The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal 9783110804737

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de Gruyter Series in Logic and Its Applications 1 Editors: W. A. Hodges (London) · R. Jensen (Berlin) S. Lempp (Madison) · M. Magidor (Jerusalem)

1749

1

sr

1999

W. Hugh Woodin

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

W G DE

Walter de Gruyter Berlin · New York 1999

Author W. Hugh Woodin Department of Mathematics University of California Berkeley, CA 94720-3840 USA Series Editors Wilfrid A. Hodges School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London El 4NS, United Kingdom

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

Steffen Lempp Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388, USA

Menachem Magidor Institute of Mathematics The Hebrew University Givat Ram 91904 Jerusalem, Israel

1991 Mathematics Subject Classification:

03-02; 03E05, 03E15, 03E35, 03E50, 03E55

Keywords: Axiom of determinacy, forcing axioms, nonstationary ideal @> Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability

Library of Congress - Cataloging-in-Publication

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Woodin, W. H. (W. Hugh) The axiom of determinacy, forcing axioms, and the nonstationary ideal / W. Hugh Woodin. p. cm. - (De Gruyter series in logic and its applications ; 1) Includes bibliographical references and index. ISBN 3-11-015708-X (alk. paper) 1. Forcing (Model theory). I. Title. II. Series. QA9.7.W66 1999 511.3-DC21 99-23307 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication

Data

Woodin, W. Hugh: The axiom of determinacy, forcing axioms and the nonstationary ideal / W. Hugh Woodin. - Berlin ; New York : de Gruyter, 1999 (De Gruyter series in logic and its applications ; 1) ISBN 3-11-015708-X

ISSN 1438-1893 © Copyright 1999 by Walter de Gruyter G m b H & Co. KG, D-10785 Berlin. 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 Author's T g X files: I. Zimmermann, Freiburg - Printing and binding: WB-Druck G m b H & Co., Rieden/Allgäu - Cover design: Rainer Engel, Berlin.

Contents

1

Introduction 1.1 The nonstationary ideal on a>\ 1.2 The partial order P m a x 1.3 P m a x variations 1.4 Extensions of inner models beyond L(R) 1.5 Concluding remarks

1 2 6 10 13 14

2

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

18 18 28 31 33 37 40

3

The nonstationary ideal 3.1 The nonstationary ideal and á j 3.2 The nonstationary ideal and CH

48 48 106

4

The P max -extension 4.1 Iterable structures 4.2 The partial order P m a x

115 115 135

5

Applications 5.1 The sentence 0AC 5.2 Martin's Maximum, 2) 5.3 The sentence ^AC 5.4 The stationary tower and P m a x 5-5 PJnax 5-6 P i x 5.7 The Axiom (*) 5.8 Homogeneity properties of ^P(o)i)/l NS

184 184 187 197 204 226 237 243 279

Pmax variations

291

6

6.1

6.2

2

Pmax

292

Variations for obtaining a>\ -dense ideals

310

6.2.1

Qmax

310

6-2-2 6.2.3

QLx 2 Qmax

338 374

vi

Contents

6.3

6.2.4

Weak Kurepa trees and Qmax

6.2.5

KT

Qmax

6.2.6 Null sets and the nonstationary ideal Nonregular ultrafilters οηωι

381 388

409 427

7

Conditional variations 7.1 Susiin trees 7.2 The Borei Conjecture

432 432 447

8

φ principles for ωι 8.1 Condensation Principles

498 501

8-2 8.3

507 585

9

P*ax ; ;+ The principles, ANSc and A NSc 1

1

Extensions of L (Γ, R) 9.1 AD+ 9.2 The P max -extension of Ζ,(Γ,Κ) 9.2.1 The basic analysis 9.2.2 Martin's Maximum + + (c) 9.2.3 ο ω (ω2) 9.3 The Qmax-extension of L(T, R) 9.4 Chang's Conjecture 9.5 Weak and Strong Reflection Principles 9.6 Strong Chang's Conjecture 9.7 Ideals on o>2

617 618 626 627 631 642 646 649 664 680 697

10 Further results 10.1 Forcing notions and large cardinals 10.2 Coding into L{iP(œ\)) 10.2.1 Coding by sets, S 10.2.2 Q ® 10.2.3 P ( Z 10.2.4 pg¡¿> 10.3 Bounded forms of Martin's Maximum 10.4 Ω-logic 10.5 Ω-logic and the Continuum Hypothesis 10.6 The Axiom (*)+

769 769 776 778 783 815 846 862 886 894 908

10.7 The Effective Singular Cardinals Hypothesis

916

11 Questions

921

Bibliography

927

Index

931

Chapter 1

Introduction

The main result of this book is the identification of a canonical model in which the Continuum Hypothesis (CH) is false. This model is canonical in the sense that Gödel's constructible universe L and its relativization to the reals, L(M), are canonical models though of course the assertion that L(R) is a canonical model is made in the context of large cardinals. Our claim is vague, nevertheless the model we identify can be characterized by its absoluteness properties. This model can also be characterized by certain homogeneity properties. From the point of view of forcing axioms it is the ultimate model at least as far as the subsets of ω\ are concerned. It is arguably a completion of ¡Ρ(ωι), thepowerset of ω\. This model is a forcing extension of L{R) and the method can be varied to produce a wide class of similar models each of which can be viewed as a reduction of this model. The methodology for producing these models is quite different than that behind the usual forcing constructions. For example the corresponding partial orders are countably closed and they are not constructed as forcing iterations. We provide evidence that this is a useful method for achieving consistency results, obtaining a number of results which seem out of reach of the current technology of iterated forcing. The analysis of these models arises from an interesting interplay between ideas from descriptive set theory and from combinatorial set theory. More precisely it is the existence of definable scales which is ultimately the driving force behind the arguments. Boundedness arguments also play a key role. These results contribute to a curious circle of relationships between large cardinals, determinacy, and forcing axioms. Another interesting feature of these models is that although these models are generic extensions of specific inner models (L (R) in most cases), these models can be characterized without reference to this. For example, as we have indicated above, our canonical model is a generic extension of L(R). The corresponding partial order we denote by · In Chapter 5 we give a characterization for this model isolating an axiom The formulation of (*) does not involve P ma x, nor does it obviously refer to L(R). Instead it specifies properties of definable subsets of ίΡ(ω\).

Pmax

C).

The original motivation for the definition of these models resulted from the discovery that it is possible, in the presence of the appropriate large cardinals, to force (quite by accident) the effective failure of CH. This and related results are the subject of Chapter 3. We discuss effective versions of CH below. Gödel was the first to propose that large cardinal axioms could be used to settle questions that were otherwise unsolvable. This has been remarkably successful particularly in the area of descriptive set theory where most of the classical questions have now been answered. However after the results of Cohen it became apparent that large

2

1 Introduction

cardinals could not be used to settle the Continuum Hypothesis. This was first argued by Levy and Solovay (1967). Nevertheless large cardinals do provide some insight to the Continuum Hypothesis. One example of this is the absoluteness theorem of Woodin (1985). Roughly this theorem states that in the presence of suitable large cardinals CH "settles" all questions with the logical complexity of CH. More precisely if there exists a proper class of measurable Woodin cardinals then Σ^ sentences are absolute between all set generic extensions of V which satisfy CH. The results of this book can be viewed collectively as a version of this absoluteness theorem for the negation of the Continuum Hypothesis (->CH).

