη , whose outcome equals $Mη , j0,η , tη %, is non branch (M, ∅, θ)(Y˙ ). It follows that rdm(tη ) is smaller than j0,η (θ ). This terminal in G in turn implies that δη† , which is equal to the first Woodin cardinal of Mη above rdm(tη ), is smaller than j0,η (θ). So δη+1 , which is equal to kη (δη† ), is smaller than (kη ◦ j0,η )(θ ). Since θ is inaccessible in M it follows that the cardinality of Qη &δη+1 + ω in Qη is smaller than (kη ◦ j0,η )(θ). So it suffices to prove that (kη ◦ j0,η )(θ ) is countable in V. By assumption θ is countable in V. (This is part of the initial settings of Section 7C.) By condition (2) in Remark 6A.4 applied to Pη , j0,η preserves countability. By condition (R3) in Section 7B and Fact 3 in Appendix A, kη preserves countability. Combining # all these assertions it follows that (kη ◦ j0,η )(θ) is countable in V.
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Lemma 7C.13 (assuming that tη† is I-acceptably obstructed over Qη ). Let Pη+1 be defined with reference to the extension U+ of U determined by conditions (a) and (b) branch (M, ∅, θ)(Y˙ ), non-terminal, and played above. Then Pη+1 is a legal position in G according to branch . Proof. Recall that γ is the U + -predecessor of η + 1, and that β = lh(Pγ ). γ is a standard limit in U by condition (iii) above. Definition 7B.17 is such that: (v) Pη+1 = Pγ −−, Tγ , bγ , Eη , tη+1 . branch (M, ∅, θ)(Y˙ ) following Pγ is played subject to By Claim 7C.2, mega-round β of G the rules of the standard limit case in Section 6A. It follows from this, from condition (v), and from the fact that Eη is not “undefined,” that Pη+1 is the position which consists of: (K1) Pγ ; followed by (K2) the moves wγ , Tγ , bγ , and yγ (given by the fixed U, w, and y) for rules (L1)–(L4) in mega-round β; followed by (K3) a leap with the moves Eη , δη+1 , and tη+1 for rules (L5) and (L6). The part of Pη+1 described in conditions (K1) and (K2) is simply the position Pγ∗ of branch , Claim 7C.11. We already know that this part is legal, played according to and non-terminal. So the key to the lemma is verifying that the continuing moves described in condition (K3) are legal and non-terminal. There is no need to worry about branch for these moves since they are made by player II. consistency with jγ ,η+1
/
Mγ
kkk Sk k Sk SSSS k / Qγ _ _ _ _ _ _ Qη γ
Eη
&
Mη+1
Tγ
Diagram 7.3. The models and embeddings connected with mega-round β in Pη+1 .
The legality of the moves described in condition (K3) follows through an application of Lemma 6A.5, with: Q∗β = Qη , Eβ∗ = Eη , δβ∗ = δη† , and tβ∗ = tη† . There are some indexical difficulties in connecting Lemma 6A.5 to the current situation, since megaround β in Pη+1 is spread over several stages of U+ , and chances are that none of these stages is β. We try to ease these difficulties by illustrating in Diagram 7.3 all the models and embeddings related to mega-round β in Pη+1 . The Diagram is identical to Diagram 6.2, except for the indexing. Mega-round β in Pη+1 starts from the outcome of Pγ , which is equal to $Mγ , j0,γ , tγ %; continues with the moves wγ , Tγ , bγ , and yγ ; and ends with the moves Eη , δη+1 , and tη+1 .
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7 Games which reach local cardinals
Other than the indexical difficulties it’s easy to check that Lemma 6A.5 applies in this context. Condition (1) in the lemma follows from Claim 7C.12; condition (2) in the lemma follows from Claim 7C.10; condition (3) in the lemma follows from Claim 7C.5; condition (4) in the lemma follows from condition (iv) above, at the start of Section 7C (2); and condition (5) in the lemma follows from Claim 7C.9. So far we verified that the moves described in condition (K3) are legal in branch (M, ∅, θ )(Y˙ ). These moves could only be terminal through condition (I7) in G branch ends with a loss to II if Mη+1 is illfounded. Section 6A. The condition states that G But Mη+1 is wellfounded by Claim 7C.9. So the moves described in condition (K3) are non-terminal. # 7C (3) Summary. We are working in Section 7C with the following objects: • a transitive model M of ZFC∗ ; • an iteration strategy for M; • a Woodin limit of Woodin cardinals θ and a θ -name Y˙ in M; branch (M, ∅, θ)(Y˙ ); branch for I in G • a winning strategy • a regular tot U of successor length η + 1 on M, consistent with ; and • a U-sequence $w, y%. Definition 7B.17 breaks U and $w, y% into strands along the branches of U . We assume branch (M, ∅, θ)(Y˙ ), non-terminal, here that each of these strands is a legal position in G branch . We assume further that the moves at the endpoint and played according to branch . Our work shows how to extend U in of each strand are also consistent with a way that makes the resulting new strand, namely the strand leading to η + 1, legal, branch . Precisely we obtain: non-terminal, and consistent with Lemma 7C.14 (under assumptions (A1)–(A5) listed earlier in Section 7C). There exists an extension of U to a regular tot U+ of length η + 1.2, so that defining Pη+1 with reference to U+ yields the following conditions: branch (M, ∅, θ)(Y˙ ); (1) Pη+1 is a legal position in G branch (M, ∅, θ)(Y˙ ); and (2) Pη+1 is non-terminal in G branch . (3) Pη+1 is played according to
#
This follows from Lemma 7C.7 if tη† is obstruction free over Qη , and from Lemma 7C.13 if tη† is I-acceptably obstructed over Qη . By Claim 7C.6 there are no other cases. In the first case U+ is defined so that the U + -predecessor of η + 1 is η, and Pη+1 continues Pη . The second case is more involved. U+ is defined to that the U + predecessor of η+1 is the ordinal γ corresponding to the critical point of the I-acceptable obstruction given by the case assumption. Pη+1 in this case continues Pγ , and does so with a leap.
7D Construction
285
7D Construction Let M be a transitive model of ZFC∗ . Let be an iteration strategy for M. Let θ be a Woodin limit of Woodin cardinals in M. Let A˙ ∈ M be a Wθ -name for a set of sequences of reals of length θ . Wθ as usual denotes Woodin’s algebra defined in Section 4B. The objects above correspond to the settings in Section 7A. We work with them fixed for the rest of the section. Definition 7D.1. For expository simplicity fix some filter G which is Wθ -generic/M. Define Y˙ ∈ M to be the canonical Wθ -name for the set of θ-sequences t ∈ M[G] so ˙ that z(t) ∈ A[G]. z(t) in Definition 7D.1 is the real part of t, given by Definition 4A.21. Recall that it is equal to $t (δ) | δ ∈ dom(t) and δ is a relative successor%. This is a sequence of reals numbers. Using the fact that θ is a Woodin limit of Woodin cardinals it is easy to check that there are θ relative successors below θ. So z(t) is a sequence (of reals) of length θ . ˙ The clause z(t) ∈ A[G] in Definition 7D.1 is therefore meaningful—it is not trivially false. Let ϕini be the formula of Definition 5G.2. The next lemma begins the proof of the main result described in Section 7A. Lemma 7D.2. Suppose that: (1) θ is countable in V; and (2) ϕini [θ, Y˙ ] holds in M. local (M, , θ, A). ˙ Then (in V) player I has a winning strategy for G local (M, , θ, A). ˙ We Proof. Fix an imaginary opponent willing to play for II in G describe how to play for I, and win. The description as usual takes the form of a construction. We work to construct: (A) an ordinal α ∗ ; (B) a regular tot U+ of length α ∗ + 0.2 on M; and (C) a U-sequence $wξ , yξ | ξ ∈ K U %, where U denotes U+ α ∗ . local (M, , θ, A) ˙ through the assignment: We associate to these objects a position z in G (D) zξ = y−1+ξ +1 for each ξ so that ξ + 1 < 1 + α ∗ . In other words z is simply the sequence: $yζ | ζ < α ∗ is zero or a successor ordinal%. This is a sequence of reals by condition (C2) in Section 7B. We shall verify at the end local (M, , θ, A) ˙ which of the construction that this sequence of reals forms a run of G
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7 Games which reach local cardinals
is won by player I. To verify victory by I we shall need an iteration tree U witnessing condition (P) in Section 7A. Let us already here say that the tree we intend to use is simply the merge of U+ . The only help we get from the imaginary opponent during the construction is in the creation of z. In other words the imaginary opponent only participates in the creation of yζ , and only for ζ which is either zero or a successor ordinal. The production of all the other objects involved in conditions (B) and (C) falls squarely on us. We handle it using the tools of Chapter 6. Corollary 6G.2 and Remark 6G.3 tell us that (under the assumptions of the current lemma) player I has a winning strategy in the game branch (M, ∅, θ )(Y˙ ). Fix, for the rest of the proof, a strategy branch which witnesses G branch . this. We intend to heave much of the burden of the construction on We construct subject to the following conditions: branch (M, ∅, θ)(Y˙ ), non-terminal, and (1) for each η < α ∗ , Pη is a legal position in G played according to branch ; (2) U+ is consistent with the iteration strategy ; (3) if η < α ∗ is a successor or zero then wη , yη , Tη , and bη are consistent with branch [Pη ]; (4) if η < α ∗ is a standard limit in U then wη , Tη , bη , and yη are consistent with branch [Pη ]; (5) if η < α ∗ is either zero, a successor, or a standard limit in U, then Uη + 1.2 is obtained from Uη + 1 through an application of Lemma 7C.14; and (6) if η < α ∗ is a phantom limit in U then Tη is the iteration tree which consists entirely of “padding,” bη is the unique branch through it, and Eη =“undefined.” Conditions (2)–(6) in fact completely determine each step of the construction: ¶ If η < α ∗ is zero or a successor then wη , yη , Tη , and bη are determined through branch [Pη ], the imaginary opcondition (3) by a collaborative effort which involves ponent, and the iteration strategy . (The imaginary opponent participates in the cre branch .) Eη and the ation of yη , and chooses bη . Everything else is done by U + -predecessor of η + 1 are subsequently determined through condition (5) by an application of Lemma 7C.14. ¶ If η < α ∗ is a standard limit in U then wη , Tη , bη , and yη are determined through branch [Pη ] and the iteration stratcondition (4) by a collaborative effort which involves branch .) Eη and the U + -predecessor egy . ( chooses bη . Everything else is done by of η + 1 are again determined through condition (5) by an application of Lemma 7C.14. ¶ If η < α ∗ is a phantom limit in U then Tη , bη , and Eη are precisely specified by condition (6). The U + -predecessor of η + 1 in this case is η. There is no need to define wη and yη since η does not belong to K U .
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¶ Finally, if γ ≤ α ∗ is a limit then the branch {ζ | ζ U γ } which is used to give rise to the direct limit model Mγ is picked by subject to condition (2). branch [Pη ] Condition (1) is an inductive condition, needed to allow the references to in conditions (3) and (4). Notice that it holds trivially for η = 0. For as long as the condition continues to hold we can continue the construction. If ever it fails we must end. In other words the construction ends at the first ordinal α ∗ so that condition (1) fails for η = α ∗ , if such an ordinal is ever reached. Claim 7D.3. Let η be a limit ordinal smaller than ω1V . Suppose that condition (1) holds for all η¯ < η. Then the condition holds also for η. Proof. For a limit ordinal η, Pη is equal to the increasing union ζ U η Pζ . By assump branch (M, ∅, θ)(Y˙ ), non-terminal, and played according to tion each Pζis legal in G branch (M, ∅, θ)(Y˙ ) and played according to branch . So P is certainly legal in G ζUη ζ branch . It remains to check that it is non-terminal. Now ζ U η Pζ , being a position of limit length, can only be terminal through one of the conditions (I3) and (I4) in Section 6A. But neither condition holds here. Condition (I3) does not hold since Mη , being a model on a tot consistent with the iteration strategy , is wellfounded. Condition (I4) does not hold since by assumption η is smaller than ω1V , and therefore so is the order type of {ζ | ζ U η}. # Claim 7D.4. Let η be either zero, a successor ordinal, or a standard limit in U. Suppose that condition (1) holds for η. Then the condition holds also for η + 1. Proof. For η as in the claim, Uη + 1.2 is obtained from Uη + 1 through condition (5) of the construction, or in other words through an application of Lemma 7C.14. The conclusion of that lemma precisely secures condition (1) for η + 1. # The last two claims imply that if the construction ends before ω1V , it must be at a successor ordinal which immediately follows a phantom limit. Claim 7D.5. The construction ends before reaching ω1V . Proof. Suppose for contradiction that the construction reaches ω1V . Let r = {ζ | ζ U branch (M, ∅, θ)(Y˙ ) and played ω1V }. Each of the positions Pζ , ζ ∈ r, is legal in G branch . It follows that the increasing union according to ζ ∈r Pζ is a legal position in ˙ Gbranch (M, ∅, θ )(Y ), and played according to branch . Now r, being cofinal in ω1V , has branch (M, ∅, θ)(Y˙ ). order type ω1V . It follows that ζ ∈r Pζ is a position of length ω1V in G It is therefore lost by player I through condition (I4) in Section 6A. But this contradicts branch is a winning strategy for player I. # the fact that Remark 7D.6. By Claim 7D.5 only countable iteration trees actually appear in the construction. But to prove the claim we had to allow for the possibility of a tree of length ω1V + 1. So the claim uses the fact that acts not only on countable iteration trees but also on iteration trees of length ω1V .
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This kind of phenomenon, where an iteration strategy which acts also on trees of length ω1V is needed to in fact rule out the case of trees of length ω1V , is typical in the study of large cardinals. The classic example is the comparison argument of countable mice in inner model theory, where it is always shown at the end that the comparison has countable length, but where the possibility of an iteration tree of length ω1 must be allowed before it can be ruled out. Let α ∗ be the first ordinal for which condition (1) fails. We know from the last three claims that such an ordinal exists, that it is smaller than ω1V , and that it is a successor ordinal immediately following a phantom limit in U. Constructing up to α ∗ subject to conditions (2)–(6) we precisely obtain the objects listed in conditions (B) and (C) above. Through the assignment of condition (D) we obtain also a position z = $y−1+ξ +1 | ξ + 1 < 1 + α ∗ % local (M, , θ, A). ˙ We now work through a series of claims to verify that z is a in G local (M, , θ, A), ˙ and won by player I. complete run of G branch (M, ∅, θ)(Y˙ ), and won by player I Claim 7D.7. Pα ∗ is a terminal position in G through condition (P2) in Section 6A. Proof. Let α = α ∗ − 1. α is then a phantom limit in U. From condition (6) it follows that Eα =“undefined.” The U + -predecessor of α ∗ = α + 1 is therefore equal to α, and Pα ∗ extends Pα (by one mega-round). Let β = lh(Pα ). Since α is a phantom limit in U, Claim 7C.2 implies that branch (M, ∅, θ)(Y˙ ) following Pα is played subject to the trivial mega-round β of G rules of the phantom limit case in Section 6A. So Pα ∗ is simply the position which extends Pα with one, trivial phantom mega-round. Such an extension is legal in branch (M, ∅, θ )(Y˙ ) following Pα , and trivially consistent with branch [Pα ]. So Pα ∗ G ˙ is legal in Gbranch (M, ∅, θ)(Y ) and played according to branch . Since condition (1) branch (M, ∅, θ)(Y˙ ). The only way nonetheless fails for α ∗ , Pα ∗ must be terminal in G a position ending a phantom mega-round could be terminal is through condition (P2) in Section 6A. Pα ∗ must be won by player I through that condition, since it is played branch . # according to I’s winning strategy Following the notation of Section 7B let Mξ and jζ,ξ : Mζ → Mξ denote the models and embeddings of the tot U+ . U+ is a tot of length α ∗ + 0.2 on M. It leads to a final model Mα ∗ , and to a final iteration embedding j0,α ∗ : M → Mα ∗ . Through Definition 7B.14, or more precisely the extension of that definition to U+ , we may talk about tα ∗ , an annotated position over Mα ∗ . Claim 7D.8. There exists some G so that: G is j0,α ∗ (Wθ )-generic/Mα ∗ ; and tα ∗ belongs to j0,α ∗ (Y˙ )[G]. Proof. This is simply the payoff in condition (P2) of Section 6A, translated to the current context using the fact that the outcome of Pα ∗ is equal to $Mα ∗ , j0,α ∗ , tα ∗ %. #
7E The main theorem
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Fix from now on some G which witnesses Claim 7D.8. ˙ Claim 7D.9. z belongs to j0,α ∗ (A)[G]. Proof. Recall that z is equal to $y−1+ξ +1 | ξ + 1 < 1 + α ∗ %. By Claim 7B.16, z is precisely equal to z(tα ∗ ). The current claim now follows from the fact that tα ∗ belongs to j0,α ∗ (Y˙ )[G], and from the relationship between Y˙ and A˙ given by Definition 7D.1. # Claim 7D.9 completes the proof of Lemma 7D.2. It shows that z, which was produced with the collaboration of the imaginary opponent, is a complete run of local (M, , θ, A) ˙ and won by player I. The objects which witness I’s victory are the iterG # (Lemma 7D.2) ation tree merge(U+ ) and the generic filter G given by Claim 7D.8. Remark 7D.10. For the sake of clarity let us take note of the use of the assumptions of Lemma 7D.2 during its proof. The assumptions were used in the appeal to Corollary 6G.2, which (combined with Remark 6G.3) provided the most crucial element of branch . the construction, namely the strategy
7E The main theorem Let M be a transitive model of ZFC∗ . Let be an iteration strategy for M. Let θ be a Woodin limit of Woodin cardinals in M. Let A˙ and B˙ be Wθ names, each naming a set of sequences of reals of length θ . The following theorem, previewed in Section 7A, is local the main result of the current chapter. It is phrased with reference to the games G local defined in Section 7A. We already did most of the work needed for its proof. and H All that’s left now is to assemble the various parts. Theorem 7E.1. Suppose that θ is countable in V. Then at least one of the following three cases holds: local (M, , θ, A); ˙ (1) player I has a winning strategy in G local (M, , θ, B); ˙ or (2) player II has a winning strategy in H (3) there exists some G ∈ V which is Wθ -generic/M, and there exists some sequence ˙ nor of reals $zξ | ξ < θ% ∈ M[G], so that $zξ | ξ < θ% belongs to neither A[G] ˙ B[G]. Moreover M can distinguish this. More precisely, there are formulae ϕ and ψ so that: if ˙ then case (1) holds; if M |= ψ[θ, B] ˙ then case (2) holds; and otherwise M |= ϕ[θ, A] case (3) holds. ¯ which is Wθ -generic/M. Let Y˙ ∈ M Proof. For expository simplicity fix some G ¯ so that z(t) ∈ A[ ˙ G]. ¯ Let be the canonical name for the set of θ-sequences t ∈ M[G] ¯ so that z(t) ∈ B[ ˙ G]. ¯ Z˙ ∈ M be the canonical name for the set of θ -sequences t ∈ M[G] Let ϕini and ψini be the two formulae of Definition 5G.2.