1.1 The nonstationary ideal on α>ι We begin with the following question. Is there a family {Sa | a < ω2} of stationary subsets of ω\ such that Sa Π Sß is nonstationary whenever α φ β ? The analysis of this question has played (perhaps coincidentally) an important role in set theory particularly in the study of forcing axioms, large cardinals and determinacy. The nonstationary ideal on a>\ is ¿¿^-saturated if there is no such family. This statement is independent of the axioms of set theory. We let i N S denote the set of subsets of i)/Xns is a complete boolean algebra which satisfies the 002 chain condition. Kanamori (1994) surveys some of the history regarding saturated ideals, the concept was introduced by Tarski. The first consistency proof for the saturation of the nonstationary ideal was obtained by Steel and VanWesep (1982). They used the consistency of a very strong form of the Axiom of Determinacy (AD), see (Kanamori 1994) and Moschovakis (1980) for the history of these axioms. Steel and VanWesep proved the consistency of ZFC + "The nonstationary ideal on ω\ is ^-saturated" assuming the consistency of ZF + AD μ + " Θ is regular". A D r is the assertion that all real games of length ω are determined and Θ denotes the supremum of the ordinals which are the surjective image of the reals. The hypothesis was later reduced by Woodin ( 1 9 8 3 ) to the consistency of Z F + A D . The arguments of Steel and VanWesep were motivated by the problem of obtaining a model of ZFC

1.1 The nonstationary ideal on ω\

3

in which ωι is the second uniform indiscernible. For this Steel defined a notion of forcing which forces over a suitable model of AD that ZFC holds (i. e. that the Axiom of Choice holds) and forces both that (xn is the second uniform indiscernible and (by arguments of VanWesep) that the nonstationary ideal on ω ι is α>2-saturated. The method of (Woodin 1983) uses the same notion of forcing and a finer analysis of the forcing conditions to show that things work out over L(R). In these models obtained by forcing over a ground model satisfying AD not only is the nonstationary ideal saturated but the quotient algebra ,P(OJ\ ) / 1 s s has a particularly simple form, 3>(a>i)/lm = RO(Coll(w, < ω2)). We have proved that this in turn implies AD ¿ ( K ) and so the hypothesis used (the consistency of AD) is the best possible. The next progress on the problem of the saturation of the nonstationary ideal was obtained in a series of results by Foreman, Magidor, and Shelah (1988). They proved that a generalization of Martin's Axiom which they termed Martin's Maximum actually implies that the nonstationary ideal is saturated. They also proved that if there is a supercompact cardinal then Martin's Maximum is true in a forcing extension of V. Later Shelah proved that if there exists a Woodin cardinal then in a forcing extension of V the nonstationary ideal is saturated. This latter result is most likely optimal in the sense that it seems very plausible that ZFC + "The nonstationary ideal on ω\ is c&i-saturated" is equiconsistent with ZFC + "There exists a Woodin cardinal" see (Steel 1990). There was little apparent progress on obtaining a model in which a>2 is the second uniform indiscernible beyond the original results of (Steel and VanWesep 1982) and (Woodin 1983). Recall that assuming that for every real x, x§ exists, the second uniform indiscernible is equal to S t h e supremum of the lengths of AÍ, prewellorderings. Thus the problem of the size of the second uniform indiscernible is an instance of the more general problem of computing the effective size of the continuum. This problem has a variety of formulations, two natural versions are combined in the following: • Is there a (consistent) large cardinal whose existence implies that the length of any prewellordering arising in either of the following fashions, is less than the least weakly inaccessible cardinal? - The prewellordering exists in a transitive inner model of AD containing all the reals. - The prewellordering is universally Β aire. The second of these formulations involves the notion of a universally Baire set of reals which originates in (Feng, Magidor, and Woodin 1992). Universally Baire sets are discussed briefly in Section 10.3. We note here that if there exists a proper

4

1 Introduction

class of Woodin cardinals then a set Λ ç M is universally Baire if and only if it is °°-weakly homogeneously Suslin which in turn is if and only if it is 00-homogeneously Suslin. Another relevant point is that if there exist infinitely many Woodin cardinals with measurable above and if A ç R is universally Baire, then L(A,R)

Ν AD

and so A belongs to an inner model of AD. The converse can fail. More generally one can ask for any bound provided of course that the bound is a "specific" ωα which can be defined without reference to 2**°. For example every EÌ, prewellordering has length less than u>2 and if there is a measurable cardinal then every Σ 3 prewellordering has length less than (¿»3. A much deeper theorem of (Jackson 1988) is that if every projective set is determined then every projective prewellordering has length less than ωω. This combined with the theorem of Martin and Steel on projective determinacy yields that if there are infinitely many Woodin cardinals then every projective prewellordering has length less than ωω. The point here of course is that these bounds are valid independent of the size of 2*o. The current methods do not readily generalize to even produce a forcing extension of L ( R ) (without adding reals) in which ZFC holds and ω3 < 0 L ( M ) . Thus at this point it is entirely possible that 3 is the bound and that this is provable in ZFC. If a large cardinal admits an inner model theory satisfying fairly general conditions then most likely the only (nontrivial) bounds provable from the existence of the large cardinal are those provable in ZFC; i. e. large cardinal combinatorics are irrelevant unless the large cardinal is beyond a reasonable inner model theory. For example suppose that there is a partial order Ρ e L ( R ) such that for all transitive models M of A D + containing IK, if G ç IP is M-generic then . (R)MG]

=

(Κ)Λί>

. (¿1)*[G1 = ( ω 3 ) ^ ] ; • L(R)[G]

Ν ZFC,

where S3 is the supremum of the lengths of Δ3 prewellorderings of R. The axiom A D + is a technical variant of AD which is actually implied by AD in many instances. Assuming DC it is implied, for example, by AD m · It is also implied by AD if V = L ( R ) . Suppose that a large cardinal admits a suitable inner model theory. Then the existence of the large cardinal is consistent with ¿3 = 3. The precise assumptions the inner model theory must satisfy are given in (Woodin c). It follows from the results of (Steel and VanWesep 1982) and (Woodin 1983) that such a partial order Ρ exists in the case of more precisely, assuming L(R) Ν AD, there is a partial order Ρ e L (R) such that for all transitive models M of A D + containing R, if G c Ρ is M-generic then

1.1 The nonstationary ideal on ω\

5

• L(R)[G] Ν ZFC. Thus if a large cardinal admits a suitable inner model theory then the existence of the large cardinal is consistent with δ 2 = un- We shall prove a much stronger result in Chapter 3, showing that if δ is a Woodin cardinal and if there is a measurable cardinal above δ then there is a semiproper partial order Ρ of cardinality δ such that V P Ν δ\ = £02. This result which is a corollary of Theorem 1.1, stated below, and Theorem 2.64, due to Shelah, shows that this particular instance of the Effective Continuum Hypothesis is as intractable as the Continuum Hypothesis. Foreman and Magidor initiated a program of proving that < &>2 from various combinatorial hypotheses with the goal of evolving these into large cardinal hypotheses, (Foreman and Magidor 1995). By the (initial) remarks above their program if successful would have identified a critical step in the large cardinal hierarchy. Foreman and Magidor proved among other things that if there exists a (normal) («3-saturated ideal on a>2 concentrating on a specific stationary set then ¿2 2 then δ 2 < cû2An early conjecture of Martin is that = K„ for all η follows from reasonable hypotheses, δ), is the supremum of the lengths of Δ^ prewellorderings. The following theorem "proves" the Martin conjecture in the case of η — 2. Theorem 1.1. Assume that the nonstationary ideal on ω\ is u>2-saturated and that there is a measurable cardinal. Then ài, = a>2 and further every club in ω\ contains a club constructible from a real. • As a corollary we obtain, Theorem 1.2. Assume Martin's Maximum. Then §2 = 002 and every club in a>\ contains a club constructible from a real. • Another immediate corollary is a refinement of the upper bound for the consistency strength of ZFC -I- "For every real x, x* exists." + "2 is the second uniform indiscernible." Assuming in addition that larger cardinals exist then one obtains more information. For example, Theorem 1.3. Assume the nonstationary ideal on \ is a/i-saturated and that there exist ω many Woodin cardinals with a measurable cardinal above them all.