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Case 1. If ϕini [θ, Y˙ ] holds in M. In this case an application of Lemma 7D.2 shows that local (M, , θ, A). ˙ So case (1) of the theorem holds. player I has a winning strategy in G # ˙ holds in M. In this case an argument which precisely mirrors the Case 2. If ψini [θ, Z] development which led to case 1 (including the entire development of Chapter 6 and local (M, , θ, B). ˙ So case Section 7D) shows that player II has a winning strategy in H (2) of the theorem holds. # ˙ fail in M. In this case an application of Case 3. If both ϕini [θ, Y˙ ] and ψini [θ, Z] Corollary 5G.3 produces, in some generic extension of M, a θ -sequence t and some filter G so that: (i) G is Wθ -generic/M; (ii) t belongs to M[G]; and ˙ (iii) t belongs to neither Y˙ [G] nor Z[G]. Let D be the collection of maximal anti-chains of Wθ in M. Each anti-chain in D is a subset of M&θ which belongs to M, and has size strictly less than θ in M by Lemma 4B.19. So each anti-chain in D is an element of M&θ. It follows from this and from the inaccessibility of θ in M that the cardinality of D in M is at most θ. Theorem 7E.1 assumes that θ is countable in V. So: (iv) D is countable in V. Using the last condition one can reflect the existence of objects t and G which satisfy conditions (i)–(iii), from an arbitrary generic extension of M, to V. So we may without loss of generality assume that t and G belong to V. Since θ is an inaccessible limit of Woodin cardinals, the set of relative successors below θ has order type precisely θ. Let $δξ | ξ < θ % enumerate this set in increasing order. Following the notation of Definition 4A.21 let zξ = t (δξ ) for each ξ < θ . $zξ | ξ < θ% is a sequence of reals. It is precisely equal to the real part of t, namely to z(t). From condition (ii) it follows that: (v) $zξ | ξ < θ% belongs to M[G]. ˙ and B˙ in the definition of Y˙ and Z˙ above it From condition (iii) and the use of z(t), A, follows that: ˙ ˙ (vi) $zξ | ξ < θ% belongs to neither A[G] nor B[G]. So case (3) of Theorem 7E.1 holds.
#
The three cases above complete the proof of Theorem 7E.1. The formulae ϕ and ψ which witness the final part of the theorem are defined in the natural way, using the formulae ϕini and ψini , and using the definitions of Y˙ and Z˙ at the start of the proof. # (Theorem 7E.1)
7E The main theorem
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It is clear from the construction in Chapter 6, its use in Sections 7B and 7C, and ultimately the proof of Lemma 7D.2, that the winning strategies in cases 1 and 2 are produced definably in and a given real coding M&θ . In fact the strategies are definable from the real and the restriction of to just countable iteration trees. These are the only trees that come up in construction; trees of length ω1 only come up through the proof of Claim 7D.5. Certainly then the winning strategies belong to L(R, {countable itereation trees}). This can be refined further. For a model P let ¯ P be the restriction of to iteration trees which belong to P and are countable in P . Given a strategy σ in a long game on natural numbers, let σ¯ P be the restriction of σ to positions which belong to P and are countable in P . The strategy σ is locally definable from above a real u if it satisfies the following property: let P be any model of ZFC∗ so that u belongs to P , and so that ¯ P belongs to P ; then σ¯ P belongs to P . The following remark is then clear from the constructions which go into the proof of Theorem 7E.1: Remark 7E.2. Condition (1) in Theorem 7E.1 can be strengthened to say that the winning strategy is locally definable from over a given real which codes M&θ; and similarly with condition (2). Let us also note for the record that need only apply to iteration trees on M&θ. This is clear from the constructions, and can also be derived abstractly from Theorem 7E.1 by assuming, without loss of generality, that M has no extenders above θ. Exercise 7E.3. For each Woodin cardinal δ ∈ M let otrs(δ) denote the order type of the set {δ¯ ∈ (δ + 1) ∩ W | δ¯ is a relative successor}. Show that Theorem 7E.1 holds with condition (3) strengthened to include also the following statement: For each Woodin cardinal δ < θ of M, the sequence $zξ | ξ < otrs(δ)% belongs to a generic extension of M via the poset col(ω, δ), if δ is not a Woodin limit of Woodin cardinals in M, and via the poset Wδ if it is. Exercise 7E.4. This exercise reaches a parallel of Theorem 7E.1 for the case that θ is a Woodin cardinal in M but not a Woodin limit of Woodin cardinals. Fix such ˙ B˙ ∈ M be names for sets of a θ . Suppose that M&θ + 1 is countable in V. Let A, local (M, , θ, A) ˙ and sequences of reals of length otrs(θ) (see Exercise 7E.3). Define G local (M, , θ, A) ˙ following the specifications in Section 7A, only replacing the poset H Wθ in the payoff conditions (P) and (Q) with the poset col(ω, θ ). Prove that at least one of the following three cases holds: local (M, , θ, A). ˙ (1) Player I has a winning strategy in G local (M, , θ, B). ˙ (2) Player II has a winning strategy in H (3) There exists some G ∈ V which is col(ω, θ )-generic/M, and there exists some ˙ sequence of reals z = $zξ | ξ < otrs(θ)% ∈ M[G] which belongs to neither A[G] ˙ nor B[G].
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Moreover, the sequence z is such that for each Woodin cardinal δ < θ of M, the restriction zotrs(δ) belongs to a generic extension of M via the poset col(ω, δ), if δ is not a Woodin limit of Woodin cardinals in M, and via the poset Wδ if it is. Show further that M can distinguish this. Precisely, show that there are formulae ϕ and ˙ then case (1) holds; if M |= ψ[θ, B] ˙ then case (2) holds; ψ so that: if M |= ϕ[θ, A] and otherwise case (3) holds. Exercise 7E.5. Suppose that M has no extenders which overlap Woodin cardinals. (In particular there are no Woodin limits of Woodin cardinals in M.) Repeat Exercise 7E.4 in this case, without using any of the results in Chapters 4, 5, and 6. (This will not lead to any additional results, but you will gain some experience.) You will have to define, and work with, your own pullback operation. Since M has no extenders overlapping Woodin cardinals, the settings here are a great deal simpler than the general settings in Chapter 4. For your pullback operation you will only need the successor case from that chapter, and a substantially simplified version of the case of relative limits, more in the spirit of Chapter 2 than Chapter 3, without any records and with no reference to obstructions.
7F Determinacy Theorem 7E.1 yields an array of determinacy results. Roughly speaking it allows handling games which end at the first cardinal of some inner model built relative to the play. We illustrate this by presenting a specific application, with L as the inner model. Let C ⊂ R<ω1 be given. Glocal (L, C) is played as follows: Players I and II alternate playing natural numbers as in Diagram 7.4 to produce reals zξ . They continue until reaching the first γ which is uncountable in L[zξ | ξ < γ ]. At that point the game ends. Player I wins iff $zξ | ξ < γ % belongs to C. I II
z0 (0)
z0 (2) z0 (1)
...... ...
zξ (0) zξ (1)
zξ (2) ...
......
Diagram 7.4. The game Glocal (L, C). L[z |ξ <γ ]
The first γ which is uncountable in L[zξ | ξ < γ ] must in fact be equal to ω1 ξ . We therefore refer to the games Glocal (L, ∗) as games which “end at ω1 in L of the L[z |ξ <γ ] play.” We view ω1 ξ as a “local” cardinal, in that it is a cardinal of the inner model L[zξ | ξ < γ ]. The notation Glocal (L, C) comes from this view. Technically runs of length ω1 in Glocal (L, C) are won by player II. But even very mild large cardinal assumptions imply that the first γ which is uncountable in L[zξ | ξ < γ ] is always countable in V. The large cardinals we work with here are more than enough
7F Determinacy
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for this, as the next claim demonstrates. So in our context all runs of Glocal (L, C) are countable. Claim 7F.1. Suppose that there exists an iterable class model M with a cardinal θ so that: (a) θ is a Woodin limit of Woodin cardinals in M; and (b) θ is countable in V. Let $uξ | ξ < ω1V % be a sequence of reals of length ω1V . Then there is some α, strictly smaller than ω1V , so that α is uncountable in L[uξ | ξ < α]. Proof. Let A˙ ∈ M be the canonical Wθ -name for the set of all sequences of reals of length θ . Let B˙ name the empty set. Apply Theorem 7E.1 with these names. Neither case (2) nor case (3) of the theorem can occur. So it must be that I has a winning strategy local (M, , θ, A). ˙ Let σ be such a strategy. Play against σ by making the moves in G zξ (2i + 1) = uξ (i) for II. σ must secure victory for I at some α strictly smaller than ω1V . # Using Claim 7A.2 it is easy to see that this α is uncountable in L[uξ | ξ < α]. We should note that the large cardinal assumption in the claim is in fact a huge overkill. It is enough to assume that θ is a Woodin cardinal (rather than a Woodin limit of Woodin cardinals). One can then obtain the conclusion of the claim using Woodin’s standard extender algebra, where extender axioms are defined using all extenders, rather than just ones which overlap Woodin cardinals. Our intention next is to prove the determinacy of Glocal (L, C) for payoff sets C which are ω (<ω2 − 11 ) in the codes. Let us first be more precise on the way we code runs. Definition 7F.2. Let γ be a countable ordinal and let $zξ | ξ < γ % be a sequence of reals of length γ . By a code for $zξ | ξ < γ % we mean a sequence $xn | x < ω% in Rω so that: (1) x0 is a wellordering of a subset of ω − {0}; (2) the order type of x0 is precisely γ ; and (3) for every n ∈ dom(x0 ), xn is equal to zξ where ξ < γ is the order type of n in the wellordering x0 . Definition 7F.3. Let be a pointclass. A set C ⊂ R<ω1 is in the codes just in case that the set C ∗ of $xn | n < ω% ∈ Rω which code elements of C belongs to . Theorem 7F.4. Suppose that there exists an iterable class model M with a cardinal θ so that: (a) θ is a Woodin limit of Woodin cardinals in M; and (b) θ is countable in V. Then the games Glocal (L, C) are determined for all C which are ω (<ω2 − 11 ) in the codes. Remark 7F.5. The existence of M and θ satisfying the assumptions in Theorem 7F.4 follows from the existence, in V, of a sharp for a Woodin limit of Woodin cardinals, see Theorem 16 in Appendix A.
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Proof of Theorem 7F.4. Fix M and θ satisfying the assumptions of the theorem. Let be an iteration strategy for M. Fix C ⊂ R<ω1 which is ω (<ω2 − 11 ) in the codes. We work to show that Glocal (L, C) is determined. Let C ∗ be the set of $xn | n < ω% ∈ Rω which code elements of C. C ∗ then belongs to ω (<ω2 − 11 ). By Martin [24] there is a number k < ω and a 1 formula φ so that for any x = $xn | n < ω%: x ∈ C ∗
⇐⇒
L[ x ] |= φ[ x , c0 , . . . , ck−1 ] whenever c0 < · · · < ck−1 are Silver indiscernibles for x
Let ψ(a, γ , c0 , . . . , ck−1 ) be the formula “a is a sequence of reals of length γ ; and it is forced in col(ω, γ ) that there is a code x for a so that φ[ x , c0 , . . . , ck−1 ] holds in L[ x ].” Let u0 < · · · < uk−1 be uniform Silver indiscernibles. For expository simplicity fix some G which is Wθ -generic over M. Wθ as usual stands for Woodin’s extender algebra defined in Section 4B. Working in M[G] let A be the set of sequences $zξ | ξ < θ% of reals so that M[G] |= ψ[$zξ | ξ < γ %, γ , u0 , . . . , uk−1 ] where γ ≤ θ is the least ordinal which satisfies the condition: (∗) γ is uncountable in L[zξ | ξ < γ ]. Note that such an ordinal γ must exist, since θ is uncountable in M[G]. Let A˙ ∈ M be the canonical Wθ -name for A. Let B be the set of sequences $zξ | ξ < θ% ∈ M[G] of reals so that M[G] |= ¬ψ[$zξ | ξ < γ %, γ , u0 , . . . , uk−1 ] where γ ≤ θ is the least ordinal which satisfies the condition (∗) above. Let B˙ ∈ M be the canonical Wθ -name for B. ˙ or Claim 7F.6. Every sequence of reals $zξ | ξ < θ % in M[G] belongs either to A[G] ˙ to B[G]. Proof. Immediate. A˙ and B˙ name complementary sets.
#
˙ The last claim rules out case (3) of Apply Theorem 7E.1 with the names A˙ and B. the theorem. The remaining possibilities are either: (1) player I has a winning strategy local (M, , θ, B). local (M, , θ, A); ˙ or (2) player II has a winning strategy in H ˙ in G local (M, , θ, A). ˙ It is easy to see that this Case 1. If I has a winning strategy in G strategy induces a winning strategy for I in Glocal (L, C). The key point is that any run local (M, , θ, A) ˙ which is won by I must, by Claim 7A.2, either reach $zξ | ξ < α% of G or pass a γ which is uncountable in L[zξ | ξ < γ ]. In other words it must either equal or extend a complete run of Glocal (L, C). The definition of A˙ above is such that this run of Glocal (L, C) must be won by player I. #
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local (M, , θ, B). ˙ Then this strategy induces Case 2. If II has a winning strategy in H a winning strategy for II in Glocal (L, C). Again the key point is the fact that any run local (M, , θ, B) ˙ which is won by II must either equal or extend a complete run of of H Glocal (L, C). The definition of B˙ above is such that this run of Glocal (L, C) must be won by player II. # We already saw, through a use of Theorem 7E.1, that one of the two cases above # (Theorem 7F.4) must hold. So Glocal (L, C) is determined. To progress beyond the games in Theorem 7F.4 it is convenient to phrase the ending condition in terms of descriptive set theory. Let $zξ | ξ < γ % be a sequence of reals of length γ . let x = $xn | n < ω% code this sequence in the sense of Definition 7F.2. In particular then x0 is a wellordering of ω − {0}, of order type precisely γ . Given a function f : ω → γ let fx be the real a defined by setting a(n) equal to the unique k so that the order type of k in x0 is precisely f (n). fx then codes the function f , modulo the wellordering x0 . Let be a pointclass. We say that f : ω → γ belongs to [zξ | ξ < γ ] just in case that, for every code x of $zξ | ξ < γ %, the real fx belongs to the pointclass ( x ). We say that γ is countable in [zξ | ξ < γ ] just in case that there exists a function f from ω onto γ , which belongs to [zξ | ξ < γ ]. Remark 7F.7 (assuming every real has a sharp). If is the pointclass ω (<ω2 − 11 ) then, by Martin [24], the reals which belong to are precisely the reals in L, and similarly for any x ∈ R the reals which belong to (x) are precisely the reals in L[x]. It follows in this case that γ is countable in [zξ | ξ < γ ] if and only if it is countable in L[zξ | ξ < γ ]. Let C ⊂ R<ω1 be given. Define Glocal (, C), the game ending at “ω1 in of the play,” in the obvious way, taking the definition of Glocal (L, C) and changing “uncountable in L[zξ | ξ < γ ]” to “not countable in [zξ | ξ < γ ].” Notice then that the games Glocal (, ∗) become stronger, meaning longer, as increases. Theorem 7F.4 states the determinacy of the games Glocal (, C) for the pointclass = ω (<ω2 −11 ) and sets C in . The proof of Theorem 7F.4 involves nothing more than an application of Theorem 7E.1. That theorem in turn is general enough that it can be used in the context of pointclasses larger than ω (<ω2 − 11 ). More specifically it can be applied whenever the given model M, and its iterates, can correctly compute membership in sets which belong to . Let us be more precise. Let A be a set of reals. Let M be a model, let be an iteration strategy for M, let τ be a Woodin cardinal in M, and suppose that M&τ + 1 is countable in V. Let A˙ ∈ M be a col(ω, τ )-name for a set of reals. The name A˙ captures A over $M, τ % just in case that for every g ∈ V which is col(ω, τ )-generic over M, ˙ A[g] is precisely equal to A ∩ M[g]. A˙ fully captures A over $M, τ, % just in case that ˙ captures for any (countability preserving) -iteration embedding j : M → M ∗ , j (A) A over $M ∗ , j (τ )%. A pointclass is captured over $M, τ, % just in case that every
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set of reals in is fully captured by some name over $M, τ, %. These definitions are all due to Woodin. It is clear, using Shoenfield absoluteness, that the pointclass 12 is captured over $M, τ, % whenever M is a (wellfounded) class model. The results of Martin [24] give the same for the pointclass ω (<ω2 − 11 ), and this really was our tool in deriving Theorem 7F.4 from Theorem 7E.1. Thus working in general settings we obtain: Exercise 7F.8. Let be a pointclass, either ω-parameterized (see Moschovakis [26, p. 36]) or the union of countably many ω-parameterized pointclasses. Suppose that there is an iterable model M of ZFC∗ , a cardinal θ in M, and an iteration strategy for M so that: (a) θ is a Woodin limit of Woodin cardinals in M; (b) is captured by $M, θ, %; and (c) M&θ + 1 is countable in V. Let C be in the codes. Then Glocal (, C) is determined. Hint. Replace the names A˙ and B˙ in the proof of Theorem 7F.4 with names capturing (the set of codes for) C and its complement. You will have to add an argument in case 1 local (M, , θ, A) ˙ reach ω1 in of the play, and similarly in case 2. to show that runs of G For this you will have to argue that for any -iteration embedding j : M → M ∗ , any sequence of reals $zξ | ξ < γ % of length at most j (θ ), and any function f : ω → γ which belongs to [zξ | ξ < γ ], if $zξ | ξ < γ % belongs to an extension of M ∗ by a generic for j (Wθ ), then f belongs to the same extension. Here too you will use the fact that is captured over $M, θ, %. # The last exercise can be applied in specific settings by noticing that our determinacy proofs generally show that iterable models with enough large cardinals capture pointclasses generated by game quantifiers. The exercise and the corollary below demonstrate this in the context of the quantifiers associated to games of fixed countable length. Similar results can be proved with the quantifiers associated to the games in Chapter 3, and also with the quantifiers associated to the games here, in the current chapter. Exercise 7F.9. Let α ≥ 1 be a countable ordinal. Let M be an iterable class model and let τ < η be such that: (a) M sees that α is countable; (b) τ is a Woodin cardinal in M; (c) there are −1 + α Woodin cardinals of M in the interval (τ, η); and (d) M&η is countable in V. Let be an iteration strategy for M. Then the pointclass ω·α (<ω2 − 11 ) is captured over $M, τ, %. Hint. Go over the results in Chapter 2 and check that for each g which is col(ω, τ ) generic over M and each real x ∈ M[g], M[g] can (uniformly) identify which player wins a game of length ω · α with payoff in (<ω2 − 11 )(x). This will show that sets in ω·α (<ω2 − 11 ) are captured over $M, τ %. Then use the uniformity of the argument to obtain full capturing over $M, τ, %. # Corollary 7F.10. Let α ≥ 1 be a countable ordinal. Suppose that there exists an iterable class model M and θ < η in M so that: (a) θ is a Woodin limit of Woodin cardinals in M; (b) there are −1 + α Woodin cardinals of M in the interval (θ, η); and (c) M&η is countable in V. Let be the pointclass ω·α (<ω2 − 11 ), and let C be in the codes. Then Glocal (, C) is determined. #