6

1 Introduction (1) Suppose that A ç M, A e L(W), and that there is a sequence (Ba : a < ω\) of borei sets such that A = Ό{Βα I a < ωι}. Then A is Έ\. (2) Suppose that X is a bounded subset of®L® of cardinality ω\. Then there exists a set Y e L( E) of cardinality ωι in L(E) such that X ç Y. •

We note that assuming for every χ e Μ, χ* exists, the statement ( 1 ) of Theorem 1.3 implies that ¿2 = ¿>2; if < o>i then every Σ3 set is an ω\ union of borei sets.

1.2 The partial order P max Theorem 1.3 suggests that if the nonstationary ideal is saturated (and if modest large cardinals exist) then one might reasonably expect that the inner model L(!P{a>\)) may be close to the inner model L{M). However if the nonstationary ideal is saturated one can, by passing to a ccc generic extension, arrange that 3>(R) Ç L(3>(a> 1)) and preserve the saturation of the nonstationary ideal. Nevertheless this intuition was the primary motivation for the definition of P m a x . The canonical model for - C H is obtained by the construction of this specific partial order, P ma x· The basic properties of P m a x are given in the following theorem. Theorem 1.4. Assume AD L ( R ) and that there exists a Woodin cardinal with a measurable cardinal above it. Then there is a partial order P m a x in L(M) such that; (1) Pmax is ω-closed and homogeneous (in L(R)), (2) L(R) Pmax h ZFC. Further if φ is α Π2 sentence in the language for the structure (H(co2),e,lm) and if (H(£02),e,i NS > \=Φ then {H(a>2), e, ! N S ) L ( R ) P m a x 1= φ. The partial order P m a x is definable and thus, since granting large cardinals Th(L(M)) is canonical, it follows that Th(L(R) Pmax ) is canonical. Many of the open combinatorial questions at ω\ are expressible as Π2 statements in the structure (H(an), 6,1ns)

•

1.2 The partial order P max

7

and so assuming the existence of large cardinals these questions are eitherfalse, or they are true in L(R)Pmax. In some sense the spirit of Martin's Axiom and its generalizations is to maximize the collection of Π 2 sentences true in the structure (H(an),e) Indeed ΜΑ ω , is easily reformulated as a Π2 sentence for (H(a>2), e). By the remarks above, assuming fairly weak large cardinal hypotheses, any such sentence which is true in some set generic extension of V is true in a canonical generic extension of L(M). The situation is analogous to the situation of Σ ] sentences and L. By Shoenfield's absoluteness theorem if a Σ ] sentence holds in V then it holds in L. The difference here is that the model analogous to L is not an inner model but rather it is a canonical generic extension of an inner model. This is not completely unprecedented. Mansfield's theorem on Σ ] wellorderings can be reformulated as follows. Theorem 1.5 (Mansfield). Suppose that φ is α Π3 sentence which is true in V and there is a nonconstructible real. Then φ is true in Lr where Ρ is Sacks forcing (defined in L). • Of course the Π3 sentence also holds in L so this is not completely analogous to our situation. ->CH is a (consistent) Π2 sentence for (Η(ω2), e) which is false in any of the standard inner models. Nevertheless the analogy with Sacks forcing is accurate. The forcing notion P m a x is a generalization of Sacks forcing to ω\. The following theorem, slightly awkward in formulation, shows that any attempt to realize in Η (cú2) all suitably consistent Π2 sentences, requires at least Σ ] -Determinacy. Theorem 1.6. Suppose that there exists a model, (Μ, E), such that {M, E) t= ZFC and such that for each Π 2 sentence φ if there exists a partial order Ρ such that (Η(ω2),€)ν?

N0,

then (H(a>2),£){M'E)

ϊφ.

Assume there is an inaccessible cardinal. Then V Ν Σ2-Determinacy.

•

One can strengthen Theorem 1.4 by expanding the structure (Η(ω2), e, 1NS> by adding predicates for each set of reals in L (K). This theorem requires additional large cardinal hypotheses which in fact imply AD ¿ ( R ) unlike the large cardinal hypothesis of Theorem 1.4.

8

1 Introduction

Theorem 1.7. Assume there are ω many Woodin cardinals with a measurable above. Suppose φ is α Π2 sentence in the language for the structure (H(CO2),

e , 1 N S , Χ ; X e L(R),

Ζ ς

M)

and that (H(ù)2), €, 1 NS , Ζ; X e L(R), X c R) |= φ Then (Η(ω2), e, i N S , Χ; X e L(R), X c R)¿«)Pmax

0

We note that since Pmax is &>-closed, the structure (Η(ω2), e, iNS, Χ-, X e L(R), X Q is naturally interpreted as a structure for the language of (Η(ω2), e, 1 NS , Χ; X e L(R), X ç R). The key point is that this strengthened absoluteness theorem has in some sense a converse. Theorem 1.8. Assume AD ¿(K) . Suppose that for each Π2 sentence in the language for the structure (Η(ω2), e, 1 NS , Χ; X e L(R), X ç R> if (Η(ω2), e, 1NS, Χ; X € L(R), X ç R)¿,)) = L(R)[G] which is L(W)-generic.

•

#

If one assumes in addition that R exists then Theorem 1.8 can be reformulated as follows. For each π e ω let Un be a set which is Σι definable in the structure , of these extensions but which cannot be realized in any such structure defined in an extension of L(R). The results of Chapter 9 suggest a strengthening of the axiom (*): • Axiom (*) + : For each set X ç α>2 there exists a set Λ ç l and a filter G ç P m a x such that (1) L(A, R) 1= AD+, (2) G is L(A, R)-generic and X e L(A, R)[G]. This is discussed briefly in Chapter 10 which explores the possible relationships between Martin's Maximum and the axiom (*). One of the theorems we shall prove Chapter 10 shows that in Theorem 1.8, it is essential that the predicate, i N S , for the nonstationary sets be added to the structure. We shall show that Martin's Maximum

++

( c ) + Strong Chang's Conjecture

together with all the Π 2 consequences of (*) for the structure (Η(ω2), Y, e: Y ç R, Y e L(R)) does not imply (*). We shall also prove an analogous theorem which shows that "cofinally" many sets from S3 (R) Π L (R) must be added; for each set Κ0 e S3 (R) Π L (R), Martin's Maximum

++

( c ) + Strong Chang's Conjecture

together with all the Π2 consequences of (*) for the structure (Η(ω2), 1 NS , y 0 , e) does not imply (*). Finally, we shall also show in Chapter 10 that the axiom (*) is equivalent (in the context of large cardinals) with a very strong form of a bounded version of Martin's Maximum++.

1.5 Concluding remarks The following question resurfaces with added significance. • Assume AD L(M) . Is 0 L < R ) < ω 3 ? The point is that if it is consistent to have AD L ( R ) and © L i R ) > 3 then presumably this can be achieved in a forcing extension of L(R). This in turn would suggest there are generalizations of P m a x which produce generic extensions of L(R) in which c > ω There are many open questions in combinatorial set theory for which a (positive) solution requires building forcing extensions in which c > ωι·

1.5 Concluding remarks

15

The potential utility of P m a x variations for obtaining models in which a>3 < 0 i ( R ) is either enhanced or limited by the following theorem of S. Jackson. This theorem is an immediate corollary of Theorem 1.3(2) and Jackson's analysis of measures and ultrapowers in L(R) under the hypothesis of Theorem 1.10 (Jackson). Assume the nonstationary ideal on ω\ is (¡^-saturated and that there exist ω many Woodin cardinals with a measurable cardinal above them all. Then either: (1) There exists κ < &Lm) such that κ is a regular cardinal in L(R) and such that κ is not a cardinal in V, or; (2) There exists a set Λ of regular cardinals, above ωι, such that a) |Λ| = Ki, b) |pcf(«A)| =

•

One of the main open problems of Shelah's pcf theory is whether there can exist a set, Λ, of regular cardinals such that |«A| < |pcf(2, i. e. for the structure Η(ωτ,), and beyond. The first singular case, Η (ω+), seems particularly interesting. There is also the case of a>i but in the context of CH. One interesting (but tentative) result, with, we believe, potential implications for CH, is that there are limits to any possible generalization of the P m a x variations to the context of CH; more precisely, if CH holds then the theory of Η (ω2) cannot be finitely axiomatized over ZFC in Ω-logic. Acknowledgments. Many of the results of the first half of this book were presented in the Set Theory Seminar at UC Berkeley. The (ever patient) participants in this seminar offered numerous helpful suggestions for which I remain quite grateful. I am similarly indebted to all those willingly to actually read preliminary versions of this book and then relate to me their discoveries of mistakes, misprints and relics. I only wish that the final product better represented their efforts. I owe a special debt of thanks to Ted Slaman. Without his encouragement, advice and insight, this book would not exist. The research, the results of which are the subject of this book, was supported in part by the National Science Foundation through a succession of summer research grants, and during the academic year, 1997-1998, by the Miller Institute in Berkeley. Finally I would like to acknowledge the (generous) support of the Alexander von Humboldt Foundation. It is this support which enabled me to actually finish this book.

Chapter 2

Preliminaries We briefly review, without giving all of the proofs, some of the basic concepts which we shall require. In the course of this we shall fix some notation. As is the custom in Descriptive Set Theory, Μ denotes the infinite product space, ωω. Though sometimes it is convenient to work with the Cantor space, 2ω, or even with the standard Euclidean space, (—00, 00). If at some point the discussion is particularly sensitive to the manifestation of M then we may be more careful with our notation. For example L(R) is relatively immune to such considerations, but Wadge reducibility is not. We shall require at several points some coding of sets by reals or by sets of reals. There is a natural coding of sets in Η(ω\) (the hereditarily countable sets) by reals. For example if α e Η (ω ι ) then the set a can be coded by coding the structure

{b U ω, a, e) where b is the transitive closure of a. A real χ codes α if * decodes sets

A ç ω and E ç ω χ ω such that (bU ω, a, e) = {ω, A, E),

where again b is the transitive closure of a. Suppose that M e H (c + ) and let Ν be the transitive closure of M. Fix a reasonable decoding of a set X ç E to produce an element of ί>(Μ) χ ( R ) n i e ( A , R ) ç rl" s

where a is the least ordinal such that La (A, R) 1= ZF~.

•

We shall need the following theorem, (Woodin b). This theorem can be used in place of the Martin-Steel theorem on scales in L(R), Theorem 2.3, in the analysis of L(R) Pmax . Theorem 2.13. Assume there are ω many Woodin cardinals with a measurable cardinal above them all. Let ¿ be the supremum of the first ω Woodin cardinals. Suppose that A ç R and that A € L(R). Then A is < ¿ weakly homogeneously Suslin. • The basic machinery for establishing that sets are weakly homogeneously Suslin is developed in (Woodin b). An important application is given in the following theorem of Steel.

2.1 Weakly homogeneous trees and scales

25

Theorem 2.14 (Steel). Suppose that ¿o < ¿i are Woodin cardinals and A e Γ™. Then A has a scale in The fundamental theorem of (Martin and Steel 1989) implies that if S is a Woodin cardinal then pWH 1

S+

f -

Γ-Ή

-

1

< s -

An immediate corollary to this is the following theorem which is extremely useful in developing the elementary theory of these pointclasses. Theorem 2.15. Suppose that S is a limit of Woodin cardinals. Then pWH

r->H

1

«5 — 1 \

< ω

λ

.

Then there exists a filter F ç Ρ such that ffì ϋφβ for all D e i ) .

•

In fact Martin's Maximum is equivalent to SPFA. Theorem 2.44 (Shelah). The following are equivalent. (1) Martin's Maximum. (2) SPFA.

•

There are several variations of these forcing axioms which we shall be interested in. We restrict our attention to variations of Martin's Maximum. Definition 2.45 (Foreman-Magidor-Shelah). (1) Martin's Maximum^: that Ρ is a partial order which is stationary set preserving. Suppose that \ and so SRP does not imply Martin's Maximum. •

2.6

Generic ideals

One of the main results of Chapter 3 is that if the nonstationary ideal on a>\ is unsaturated and if there is a measurable cardinal, then there is an effective failure of CH. The force of this result is greatly amplified by the results of (Foreman, Magidor, and Shelah 1988) and Shelah (1987) which show that if suitable large cardinals exist then there is a semiproper partial order Ρ such that in Vthe nonstationary ideal on ω\ is ÛJ2-saturated. Combining these results yields that the effective version of the Continuum Hypothesis is as intractable a problem as the Continuum Hypothesis itself. We review briefly the results of (Foreman, Magidor, and Shelah 1988) and (Shelah and Woodin 1990). We begin with the key definition. Suppose that Λ ç ί>(ωι)\1 Ν 5 is nonempty. Let Ρ,Α denote the following partial order. Conditions are pairs ( / , c) such that (1) for some a < ω\, f : a (2) c ç

Λ,

is a countable closed subset such that for each β € c, if β € dom(/) then β 6

for some η < β, and such that c φ 0.

/(η)

2.6 Generic ideals

41

The ordering on P ^ is by extension. Suppose that (/,,Cι) € Ρ Λ and that (/ 2 , c 2 ) e Ρ,a· Then (/2,C 2 ) < ( / i , c i ) if f\ ί h and c\ = C2 Π (max(ci) + 1). We note that if (/, c) € P ^ then necessarily sup(c) e c. This is because c is closed in ω\ and not cofinal. One of the key theorems of (Foreman, Magidor, and Shelah 1988) is that if Λ c

Ρ(ωι)\1NS

is predense in (¿Ρ(ωι)\1 Ν5 , ç ) then forcing with P ^ preserves stationary subsets of ω\. It is not difficult to show that P ^ is proper if and only if there exists a sequence (Aa : a < ω\) of elements of Λ and a closed cofinal set C ç ω\ such that for all a e C, a e Aß for some β < a. The question of when the partial order P,_a is semiproper is more interesting. This isolates a fundamental combinatorial condition on the predense set Λ which we define below. This condition is implicit in (Foreman, Magidor, and Shelah 1988). Definition 2.56. Suppose that Λ c

¿p(coi)\lNS.

Then Λ is semiproper if for any transitive set M such that Mrm»2))

ç m,

if X i)\íNS is nonempty. Then the following are equivalent. (1) Λ is semiproper. (2) The partial order P^ is semiproper.

Π

The nonstationary ideal on ω\ is presaturated if for any A e ¿Ρ(ωι)\1 Ν5 andforany sequence (Λ, : i < ω) of maximal antichains in !Ρ(ω\)\1NS there exists Β ç. A such that Β φ. 1NS and such that for each i < ω, [Χ e A¡ \ Χ Π Β φ. 1 NS } has cardinality at most o)\. Theorem 2.58 (Foreman-Magidor-Shelah). Suppose that for each predense set Λ ç J>(a)i)\lNS, Λ is semiproper. Then the nonstationary ideal on ω\ is precipitous.

•

Theorem 2.59 (Foreman-Magidor-Shelah). Suppose that κ is a supercompact cardinal and that G ç Coll(a>i, < κ) is V-generic. Then in V[G], (1) each predense set Λ ç

Ρ(ωi)\lNS

is semiproper, (2) the nonstationary ideal on a>\ is presaturated.

•

The large cardinal hypothesis of Theorem 2.59 can be reduced, this yields the following theorem. Theorem 2.60. Suppose that S is a Woodin cardinal and that G c Coll (ω ι, < 5) is V-generic. Suppose that (Λη : η < S) e V[G] is a sequence such that in V[G], for each η < δ, Λη Ç ^(Wl)\lNS and Λ η is predense. Then there exists α γ < δ such that γ is strongly inaccessible in V, such that (Λη : η < γ) e V[G\Yl and such that in V[G\y], for each η < γ, Λη Ç P(coi)\lNS, Λ η is predense, and Λη is semiproper.