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Exercise 7F.11. Formulate and prove a result similar to the last corollary, for cont instead of ω·α . It is natural to ask whether the results above, on capturing and subsequently on determinacy, can be extended to universally Baire sets. Exercise 2E.5 shows how universally Baire sets can be captured over models which embed into a rank initial segment of V. The argument there is sufficiently uniform that it would produce full capturing over a model and an iteration strategy, provided that the iteration strategy picked realizable branches (see the hint for Exercise 2E.6), that is branches whose direct limits can be embedded into a rank initial segment of V. The weak iterability necessary for the results in Chapter 2 was obtained through Theorem 12 in Appendix A, and the theorem produced realizable branches through the relevant iteration trees. This allowed us to bring the determinacy proved in Chapter 2 to bear on universally Baire sets. The mild iterability necessary for the results in Chapter 3 was obtained through Claim 14 in Appendix A, which similarly produced realizable branches through the relevant trees. Again this allowed proving determinacy in the case of universally Baire payoff. The results here on the other hand require full iterability, obtained through Theorem 16 in Appendix A. This theorem is proved indirectly, using fine structure among other things. It does not show that models which embed into initial segments of V are iterable, let alone iterable by a strategy which picks realizable branches. It therefore does not lead to applications of our techniques to universally Baire sets. Still it is possible, assuming sufficient large cardinals, to obtain fully iterable models which capture a given ∞-universally Baire set and have a Woodin limit of Woodin cardinals. The proof of this is due to Woodin, and combines the iterability proof in Neeman [31] (the source of Theorem 16 in Appendix A) with techniques from the fine structure theory associated to AD+ . Woodin used his models and Theorem 7E.1 to obtain the following determinacy: Theorem 7F.12 (Woodin). Suppose that there is a Woodin limit of Woodin cardinals, and a proper class of inaccessible limits of Woodin cardinals above it. Let A be ∞universally Baire and let be the pointclass of recursive preimages of A. Let C be in the codes. Then Glocal (, C) is determined. # Woodin obtained this theorem as part of his work on !-logic. He used it to characterize the !-logic completeness of the 12 -in- ∞ consequences of the CH, where ∞ is the pointclass of ∞-universally Baire sets, in terms of the determinacy of open length ω1 games with ∞ payoff. Let us end with an application of Theorem 7F.4, also due to Woodin. We present this application through exercises with extensive hints. Ultimately they lead to a proof of the consistency of determinacy for all ordinal definable games of length ω1 on natural numbers. Exercise 7F.13. Suppose that every real has a sharp. Let N be a countable model of ZFC∗ and let be an iteration strategy for N . Prove that there exists a sequence of reals $aξ | ξ < α% so that:
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(1) N belongs to the model P = L[aξ | ξ < α], and N is countable in P ; (2) ω1P is precisely equal to α; L[aξ |ξ <β]
(3) α is minimal, in the sense that ω1
= β for each β < α; and
(4) ¯ P belongs to P . ¯ P in condition (4) is the restriction of to iteration trees which belong to P and are countable in P . Hint. Construct $aξ | ξ < α% so that a0 is a real coding N, and for each β > 0, aβ is a real coding the restriction of to countable iteration trees in L[aξ | ξ < β]. Continue L[a |ξ <α] = α. # until reaching the first α so that ω1 ξ Exercise 7F.14 (Woodin). Suppose that there exists an iterable class model M with a cardinal θ so that: (a) θ is a Woodin limit of Woodin cardinals in M; and (b) θ is countable in V. Prove that there is a class model P so that, in P , every definable (over P , with no parameters) length ω1P game on natural numbers is determined. Hint. Let be an iteration strategy for M. Let P = L[aξ | ξ < α] be obtained by Exercise 7F.13, applied to N = M&θ and the restriction of to trees on N. Let ϕ(v) be a formula in the language of set theory, with one free variable. Let G be the following game: Players I and II alternate playing natural numbers in the usual fashion, to produce x ] then player I a sequence of reals x = $xξ | ξ < α% (of length α = ω1P ). If P |= ϕ[ wins; and otherwise player II wins. You will show that G is determined in P . Let G∗ be the following game: Players I and II alternate playing natural numbers in the usual fashion, to produce reals zξ . They continue until reaching the first γ which is uncountable in L[zξ | ξ < γ ]. At that point the game ends. Player I wins if L[zξ | ξ < γ ] |= ϕ[zmain ], where zmain is the sequence $zξ | ξ < γ and ξ is even%; and otherwise player II wins. Runs of G∗ thus give rise to a “main” part zmain , and an “auxiliary part,” consisting of the reals zξ for odd ξ . The auxiliary part allows each of the two players to put as much information as she wants into the model L[z] consulted in the payoff condition. By Theorem 7F.4, G∗ is determined. Using Remark 7E.2 (and the comment which follows it) get a winning strategy σ ∗ in G∗ so that σ¯ P∗ belongs to P . Use this to produce a strategy σ in G, which belongs to P and wins against all plays in P . # Exercise 7F.15 (Woodin). Work under the assumptions of the previous exercise. Prove that ZFC+“all ordinal definable games of length ω1 on natural numbers are determined” is consistent. In fact prove the consistency of ZFC+“all length ω1 games on natural numbers, definable from real and ordinal parameters, are determined.” Hint. There are standard ways to derive determinacy for games with parameters from determinacy for games without parameters. Ordinal parameters can be added by looking at the “least” non-determined game, where “least” involves a definable wellordering
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of all the ordinal definable games. Real parameters can be added by asking player I to play, in her first ω moves, a real which defines a non-determined game, and then continue to play in the game defined by that real. #
Appendix A
Extenders, generic extensions, and iterability Let L∗ be the language of set theory with an additional unary relation symbol K. Let ZFC∗ consist of the axioms of ZFC with replacement and comprehension extended to all formulae in L∗ . We work throughout the book with the language L∗ and with models of ZFC∗ . The addition of the symbol K has to do with our treatment of extenders and iteration trees in connection with generic extensions, to be explained below.
Extenders Definition 1. By a (κ, λ) extender we mean an object E = $Ea | a ∈ [λ]<ω % so that: (1) each Ea is a κ-complete measure on [κ]|a| ; (2) the measures $Ea | a ∈ [λ]<ω % are compatible; (3) E is normal; (4) E is ω-complete; and (5) K(E). Conditions (1)–(4) in Definition 1 follow Martin–Steel [19, §1A] and we refer the reader to that paper for more precise details. Condition (5) is phrased in the language L∗ . It places the demand that E belongs to the class {x | K(x)}. This class may encompass the entire universe, or all objects in the universe which satisfy conditions (1)–(4), and in such cases condition (5) is vacuous. But there are cases where we have to make the class more restrictive. If E is an extender in M then we use Ult(M, E) to denote the ultrapower of M by E and use iEM : M → Ult(M, E) to denote the ultrapower embedding. We shall not go into the details of the ultrapower construction here. The reader may find these in Martin–Steel [18]. Note that the construction must be adapted to L∗ , but the adaptation is trivial: for x = [f, a] ∈ Ult(M, E) set K Ult(M,E) (x) iff K M (f (u)) for Ea a.e. u. Given a ZFC∗ model M and an ordinal ρ we use M&ρ to denote VρM . We say that two models M and N agree to ρ just in case that M&ρ = N&ρ. If E is a (κ, λ) extender in M, and N agrees with M to κ + 1, then Ult(N, E) makes sense and we use iEN : N → Ult(N, E) to denote the corresponding embedding. We use StrengthM (E), the strength of E in M, to denote the largest ρ so that M and Ult(M, E) agree to ρ. E is α-strong if its strength is at least α.