•

2.6 Generic ideals

43

The conclusion of Theorem 2.60 is weaker than that of Theorem 2.59, nevertheless it is sufficient to prove 1 NS is presaturated in V[G]. Theorem 2.61. Suppose that S is a Woodin cardinal and that G ç Coll (ω ι, < 5) is V-generic. Then in V[G], iNS is presaturated.

•

Suppose that Λ ç ^>( ω ,)\1 Ν 5 and that Λ is predense and not semiproper. Let Τ^ be the set of countable X
(H{co2)) such that X ç γ, Χ η ω\ — Υ Γ\ω\, and such that Υ (Λω\ e S for some S e y Π «Α. Since Λ is not semiproper, the set TA Ç is stationary in ¿Ρωι (¿Ρ(Η(ω2))). Shelah has generalized Theorem 2.60 obtaining the following theorem. For the statement of this theorem we require a definition. Suppose Ν ç M are transitive models of ZFC such that ,jv _ ft)] = (l)\l NS , Αη is predense, and Αη is semiproper. One corollary of Lemma 2.57 is the following. Lemma 2.63. Let p = np·* be the product with countable support of all the partial orders P^ such that A ç ¿P(£ÜI)\INS and such that A is semiproper. Then the partial order Ρ is semiproper. Suppose that G ç Ρ is V-generic. Then V[G] is a good extension ofV. Proof. Let M be a transitive set such that mH(k) ς

m

where κ is a regular cardinal such that \5>(3>{ω\))\ι = Χ ο Π ω ι . Then X e ΤΛο. By constructing an elementary chain, there exists X < M such that (1.1) X 0 Ç X , (1.2) Χ Π ω ι = Χ 0 Π ω ι , (1.3) for each predense set Λ ç such that Λ € X and such that Λ is

semiproper, there exists e Χη Λ

S with Χ η ωι e S. Now suppose that g ç Χ η Ρ is a filter which is X-generic. By (1.3) it follows that there is a condition p e P such that ρ \ such that ι η ν = (iNS)v

and such that I is unsaturated

in V[G],

Proof. We sketch the proof. The ideal I is rather easy to define, it is the normal ideal (in V[G]) generated by the following set. Let Io e V[G] be the set of A ç ©ι such that for some / : ωι

^(íüi)\1ns,

(1.1) A — {β < ω\ I β i / ( α ) for all α < β}, (1.2) if Λ = { / ( α ) I a < ω\} then for some γ < 5, γ is strongly inaccessible in V, Λ 6

V[GDVy],

and Λ is semiproper in V[G Π Vy]. Let / be the normal ideal generated by Io- The only difficulty is to verify that / is a proper ideal. Granting this, it is easy to prove using Theorem 2.60 that I is a saturated ideal in V[G\. Suppose that Ao Ç 5>{ωχ)\Ι is a maximal antichain. Let Λ = ω\ then Vy Ν ZFC*. Also, assuming ZFC, L(¿P(a> ι)) Ν ZFC* as does the transitive set Η (ω2). • The following lemma is a standard variation of Los' theorem. Lemma 3.2. Suppose M is a transitive model of ZFC* and that U is an ultrafilter on .Picú^) Π M. Let (Ν, E) be the model obtained from the M-ultrapower, (Μωι)Μ

/U

where (Μω')Μ

= {/ : ω" ^ Μ \ f e M).

Then (Ν, E) t= ZFC* and the natural map j :Μ ^

Ν

is an elementary embedding from the structure (M, e) into {Ν, E).

•

Let 4 be the set of stationary subsets of ω\. The partial order (S, ç ) is not separative. It is easily verified that RO(4,ç) = RO(^(Wl)/lNS). Definition 3.3. Suppose M is a model of ZFC*. (1) (P(û>i)\1 n s ) m denotes the partial order (4, ç ) computed in M.

50

3 The nonstationary ideal

(2) A filter G ç (,¡P(ι) 1= "The nonstationary ideal on ω\ is ω2 saturated". (3) Suppose the nonstationary ideal on ω\ is ¿^-saturated, M is a transitive set, M 1= ZFC*, and 3>{ω\) ç M. Then M 1= "The nonstationary ideal on ω\ is ari saturated". (4) Suppose that M is a transitive model of ZFC*, (¿Ρ(ωι))Μ € M, and that G ç (3>(ωι)\1Ν5)Μ is a filter such that G η D φ 0 for all dense sets D e M. Then G is M-generic. • Definition 3.5. Suppose that M is a countable model of ZFC*. A sequence (Mß, Ga, jatß : a < β < γ) is an iteration of M if the following hold. (1) M 0 = M. (2) ja, β '• Ma —> Mß is a commuting family of elementary embeddings. (3) For each η + 1 < y, Gn is M^-generic for (i)\lNS)Mi, Μη+\ is the Mnultrapower of Μη by Gn and ]η,η+ι : Μη Μη+\ is the induced elementary embedding. (4) For each β < γ if β is a (nonzero) limit ordinal then Mß is the direct limit of {Ma I a < β) and for all a < ß, jUtß is the induced elementary embedding. If y is a limit ordinal then γ is the length of the iteration, otherwise the length of the iteration is S where δ + 1 = γ. A model Ν is an iterate of M if it occurs in an iteration of M. The model M is iterable if every iterate of M is wellfounded. • Remark 3.6. (1) In many instances a slightly weaker notion suffices. A model M is weakly iterable if for any iterate Ν of Μ, ω^ is wellfounded. For elementary substructures of H(üy¿) weak iterability is equivalent to iterability.

3.1 The nonstationary ideal and

51

(2) Suppose M is a countable iterable model of ZFC. Then: M 1= "The nonstationary ideal is precipitous". (3) It will be our convention that the assertion, • j :M

M* is an embedding given by an iteration of M of length y,

abbreviates the supposition that there is an iteration {Mß, Ga, jatß : a < β < γ + 1) of M such that My =

M*

j -

hv

and such that (4) Suppose M is a countable model of ZFC*. Then any iteration of M has length at most ω\. (5) The assertion that a countable transitive model M is iterable is a Π^ statement about M and therefore is absolute. (6) Suppose M is iterable and Ν -< M is an elementary substructure then in general Ν may not be iterable. This will follow from results later in this section. In fact here are two natural conjectures. a) Suppose there is no transitive inner model of ZFC containing the ordinals with a Woodin cardinal. Suppose M is a countable transitive model of ZFC and that M is iterable. Suppose X < M. Then the transitive collapse of X is iterable. b) Suppose there is no transitive inner model of ZFC containing the ordinals with a Woodin cardinal for which the sharp of the model exists. Suppose M is a countable transitive model of ZFC and that M is iterable. Suppose M t= "The nonstationary ideal on ω\ is ω2 saturated". Suppose X < Μγ, Νχ e Μ and Νχ is countable in M where Νχ is the transitive collapse of Χ, γ e M and where Μγ 1= ZFC*. Then Νχ is not iterable. • Remark 3.7. We shall usually only consider iterations of M in the case that in M, 1NS is saturated. We caution that without this restriction it is possible that M be iterable but that H(a>2)M not be iterable. If in M, 1 NS is saturated and if M is iterable then H(a>i) M is also iterable. This is a corollary of the next lemma. The correct notion of iterability for those transitive sets in which 1 NS is not saturated is slightly different, see Definition 4.23. • The next two lemmas record some basic facts about iterations that we shall use frequently. These are true in a much more general context.

52

3 The nonstationary ideal

Lemma 3.8. Suppose that M and M* are countable models of ZFC* such that (i) (of = œ f , (ii) Ρ{ω\)Μ

=

Ρ(ω\)Μ*.

Suppose that either (iii) Ρ2{ω\)Μ

= ΰ>2{ω\)Μ\

or

(iv) M* 1= The nonstationary ideal on ω\ is ω2 saturated, and that {Mß, Ga, ja,β : a < β < γ) is an iteration of M. Then there corresponds uniquely an iteration (M*ß, G*a, Hß : a < β < γ) of Μ* such that for alla < β < γ: M M η\ (1) tu, ß = ω, ß ,

(2) 3>(ωχ)ΜΡ =

Ρ(ωι)Μβ;

(3) Ga = G*. Suppose further that M e M*. Then for all β < γ, elementary embedding kß : Μ β i l ß { M ) such that

ß{M)

e Mß and there is an

ß\M — kß o joß.