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We use crit(E), the critical point of E, to denote the critical point of iEM , namely the least ordinal γ so that iEM (γ ) = γ . Following Martin–Steel [19], Definition 1 has a built-in restriction to short extenders—that is extenders measuring subsets of their critical point and no more. The restriction comes in through condition (1) which forces the critical point of a (κ, λ) extender to be precisely κ, never less. A more general definition can be found in Kanamori [11, §26], but we do not use these more general extenders at all in this book. The built-in restriction to short extenders means that the level of agreement needed between M and N to make sense of the ultrapower of N by an extender E ∈ M is crit(E) + 1. The restriction also forces our extenders to always be at or below superstrong, in the sense given by: Fact 2. Let E be an extender in M and let κ be the critical point of E. Then StrengthM (E) ≤ iEM (κ). Let E be a (κ, λ) extender of a model M. We say that E is countable in V if λ and (2κ )M are both countable in V. Literally this is equivalent to the statement that E consists of countably-in-V many measures, and each of these measures is itself countable in V. An embedding h : N → Q is said to preserve countability just in case that: (1) for every θ which is inaccessible in N , if θ is countable in V then so is h(θ ); and (2) for every δ ∈ N , if (2δ )N is countable in V then so is (2h(δ) )Q . Clause (1) implies that if θ is Woodin in N and countable in V then h(θ ) is countable in V. Clause (2) implies that if δ is Woodin in N and N&δ + 1 is countable in V then Q&h(δ) + 1 is countable in V. All the uses of preservation of countability in the book are made through these two consequences. Fact 3. Let M and N be models of ZFC∗ . Let E be an extender of M. Suppose that M and N agree past crit(E). Suppose that E is countable in V. Then iEN : N → Ult(N, E) preserves countability. Claim 4. The family of embeddings which preserve countability is closed under compositions and countable direct limits. # All the ultrapower embeddings we work with in this book match the settings of Fact 3. It follows that these embeddings, and their compositions and countable direct limits—for example the ones obtained through iteration trees, see below—preserve countability.
Interpretation of K in generic extensions Let M ⊂ M ∗ be models of (some sufficiently large part of) ZFC∗ . Let E = $Ea | a ∈ [λ]<ω % be an extender in M. The fattening of E to M ∗ is the object E ∗ = $Ea∗ | a ∈
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[λ]<ω % defined by setting, for each a ∈ [λ]<ω and each A∗ ∈ M ∗ , 1 if (∃A ∈ M)(A ⊂ A∗ ∧ Ea (A) = 1), and Ea∗ (A∗ ) = 0 otherwise. Let P be a forcing notion in M. Let G be P-generic/M. We adapt the definition of a forcing extension to the language L∗ by setting K M[G] (x) iff x is the fattening to M[G] of an extender of M. Remark 5. We say that K has a trivial interpretation in M just in case that {x ∈ M | K M (x)} contains all x which satisfy conditions (1)–(4) of Definition 1 in M. Note that even if K has a trivial interpretation in M, it need not have a trivial interpretation in M[G]. There may very well be objects E ∗ ∈ M[G] which satisfy conditions (1)–(4) in M[G] yet fail to be fattenings of extenders of M. Claim 6. Let E be an extender in M with crit(E) > cardM (P). Let E ∗ be the fattening of E to M[G]. Then E ∗ is an extender in M[G]. Moreover, Ult(M[G], E ∗ ) = extends iEM . Ult(M, E)[G], and iEM[G] ∗ Proof. Condition (5) of Definition 1 holds for E ∗ trivially through our definition of K M[G] . The argument showing that E ∗ satisfies conditions (1)–(4) in M[G] is standard using the fact that P is a small forcing relative to crit(E). The same standard argument shows that Ult(M[G], E ∗ ) and Ult(M, E)[G] have the same universe, and that iEM[G] ∗ extends iEM . Using this and Ło´s Theorem it’s easy to check further that K Ult(M[G],E K Ult(M,E)[G] .
∗)
= #
Claim 7. Let E be an extender in M with crit(E) > cardM (P). Let N be a model of ZFC∗ and suppose that N agrees with M past crit(E). (In particular P ∈ N, G is P-generic/N , and N[G] agrees with M[G] past crit(E).) Let E ∗ be the fattening of E to M[G]. Then Ult(N [G], E ∗ ) = Ult(N, E)[G] and iEN∗[G] extends iEN .
Proof. Similar to the proof of Claim 6.
#
Claim 8. Every extender in M[G] is a fattening of an extender in M. Proof. Immediate using the definition of K M[G] , and using condition (5) in Definition 1. # Claim 8 would fail if it weren’t for the addition of condition (5) to Definition 1 (and implicitly the addition of K to our language).
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Iteration trees An iteration tree T of length α on M consists of a tree order T on α,1 and objects $Eξ | ξ + 1 < α% and $Mξ , jζ,ξ | ζ T ξ < α% satisfying the following conditions: (1) M0 = M. (2) For each ξ so that ξ + 1 < α, Eξ is either equal to “pad” or else it is an extender of Mξ . (3) If Eξ is an extender of Mξ then Mξ +1 = Ult(Mζ , Eξ ) and jζ,ξ +1 : Mζ → Mξ +1 is the ultrapower embedding, where ζ ≤ ξ is the T -predecessor of ξ +1. (Implicit in this definition of Mξ +1 and jζ,ξ +1 is the demand that Mξ and Mζ agree past crit(Eξ ).) If Eξ =“pad” then the T -predecessor of ξ + 1 is ξ , Mξ +1 = Mξ , and jξ,ξ +1 is the identity. (4) For limit λ < α, Mλ is the direct limit of the system $Mζ , jζ,ξ | ζ T ξ < λ%, and jζ,λ : Mζ → Mλ for ζ T λ are the direct limit embeddings. (5) The remaining embeddings jζ,ξ for ζ T ξ < α are obtained through composition. Notice that we allow padding in iteration trees. This provides some extra flexibility that is often useful for indexing in our constructions. Of course the padding can always be removed, possibly making the iteration tree shorter. A cofinal branch through T is a set b ⊂ α, linearly ordered by T , downward closed under T , and cofinal in α. Given a cofinal branch b we use Mb to denote the direct limit of the system $Mζ , jζ,ξ | ζ T ξ ∈ b% and use jζ,b : Mζ → Mb to denote the direct limit embeddings. jb stands for j0,b , embedding M = M0 into Mb . We refer to Mb as the direct limit along b, and to jb as the direct limit embedding along b. b is a wellfounded cofinal branch just in case that Mb is wellfounded. These definitions are all taken from Martin–Steel [19], and we refer the reader to that paper for more details. Remark 9. For the most part in this book we construct iteration trees of length ω. We describe these trees informally, by specifying En and the T -predecessor of n + 1 for each n < ω. These objects, and the starting model M, determine the tree completely. Let P be a forcing notion in M and let G be P-generic/M. Let T ∗ be an iteration ∗ . tree on M[G] with tree order T ∗ , extenders Eξ∗ , models Mξ∗ , and embeddings jζ,ξ T ∗ is called a fattening of T just in case that T ∗ =T , Mξ∗ = Mξ [G] for each ξ < lh(T ), Eξ∗ equals the fattening of Eξ to Mξ [G] for each ξ so that ξ + 1 < lh(T ), ∗ ⊃j and jζ,ξ ζ,ξ for all ζ T ξ . Using Claims 7 and 8 we obtain the following: Claim 10. Suppose that all the critical points used in T are above cardM (P). Then T can be fattened to an iteration tree on M[G]. # 1 See Section 7B for the definition of a tree order.
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Claim 11. Every iteration tree on M[G] with critical points above cardM (P) is a fattening of an iteration tree on M. In particular, if G collapses cardM (P) to ω then all iteration trees on M[G] are fattenings of iteration trees on M. # These two claims allow us to regard iteration trees on generic extensions of M as iteration trees on M and vice versa. We switch between the two viewpoints as needed throughout the book, and often use the same letter T to stand for both an iteration tree on M and the fattened tree on M[G].
Iterability The iteration trees constructed in this book are all normal and plus 2 in the sense of Martin–Steel [19]. These technical restrictions on the trees are needed for the following theorem: Theorem 12 (Martin–Steel [19]). Let π : M → Vθ be elementary where M is countable and Vθ is some rank initial segment of V.2 Let T be a normal plus 2 iteration tree of length ω on M. Then there is a cofinal branch b through T , and an embedding σ : Mb → Vθ , so that σ ◦ jb = π. (Note that b is then a wellfounded branch, since Mb embeds into Vθ .) # A weak iteration of M of length α consists of objects Mξ , Tξ , bξ for ξ < α and embeddings jζ,ξ : Mζ → Mξ for ζ < ξ < α, so that: (1) M0 = M. (2) For each ξ < α, Tξ is a normal, plus 2 iteration tree of length ω on Mξ ; bξ is a cofinal branch through Tξ ; Mξ +1 is the direct limit along bξ ; and jξ,ξ +1 : Mξ → Mξ +1 is the direct limit embedding along bξ . (3) For limit λ < α, Mλ is the direct limit of the system $Mξ , jζ,ξ | ζ < ξ < λ% and jζ,λ : Mζ → Mλ are the direct limit embeddings. (4) The remaining embeddings jζ,ξ are obtained by composition. A weak iteration is thus a linear composition of blocks. Each block is generated by a length ω iteration tree and a cofinal branch through it. In the weak iteration game on M players “good” and “bad” collaborate to produce a weak iteration of M, of length ω1V . “Bad” plays the iteration trees Tξ and “good” plays the branches bξ . (These moves determine the iteration completely.) If ever a model Mξ , ξ < ω1 , is reached which is illfounded, then “bad” wins. Otherwise “good” wins. M is weakly iterable if “good” has a winning strategy in the weak iteration game. 2 The actual universe V is expanded to the language L∗ through the trivial assignment K V (x) for all x.