Proof This is immediate by induction on y.

•

Remark 3.9. The Lemma 3.8 has an obvious interpretation for arbitrary models. We shall for the most part only use it for wellfounded models. • For the second lemma we need to use a stronger fragment of ZFC. There are obvious generalizations of this lemma, see Remark 3.11. Lemma 3.10. Suppose M is a countable transitive model of ZFC* + Powerset + AC + Σι-Replacement in which the nonstationary ideal on ω\ is C02-saturated. Suppose (Mß,

Ga, ja,β ' a < β < γ)

is an iteration of M such that γ < Μ Π Ord. Then Μβ is wellfounded for all β < γ.

3.1 The nonstationary ideal and ¿i

53

Proof. Let (yo, κο, ηο) be the least triple of ordinals in M such that: (1.1) M

"cof(ico) > û>i";

(1.2) η0 < κ0; (1.3) there is an iteration, (Νβ, G α , ja,β : a < β < y0 + 1),

of VKo Π Μ such that jo,ro(vo) not wellfounded. Choose (yo, κο, ηο) minimal relative to the lexicographical order. Thus yo and ηο are limit ordinals. Let {Nß, Ga, jo,,β : a < β < y0 + 1) be an iteration of VK0 DM of length yo such that jo.yoivo) is not wellfounded. Choose β* < yo and η* such that η* < jo,β* (ηo) and such that jß*,Y0(r}*) is not wellfounded. Let {Mß, Ga, ka,ß : α < β < y0 + 1) be the induced iteration of M. By the minimality of yo it follows that Mß is wellfounded for all β < yo. The key point is that for any β e Μ Π Ord if G c Coll (ω, β) then the set M [G] is Σ ¡-correct. Thus (yo, /co, ηο) can be defined in M. More precisely (yo, /co. ηο) is least such that: (2.1) M t= "cof(/co) > ω\"\ (2.2) ηο
• Ν of Νχ such that j(cof) = ω\ and such that forali A 6 Χ Π Η {ω2) j(Ax)

= A

where Αχ is the image of A under the collapsing map. Proof Define an ω\ sequence (Xa : a < ω\) of countable elementary substructures of M by induction on a: (1.1) X0 = X\

(1.2) for each a < a>\, Xa+x = {f(.Xanco{)

\ f e Xa}\

(1.3) for each limit ordinal α < ω\, Xa = U {Χβ I β < α } . Let Χ ω , = U I a < ω\}. For each a < ωχ let Na — collapse(Xa) and for each a < β < ω\ let ja,ß : Na -*• Nß the elementary embedding obtained from the collapse of the inclusion map Xa ç Χβ. Thus No = Νχ and by induction on α < a>\ using Lemma 3.12, it follows that for each a < ω\, Na+\ is a generic ultrapower of Na and ja,a+1 : Na is the induced embedding. Therefore

Na+1

j0.au No ->• Νωι is obtained via an iteration of length ωχ. Finally ω\ ç Χωχ. Hence

ίο.ω i(wf) = ωι and y'o.wi ( ^ x ) = A for each set Λ e Χ Π Η {ω2).

•

Lemma 3.14. Suppose that the nonstationary ideal on ω\ is a>2-saturated. Let M be a transitive set such that M \= ZFC* and such that ίΡ(ω\) ç M. Suppose M # exists. Then {X < M \ X is countable and Μ χ is iterable} contains a club in 3>ωχ (M). Here Μχ is the transitive collapse ofX.

56

3 The nonstationary ideal

Proof. Fix a stationary set It suffices to find a countable elementary substructure X < M such that X e S and such that Μχ is iterable. Fix a cardinal γ such that M e VY and such that Vy t= Z F C - . Thus M* e Vy. Let Y < VY be a countable elementary substructure with M € Y and such that Υ Π M e S. Let Χ = Υ Π Μ. We claim that Μχ is iterable. To see this let Ny be the transitive collapse of Y and let π :Y Ny be the collapsing map. X = Υ Π M and M* e Y and so π (M*) — (Μ χ)*. Ny Ν ZFC". Let G c Coll(û), Μχ) be Ny-generic. Let XG e R be the code of Μχ given by G, this is the real given by {(*> j) I P(i) e p(j) for some ρ e G]. Thus XQ e Ny [G] and so Ν y [G] is correct in V for Π2 statements about XG • Therefore if Μχ is not iterable then Μχ is not iterable in Νy [G]. Assume toward a contradiction that β e Ny and that there is an iteration in of Μχ of length β which is not wellfounded. Then by Lemma 3.8 this defines an iteration of Ny of length β which is not wellfounded, a contradiction since β € Ny. • The next lemma gives the key property of iterable models. For this we shall need some mild coding. There is a natural partial map π : R - + Η(ωι) such that: (1) π is onto; (2) (definability) π is Δ ι -definable; (3) (absoluteness) If χ e dom(7r) and π(χ) = a then Μ Ν "π(χ) = a" where M is any ω model of ZFC* containing χ and a; (4) (boundedness) if A ç dom(7r) is Σ] then {rank(π(χ)) | χ e A} is bounded by the least admissible relative to the parameters for A. For example one can code a set X e Η (ω ι) by relations Ρ ç ω and E ç ω χ ω where • {ω, Ρ, E) = {Y U ω, Χ, e), • Υ is the transitive closure of X. Lemma 3.15. Suppose M is an iterable countable transitive model of ZFC*. Suppose Ν is an iterate of M by a countable iteration of length a. Suppose χ is a real which codes M and a. Then rank(N) < γ where γ is least ordinal which is admissible for x.

3.1 The nonstationary ideal and ¿2

57

Proof. Let j e l code M and let y e Κ code a. Then by the properties of the coding map π , the set of ζ e dom(7r) such that π (z) is an iteration of M of length α is Σ}(χ, y). The result now follows by boundedness. • Theorem 3.16. Suppose that the nonstationary ideal on ω\ is cû2-saturated. The following are equivalent. (1) = ω2. (2) There exists a countable elementary substructure X < Η (ω2) whose transitive collapse is iterable. (3) For every countable X -< Η (ω2), the transitive collapse ofX is iterable. (4) If C ç ω\ is closed and unbounded, then C contains a closed unbounded subset which is constructible from a real.

Proof We fix some notation. Suppose χ is a real and x # exists. For each ordinal γ let M(x#, γ ) be the γ model of χ#. (1 =>· 3)FixX -< Η(ω2). Fix an ω sequence (y, : i < ω) of ordinals in XC\a>2 which are cofinalinXrto2. Foreachi < . This is expressible in Η{ω2) as a first order sentence. This is the second key point. Thus: (2.1) If M is a wellfounded iterate of Ν and if M* is a generic ultrapower of M then M* is wellfounded.