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Claim 13 (Martin–Steel [19]). Let π : M → Vθ be elementary where M is countable and Vθ is some rank initial segment of V. Then “good” has a winning strategy in the weak iteration game on M. Proof. Immediate through iterated applications of Theorem 12. The good player should simply keep choosing branches given by Theorem 12, successively embedding each # Mξ +1 into Vθ , and preserving commutativity which is needed for the limits. A mild iteration of M of length α consists of objects Mξ , Tξ , bξ , Qξ , kξ , Fξ for ξ < α and embeddings jζ,ξ : Mζ → Mξ for ζ < ξ < α, so that: (1) M0 = M. (2) For each ξ < α, Tξ is a normal, plus 2 iteration tree of length ω on Mξ ; bξ is a cofinal branch through Tξ ; Qξ is the direct limit along bξ ; and kξ : Mξ → Qξ is the direct limit embedding along bξ . (3) Fξ is either an extender of Qξ with crit(Fξ ) < crit(kξ ), or equal to “undefined.” If Fξ =“undefined” then Mξ +1 is equal to Qξ and jξ,ξ +1 is equal to kξ . If Fξ is an extender then Mξ +1 = Ult(Mξ , Fξ ) and jξ,ξ +1 : Mξ → Mξ +1 is the ultrapower embedding. (4) For limit λ < α, Mλ is the direct limit of the system $Mξ , jζ,ξ | ζ < ξ < λ% and jζ,λ : Mζ → Mλ are the direct limit embeddings. (5) The remaining embeddings jζ,ξ are obtained by composition. A mild iteration is thus a linear composition of blocks. Each block is either generated simply by a length ω iteration tree and a branch through it, or generated by a length ω iteration tree, a branch through it, and an additional ultrapower. The two possibilities are illustrated in Diagram 1. jζ,ζ +1 M0
kk Sk / Mζ k Sk SSSk / ζ Tζ
Qζ
Fζ
#
Mζ +1
k / Mξ kSkSkSkS / Qξ S kξ Tξ
=
Mξ +1
/
Diagram 1. A mild iteration.
In the mild iteration game on M players “good” and “bad” collaborate to produce a mild iteration of M, of length ω1V . “Bad” plays Tξ and Fξ for each ξ , and “good” plays the branches bξ . (These moves determine the iteration completely.) If ever a model Mξ , ξ < ω1 , is reached which is illfounded, then “bad” wins. Otherwise “good” wins. M is mildly iterable if “good” has a winning strategy in the mild iteration game. Claim 14. Let M, π , T , and b be as in Theorem 12. Let Q be the direct limit along b and let k : M → Q be the direct limit embedding. Let F be an extender of Q with crit(F ) < crit(k). Let M ∗ = Ult(M, F ) and let j : M → M ∗ be the ultrapower embedding. Then there is an embedding σ ∗ : M ∗ → Vθ so that σ ∗ ◦ j = π .
A Extenders, generic extensions, and iterability
307
Proof. Let σ : Q → Vθ be given by Theorem 12. We have σ ◦ k = π. It follows that π and σ agree through crit(k), and therefore past crit(F ). Using this agreement, a standard argument allows copying the ultrapower of M by F to an ultrapower of Vθ by σ (F ). The copying argument produces the embedding τ illustrated in the left part of Diagram 2 with the commutativity τ ◦ j = h ◦ π where h is the ultrapower embedding of Vθ by σ (F ). Using τ and the ω-completeness of σ (F ) in V a standard argument now produces σ ∗ . / Ult(Vθ ,σ (F )) cGG GG τ σ GG G GG G T kk G k Sk k SSSS / Q M∗ k F <
O cG
Vθ π M
G
j
O hQQQQ QQQσ ∗ QQQ π QQQ T kk Q k k S / k S M M∗ SSS k Q F < Vθ
j
Diagram 2. The proof of Claim 14 (left) and the end result (right).
#
Claim 15. Let π : M → Vθ be elementary where M is countable and Vθ is some rank initial segment of V. Then “good” has a winning strategy in the mild iteration game on M. Proof. Immediate through iterated applications of Theorem 12, in cases where # Fξ =“undefined,” and Claim 14 in cases where Fξ is defined. In the ( full ) iteration game on M players “good” and “bad” collaborate to construct an iteration tree T of length ω1V +1 on M. “bad” plays all the extenders, and determines the T -predecessor of ξ +1 for each ξ . “good” plays the branches {ζ | ζ T λ} for limit λ, thereby determining the limit models Mλ . (Note that “good” is also responsible for the final move, which determines MωV .) If ever a model along the tree is reached which is 1 illfounded then “bad” wins. Otherwise “good” wins. M is ( fully) iterable if “good” has a winning strategy in this game. An iteration strategy for M is a strategy for the good player in the iteration game on M. Weak and mild iteration strategies are defined similarly. Notions of iterability including weak and full were introduced by Martin–Steel [19]. Their results summarized above show how to obtain weakly and mildly iterable models. Proofs of full iterability are substantially more complicated. They seem to require fine structure, and to date they require also some limitation on the large cardinals involved. There have been several results progressively pushing this limitation further up, starting with the work of Martin–Steel [19] which obtains iterability for models with a Woodin cardinal. For our purposes in this book the following theorem of Neeman [31] reaches a sufficiently large cardinal. Theorem 16 (Neeman [31]). Suppose that there is a Woodin limit of Woodin cardinals in V, say δ, and suppose that (Vδ ) exists. Then there is a class model M of ZFC∗ and
308
A Extenders, generic extensions, and iterability
some δ¯ ∈ M so that: δ¯ is a Woodin limit of Woodin cardinals in M; δ¯ is countable in V; and M is fully iterable. # We conclude the appendix with a brief index to the uses of iterability in this book. Let X be a large cardinal axiom. Suppose that X holds in some rank initial segment Vθ of V. Using Claim 13 it follows that there is a countable model M so that X holds in M and M is weakly iterable. If in addition (Vθ ) exists in V then one can expand the argument and produce a class model M of ZFC∗ so that: ¯ • M = L(VθM ¯ ) for some θ ∈ M; • X holds in M (in fact in VθM ¯ ); • VθM ¯ is countable in V; and • M is weakly iterable. Models of this kind, with X standing for the existence of a certain number of Woodin cardinals, suffice for the determinacy proofs in Chapter 2. The constructions in Chapter 3 require mild iterability. Models satisfying the assumptions in Chapter 3 can be obtained from sharps for appropriate large cardinals in V, using Claim 15. For the determinacy in Chapter 7 even mild iterability is not enough. The iteration trees there are non-linear compositions of blocks, similar to the ones in mild iterations except that Fξ may be applied to Mζ for ζ ≤ ξ , not only to Mξ . Such trees are as complicated as the ones in the full iteration game. The construction therefore demands a model which is fully iterable. For the large cardinal assumption in Theorem 7F.4 the necessary model can be obtained assuming a sharp for a Woodin limit of Woodin cardinals in V, using Theorem 16.
Bibliography
[1] John W. Addison and Yiannis N. Moschovakis. Some consequences of the axiom of definable determinateness. Proc. Nat. Acad. Sci. U.S.A., 59:708–712, 1968. [2] David Blackwell. Infinite games and analytic sets. Proc. Nat. Acad. Sci. U.S.A., 58:1836–1837, 1967. [3] Douglas Burke and Ernest Schimmerling. Handwritten notes from a course taught by W. Hugh Woodin at UC Berkeley in the Spring of 1990. [4] Morton Davis. Infinite games of perfect information. In Advances in game theory, pages 85–101. Princeton University Press, Princeton, N.J., 1964. [5] Qi Feng, Menachem Magidor, and Hugh Woodin. Universally Baire sets of reals. In Set theory of the continuum (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ. 26, pages 203–242. Springer-Verlag, New York, 1992. [6] Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988. [7] Matthew Foreman. Potent axioms. Trans. Amer. Math. Soc., 294(1):1–28, 1986. [8] David Gale and Frank M. Stewart. Infinite games with perfect information. In Contributions to the theory of games, vol. 2, Annals of Mathematics Studies 28, pages 245–266. Princeton University Press, Princeton, N. J., 1953. [9] Leo Harrington. Analytic determinacy and 0 . J. Symbolic Logic, 43(4):685–693, 1978. [10] Thomas Jech. Set theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. [11] Akihiro Kanamori. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. [12] Alexander S. Kechris and W. Hugh Woodin. Equivalence of partition properties and determinacy. Proc. Nat. Acad. Sci. U.S.A., 80(6 i.):1783–1786, 1983. [13] Alexander S. Kechris. Homogeneous trees and projective scales. In Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79), Lecture Notes in Math. 839, pages 33–73. Springer-Verlag, Berlin, 1981. [14] Kenneth Kunen. Saturated ideals. J. Symbolic Logic, 43(1):65–76, 1978.
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[15] Kenneth Kunen. Set theory. An introduction to independence proofs. Studies in Logic and the Foundations of Mathematics 102. North-Holland Publishing Co., Amsterdam, 1980. [16] Casimir Kuratowski. Sur les théorèmes de séparation dans la Théorie des ensembles. Fund. Math., 26:183–191, 1936. [17] Menachem Magidor. Precipitous ideals and 14 sets. Israel J. Math., 35(1–2): 109–134, 1980. [18] Donald A. Martin and John R. Steel. A proof of projective determinacy. J. Amer. Math. Soc., 2(1):71–125, 1989. [19] Donald A. Martin and John R. Steel. Iteration trees. J. Amer. Math. Soc., 7(1):1–73, 1994. [20] Donald A. Martin. The axiom of determinateness and reduction principles in the analytical hierarchy. Bull. Amer. Math. Soc., 74:687–689, 1968. [21] Donald A. Martin. Measurable cardinals and analytic games. Fund. Math., 66:287–291, 1969/1970. [22] Donald A. Martin. Borel determinacy. Ann. of Math. (2), 102(2):363–371, 1975. [23] Donald A. Martin. Infinite games. In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 269–273, Helsinki, 1980. Acad. Sci. Fennica. [24] Donald A. Martin. The largest countable this, that, and the other. In Cabal seminar 79–81, Lecture Notes in Math. 1019, pages 97–106. Springer-Verlag, Berlin, 1983. [25] Donald A. Martin. A purely inductive proof of Borel determinacy. In Recursion theory (Ithaca, N.Y., 1982), Proc. Sympos. Pure Math. 42, pages 303–308. Amer. Math. Soc., Providence, RI, 1985. [26] Yiannis N. Moschovakis. Descriptive set theory. Studies in Logic and the Foundations of Mathematics 100. North-Holland Publishing Co., Amsterdam, 1980. [27] Jan Mycielski and Hugo Steinhaus. A mathematical axiom contradicting the axiom of choice. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 10:1–3, 1962. ´ [28] Jan Mycielski and Stanisław Swierczkowski. On the Lebesgue measurability and the axiom of determinateness. Fund. Math., 54:67–71, 1964. [29] Itay Neeman. Optimal proofs of determinacy. Bull. Symbolic Logic, 1(3):327–339, 1995. [30] Itay Neeman. Large cardinals and the determinacy of long games. PhD thesis, UCLA, 1996.