58

3 The nonstationary ideal

From (1.1) and (2.1) it follows that Ν is iterable. (2 => 4) Fix X < H{a>2) such that Νχ is iterable where Νχ is the transitive collapse of X. It suffices to show that if C e X and if C c ω\ is closed and unbounded then C contains a closed unbounded subset which is constructible from a real. This is because if (4) fails then there must be a counterexample in X. Fix C e X such that C is a club in ω\. Let ζ be a real which codes Νχ. Let Cx = C Π X. By Corollary 3.13 there is an iteration of length a>\ j : Νχ

Ν

such that j(Cx) = C. By Lemma 3.15, if a is admissible relative to ζ and if k : Νχ —> M is any iteration of length a then k(a>ix) = a. Therefore if a < ω\ is admissible relative to ζ then a e C . Thus D = {α 1) This is a standard fact. The only additional hypothesis required is that for all χ € R, x# exists and this is an immediate consequence of the assumption that the nonstationary ideal on ω\ is saturated. Suppose ω\ < a < o>i. Fix a wellordering < a of ω\ of length a. Choose a club C Q ωι such that for all γ € C, rank(2-saturated and that Ρ(ω\ )# exists. Then = &>2· Proof. ίΡ(ω\)* exists and so Η {ω2)# exists. By Lemma 3.14, there exists a countable elementary substructure X < Η {ω2) whose transitive collapse is iterable. The theorem follows by Theorem 3.16. •

•

3.1 The nonstationary ideal and ¿2

59

There is a version of Theorem 3.16 which does not require the hypothesis that the nonstationary ideal is saturated. Remark 3.18. The proof that (2) follows from (4) in Theorem 3.19 plays a fundamental role in the analysis of the P max -extension and its generalizations. This analysis is of course the main subject of this book. • Theorem 3.19. The following are equivalent. (1) There exists a countable elementary substructure X < Η (ω2) whose transitive collapse is iterable. (2) For every countable X < Η (an), the transitive collapse ofX is iterable. (3) For all reals x, x# exists and ifC Ç ω\ is closed and unbounded, then C contains a closed unbounded subset which is constructible from a real. (4) / / C Ç î w 1 is closed and unbounded, then there exists

such that

{a \ such that Β is constructible from a real. For this one uses the sequence (zi : i < ω). It is straightforward to verify that for Β e Ζ Π Χ Π ίΡ(ω\), πΜ(Β) - j (Β) η u^ where j

L[zJ(m

is the ultrapower embedding as computed in L[z, Fix m < ω, Β ç. um, A Q ω\ and χ e R such that Β e Ja+\[π, ί], A e L[B, χ] and Α φ Ζ. We may assume that {A, B, x] ç. X. Let Β m be the image of Β under the collapsing map and let Am be the image of A. Since ω\ ç Χ, A = Am-

3.1 The nonstationary ideal and ¿I

65

Thus Bm e Jau+l[nM,t] and A e L[BM,xl But JaM+x\nM, f] € L[z, F ] and + L[z, y ] c Therefore A e ¿[z ], a contradiction since Α φ Ζ and so A is not constructible from a real. We now prove (2). The key is to represent π as the embedding derived from an ultrapower. For each l e l w e abuse notation slightly and let

Ζ[π, * Γ

ω[

= U {L[n, χ]

Π L[n, y] \ y 6 R } .

By (1) we can form the ultrapower ΐ[π, χ]ωι /F where !F is the club filter on ω\. The point of course is that by (1), !F is an ultrafilter on υ { 5 > ( ω , ) η L[n,y] \ y e R}. The filter Τ is countably complete and so the ultrapower is wellfounded. For each χ e R let jx • L[n, χ] Mx be the induced elementary embedding. It follows that for all X e Ζ Π L[τι, χ], jx(X) = *(X). For each Jt e R let Ex be the (u\,uw) extender derived from jx. Thus Ex e L[n,x], Let Nx=\]\t{L[n,x],Ex) and let j0x-.L[n,x}-+

Nx

be the corresponding embedding. The ultrapower Ult (L[n, x], Ex) is wellfounded since it embeds into Mx. Since Ex e L[:r, jc] it follows that jx\La[n,

x] e

L[π,χ]

for all a € Ord. Further by the definition of Ex it follows that jx = j® when restricted to VUu> Π L[π, χ]. • The proof of part 2 of Lemma 3.23 shows that assuming that «2 = &>2, the map π is obtained from a restricted ultrapower. Theorem 3.24. Assume the nonstationary ideal on ω ι is ωι-saturated. lowing are equivalent.

Then the fol-

(1) & = 0>2. (2) There is an inner model Ν of ZFC containing the ordinals and an elementary embedding k :Ν N* such that if G ç (^Ρ(ωι)\1 Ν5 , c ) is V-generic and if J:V

M

66

3 The nonstationary ideal is the associated generic elementary embedding then a) k\Na e Ν for all a; b) j\NKa = k\NKa).

where κω = sup {κη\η

< ω} and {κη | η < ω) is the critical sequence ofk.

Proof. (1) implies (2) by the previous lemma. We now prove that (2) implies (1). Fix a cardinal such that | | = ω\. Thus V¿ 1= ZFC* and (NKm,k\NKJeVs. LetX κ Vá be a countable elementary substructure such that ΝΚω, k | ΝΚω e X. We show that X(~)H(cü2) is iterable. The relevant point is that (NKoj, k\NKoj) is naturally a structure that can be iterated and further all of its iterates are wellfounded. Let kKm = k \ ΝΚω. The fact that (NKu>, kKw) is iterable is a standard fact, k ç Ν and Ν contains the ordinals, therefore (N, k) is iterable; i. e. any iteration of set length is wellfounded. The image of (NKa>, kKa) under an iteration of (N, k) of length a is simply the a 0 1 iterate of (ΝΚω, kK(0). Let Μχ be the transitive collapse of X. Let Ν*ω be the image of ΝΚω under the collapsing map and let kbe the image of kKaJ. We claim that ( Ν * , is iterable. This too is a standard fact. Any iterate of (Ν* , k^) embeds into an iterate of (NKa>, kKa) which is wellfounded since (NKai, kKoj) is iterable. The image of under any iteration of Μχ is an iterate of ( Λ ^ , k^). This is an immediate consequence of the definitions and the hypothesis, (2), of the lemma. Therefore the image of a>2 under any iteration of Μχ is wellfounded and so by Lemma 3.8, the transitive collapse of Χ Π H {an) is iterable. But then by Theorem 3.16, = ω2.

•

Combining Shelah's theorem with Theorem 3.17 yields a new upper bound for the consistency strength of ZFC + "For every real x, x# exists" + " á j = an"· With an additional argument the upper bound can be further refined to give the following theorem. One corollary is that one cannot prove significantly more than A^-Determinacy from the hypothesis of Theorem 3.22. It is proved in (Woodin b) that A\-Determinacy implies that there exists an inner model with two Woodin cardinals. Therefore Theorem 3.22 cannot be improved to obtain A\-Determinacy. Theorem 3.25. Suppose δ is a Mahlo cardinal and that there exists S* < δ such that: (i)

L(Ag,RG)

such that j (A) = AQ and j is the identity on the ordinals. Proof. Let **uniformization denote the following assertion: • Suppose Ζ ç Μ χ 1 is a set such that dom(Z) is comeager. Then there exist a comeager set Y ç R and a borei function / : Y R such that Y ç dom(Z) and such that for all λ e Y, f(x))

e Ζ.

It is a standard fact that AD implies **uniformization and that **uniformization implies that every set of reals has the property of Baire. Assume * * uniformization and that V = L(A, R) where A ç R. Then **AC holds; i. e. if F : R ^ V\{0} is a function then there exists a function H :R

V

such that {x | H(x) e F (χ)} is comeager in R. This easily generalizes as follows. Suppose Q is a partial order with a countable dense set and let r e V1® be a term for a real. For each condition ρ e Q define an ideal i f as follows. Let S be the Stone space of Q. i f = {Z c R I {G ç Q I ρ e G, IG (τ) e Ζ] is meager in S] where if G ç Q is a filter then IG is the associated (partial) interpretation map. For a comeager collection of filters G ç Q, IG (τ) is defined; i. e. for each η ç. ω there exists m e ω such that for some q e G, q II· r (η) = m.