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Index A (auxiliary games map), 15, 23 definition, 21–23 Amix (mixed pivot games map), 43, 50 definition, 43–44 exact rules, 44–46 first fifth of a mega-round, 222 useful positions, 48 Apiv (pivot games map), 28, 30 definition, 28–30 acceptable obstruction, 156, 157 I-acceptable, 168 II-acceptable, 167, 168 agreement between models, 301 amenable witness to t when λ = rdm(t) is a limit, 140 when λ = rdm(t) is a successor, 139 annotated position, 137 auxiliary games map, 23 mirrored, 38 avoids C˙ and D˙ (δ-sequence which) for δ a Woodin limit of Woodin cardinals, 194 for δ ∈ W , 176 B (mirrored auxiliary games map), 37 Bpiv (mirrored pivot games map), 38 Back(δ, δ ∗ )(C˙ ∗ ), see pullback Back I and Back II , 167 Baire, see universally Baire set basic axiom, 146 basic identity, 141–142 branch through a tree order, 271 captured pointclass, 295 clear annotated position, 151 code (and λ-code), 138–139 code (in Chapter 3) for a ν-position, 88 cofinal branch through a tree order, 271
cofinal branch through an iteration tree, 304 continuously coded length, 87 countability preserving embeddings, 302 countable extender, 302 critical point (of an extender), 302 (δ, δ ∗ )-pullback, see pullback δ-name for δ a Woodin limit of Woodin cardinals, 162 for δ ∈ W , 155 δ-sequence for δ a Woodin limit of Woodin cardinals, 162 for δ ∈ W , 155 determinacy games ending at ω1 in L of the play, 293 games ending at ω1 in of the play, 296, 297 games of continuously coded length, 119, 120, 127, 128 games of fixed countable length, 53, 73, 85, 86 games with homogeneously Suslin payoff, 86, 128 games with universally Baire payoff, 83, 85, 127, 297 ordinal definable games of length ω1 (consistency result), 298 12 sets, 39 e(δ), 135 early end branch , in a limit mega-round of G 213 in a successor mega-round of skip , 218 G
314
Index
elastic type, 18 even branch, 15, 27 in runs of Amix , 47 exceeding type, 18 expansion of a reduced position, 249 extender, 301 countable, 302 critical point, 302 fattening, 302 short, 302 strength, 301 with respect to H , 19 extender algebra, 148 chain condition, 148 definition, 143–148 generics, 151 history, 8, 154 extender axiom, 147 extension (and a-extension) for steps, 177 extension (and n-extension) for witnesses, 137, 140 extension of a tot to length α + 0.2, 276 fattening of an extender, 302 of an iteration tree, 304 first fifth of a mega-round of Rmix [x], 222 full iterability, 307–308 functor, 156 (game quantifier), 83, 85, 127 branch G definition, 209–214 phantom limit case, 212 standard limit case, 212–214 successor case, 210–211 early end in a limit mega-round, 213 history of a position, 215 leap, 213 outcome of a position, 215 position, 215
cont , 124–126 G Gfixed , 52 local , 268–269 G skip G definition, 218–219 early end in a successor mega-round, 218 expansion of a reduced position, 249 position, 249 reduced position, 249 skip, 219 G(t) definition, 150 genericity, 151 games ending at ω1 in L of the play definition, 292 determinacy, 293 games ending at ω1 in of the play definition, 295 determinacy, 296, 297 games of continuously coded length definition, 87 determinacy, 119, 120, 127, 128 games of fixed countable length definition, 51 determinacy, 53, 73, 85, 86 generic run, 15, 23 in mirrored auxiliary games, 38 generic strategies map, 27 mirrored, 38 genericity iteration, 43, 154 cont , 126 H fixed , 53 H Hlocal , 269 height of an identity, 142 branch , 215 history of a position in G hopeful δ-sequence wrt Y˙ , 162 identity, 142 indiscernibles, see local indiscernibles inference (on identities), 143–144 interpretation of K
Index
in generic extensions, 302 in ultrapowers, 301 iseq (identities sequence), 151 realization, 151 iterability, 305–308 full, 307 mild, 306 uses of, 308 weak, 305 iteration game full, 307 mild, 306 weak, 305 iteration strategy, 307 iteration tree, 304 fattening, 304 nice, 27 normal, 29 padding, 304 using only extenders from below δ, 29 jumpM (δ), 20 K, 301 interpretation, see interpretation of K K U , 273 (κ, n)-type, see type L, 136 L(t (κ)), see L(w) L(w), 136 leap, 213 length of a skip frame, 250 <ω2 − 11 sets, 51 linear strand, see strand local indiscernibles, 20 locally definable strategy, 291 Martin’s auxiliary game, 74–75 related facts, 75, 80 merge of a tot, 272 mild iterability, 306–307
315
minimal obstruction, 152 mirrored auxiliary games map, pivot games map, and strategies map, 37–38 pullback, 167–174 mixed pivot, 43, 47 residing above λ, 50 useful, 48 mixed pivot games map, 50 mixed pivot strategies map, 50 -name, see δ-name nice annotated position, 176 nice δ-sequence, 183 nice iteration tree, 27 normal iteration tree, 29 ν-position, 87 ν-run, 88 obstruction, 151 acceptable, 156, 157 minimal, 152 obstruction free annotated position, 151 odd branch, 15, 28 in runs of Amix , 47 One Step Lemma, 20 ⊕, see u ⊕ v otrs(δ), 291 branch , 215 outcome of a position in G padding (in iteration trees), 304 phantom limit in a regular tot, 273 branch , 212 in G ˙ 175, 208 ϕini (δ, C), 12 determinacy, 39 pivot, 15, 28 in mirrored auxiliary games, 38 mixed, see mixed pivot residing above λ, 36 pivot games map, 30 mirrored, 38 pivot strategies map, 35
316
Index
mirrored, 38 branch , 215 position in G position in Gskip , 249 preservation of countability (for embeddings), 302 (u), 17 projm τ ˙ 175, 208 ψini (δ, D), pullback, 155 compositions and summary, 166–167 mirrored, 174 from a relative limit, 158–162 mirrored, 169–171 from a relative successor, 165 mirrored, 173–174 from a Woodin limit of Woodin cardinals, 162–165 mirrored, 172–173 rdm(t), 137 real part of an annotated position, 141 realizable branch through an iteration tree, 83 red(P , f ), 251 skip , 249 reduced position in G regular tot, 272, 277 relative domain, 135, 137 relative limit, 135 relative successor, 135 residing above λ mixed pivot, 50 pivot, 36 root, 47 route, 251 saturated annotated position, 177–178 saturated δ-sequence, 183 seals Y˙ (ordinal which), 154 -sequence, see δ-sequence, see U-sequence short extender, 302 σgen (generic strategies map), 27 σmix (mixed pivot strategies map), 48, 50 construction, 48–49 properties, 49–50
σpiv (pivot strategies map), 28, 35 construction, 30–35 properties, 35–36 skip, 219 skip frame, 250 skipping game, 218 standard limit in a regular tot, 273 branch , 212 in G , see x y step, 177 strand, 276 strength (of an extender), 301 with respect to H , 19 StretchE λ (u), 18 subtype, 17 suitable annotated position, 156 I-suitable, 168 II-suitable, 167 supernice δ-sequence, 201 t˙[G], 154 t, w, see code t−−, w, y, 138 τgen (mirrored generic strategies map), 38 τpiv (mirrored pivot strategies map), 38 tot (tree of trees), 271 consistent with , 272 extended to length α + 0.2, 276 regular, 272 tree order, 270 trunc(P, b, n), 227 truth value of an identity, 142–143 type, 16 domain, 16 elastic, 18 exceed, 18 of x0 , . . . , xn−1 in Vη , 17 projection, 17 realizable, 17 stretch, 18 sub-, 17 u− , 17
Index
u− (in connection with types), 17 u ⊕ v, 161 U-sequence, 275 universally Baire set, 82 propagation of u.B. representations through the projective hierarchy, 85 under applications of cont , 127 under applications of ω , 84 under applications of ω·α , 85 used in a witness, 136 useful runs and positions in Amix , 48 W , 135 Wθ (see also extender algebra), 148
317
weak iterability, 305–306 wellfounded branch through an iteration tree, 304 witness, 136–137 amenable, limit λ, 140 amenable, successor λ, 139 n-extension, 140 Woodin cardinal, 19 x y, 139 z(t), 141
ZFC∗ (see also K, interpretation of K),
301