3.1 The nonstationary ideal and ¿.2

77

We say Ζ ç R is if-positive if Ζ £ i f . The set Ζ is if-large if X e Ι ρ τ where X -- R \ Z . The following facts are easily verified. The ideals i f are countably complete and for all Ζ ç Μ, Ζ is if-positive if and only if there exists q < ρ such that Ζ is -large. Now assume Q is a partial order with a countable dense set, τ is a term for a real, ρ € Q, V = L(A, R) and **uniformization. Suppose F :

R - » V\{0}

is a function. Then there is a function H :R

V

such that I H(x) 6 F(x)} is if-large. Suppose g ç Q is L(A, R)-generic and let ζ € L(A, R)[g] be the interpretation of r by g. Let Ug = {Z ç R I Ζ e L(A, R), Ζ is l^-large for some

peg}

and so UG is an L(A, R)-ultrafilter on L(A, R) Π ^P(R). Let Ν = Ult(L(A, R), Ug). It is easily verified that Ν is wellfounded (use DC in L(A, R)) and we identify Ν with its transitive collapse. Let jT : L(A, R)

Ν

be the associated generic elementary embedding. It follows that jT is the identity on the ordinals, R N = R l m - R ) | : : 1 , and that jT(A) = u { X v l z ! I X € V, X is borei and X ç Λ} where if X e V is a borei set then Xv^ denotes its interpretation in V[z]· Now suppose that G ç Coll (ω, < ω\) is V-generic. Let R c = Rv/[G]. Suppose ζ E Rg· Then there exists a < ω\ such that ζ e V[G|a] where G\a = G Π Coll (ω, < a). By the remarks above, setting Q = Coll (ω, < α), there exists an elementary embedding jz:V

^

L(jz(A),

jz(R))

such that jz is the identity on the ordinals, j z ( R) = R ^ ' 1 , and such that jz(A) = U{X VU] I X e V, X is borei and X ç A}. This defines a directed system of elementary embeddings and the limit yields an elementary embedding j : L(A, R) -> L(AG, R g ) as desired. This proves (3).

78

3 The nonstationary ideal For each a < ω\ let a = Coll (ω, < α).

For each a < β < ω ι let Q\-**uniformization. Assume ω\-**uniformization and that V — L(A, R) where A Ç R . Then ω\-**AC holds; i. e. if F : Η (ω ι)

V\ {0}

is a function then there exists X ç Η{ω\) and there exists a function Η : X ->· V such that for all g e Χ, H (g) e Fig) and such that for each a < ω ι, Χ Π ·8α is comeager in S a . In fact, both assertions (1) and (2) in the statement of theorem follow simply from the assumption that a)\-**uniformization holds in L(A, R) and we shall prove (1) and (2) from only this weaker assumption.

3.1 The nonstationary ideal and ¿2

79

Let S be the following partial order. Conditions are triples (N, g, a) such that

(2.1) Ν ç ωχ, (2.2) a < ωι, (2.3) g e Sa. Suppose (Ni, g\, a\) e S and (N2, g2, a2) e S. Then (N2,g2,a2)


, < u>\) is ccc and of cardinality ω\. (4) follows from (2) and the elementary embedding given by Theorem 3.36(3). We prove (3). For each χ e M, let kx : L[x]

L[x]

be the canonical elementary embedding where kx has critical point ω J7 and

These conditions together with the condition I h e L[x] anda
i)}

uniquely specify kx. Suppose χ e L[y], then for any Ζ e

P(ú)\)nL[x],

kX(Z)=ky(Z). Fix f

: ω\ —*• S with / e L(AG,

Rg)[G].

Clearly there exists a set X e L(A, R ) such that (1.1)

X c s,

(1.2) \X\ — ω\ in L ( A , R), (1.3) ran(/) ç X. Therefore we may assume that / : ω\ Let r be a term for /. We assume that

ω\.

1 II· " τ is a function from ω\ to ω{\ Let Τ = {(ρ, α, η) I ρ e Coll (ω, < ω\), ρ II· τ ( α ) =

η].

Since AD holds in L(A, R), there exists χ e M such that Τ e L[x], Let T = Thus Γ is a term for a function /

:

§2

in the forcing language for Coll (ω, < á^)·

kx(T).

86

3 The nonstationary ideal Let A ç Coll(o), < ¿J,) be a maximal antichain such that A e L[x] and such that

for all ρ e A there exists η
\. Otherwise, working in L(A, M), there exists a closed, unbounded, set C ç ω\ and a condition qo € Coll(û>, < ω\) such that for all γ € C, qoW-yia", which contradicts that σ is a term for a stationary set. Thus ω^ e kz(Z). Suppose G ç Coll (ω, < áj,) is a V-generic filter such that G = G Π Coll (ω, < ωι) and such that ρ € G for some ρ e Coll (ω, < á^) with {ρ, ω\) € S. Then kz lifts to kz • L[z\[G]

L[z][G]

Λ is a maximal antichain and so there exists a < co\ such that pa e G. Therefore = ηα

=kz{ga)(œ\)

and so it follows that ω\ e

kz(4).

However ω\ 6 kz(E) and so by elementarity, EDS φ β. Therefore in L(Ag, ®G)[G], S contains a closed unbounded subset of ω\.

•

Lemma3.39. Suppose A CM, L(A,

and that for all Β e

R) 1= AD

(Μ) Π L(A, R) the set

{Χ < {Η (ω 2), Β, e) \ Μχ is B-iterable and Χ is countable} is stationary where Μχ is the transitive collapse of X. Suppose G ç Coll (ω, < ωι) 1/(01

is V-generic. Let Mc = Κ

and let V[G]

AG = U{X I X e V,X is borei and X ç Λ} where if Χ € V is a borei set then Xv^ denotes its interpretation in V[G]. Then in V[G\ for all Β e tP(Rc) Π L(RG, Ac) the set {X < {Η(ω2)ν[°\ is stationary.

Β, e) | Μχ is B-iterable and X is countable]

88

3 The nonstationary ideal

Proof. Suppose G ç Coll(o), < ω\) is V-generic. For each set C e ¿P(R) Π L(A, M) let C G = u { X v l G 1 I X e V, X is borei and X ç

C).

By Theorem 3.36, there is an elementary embedding j : L(A,

R)

L(AG,WG)

such that j is the identity on the ordinals. It follows that for each C e ^P(R) Π L(A, R), j(C)

=

CG.

Therefore if Β e ¿P(K G ) Π L(AC, Rc) there exists C e Í ' ( R ) Π L(A, R) such that f-][CG]

B =

for some continuous function / : RG -»· RG with / € V[G]. Thus it suffices to prove that for all C e ¿P(R) Π L(A, R), the set {X -< (Η(ω2)ν[0],

CG, e) I MX is C G -iterable and X is countable}

is stationary.

Fix C € 3>(Μ.) Π L(A, R ) . L e t U C R χ R

be a universal Σ !1 set. For each ζ € M let Uz = {y e R ι (z, y) e

U}.

M

If M is a transitive model of ZFC* we let U = U Π M. By absoluteness UM is defined in M by the same Σ,1 formula which defines U in V. Suppose X -< (Η(ω2),

C, E)

is a countable elementary substructure in V such that Μχ is Z)-iterable and Μχ is £-iterable where {Z\UZ£C)

D =

and Ε = {ζ I Uz ç

R\C}.

Let Y =

X[G]

and let Ν be the transitive collapse of Y. Therefore Ν = Mx[G Γ) Coll (ω, < wf )] where œf = Χ Π ω\. Since **AC holds in L(A, R) (see the proof of Theorem 3.36) it follows that Y -
z i ) | zi e Λ 0 }

and so since Γ is closed under finite unions (and contains all Π j sets), ωω\Α*

e Γ.

Therefore A* e r n f But for each (xo, x\) E Ζ ,

(JT(JCO))_1[AO]

= Δ.

is a continuous preimage of A* and so

κ Β < ka* < δ A where Β = ( π ( χ ο ) ) ~ ' [ Α ο ] . Therefore s u p { p ( * o , * i ) I (·*ο,*ι) e Ζ } < κ A* < δΑ, and so Ζ is bounded.

•

We begin with a technical lemma. Lemma 3.41. Suppose that M is a transitive inner model such that M t= ZF + DC + AD, and such that (i)

RÇM,

(ii) Ord ç M, (iii) for all A e Μ Γ) ¿P(M), the set { Χ χ (Η(ω2),

e ) I Μχ is A-iterable and Χ is countable}

is stationary where Μ χ is the transitive collapse of X. Suppose δ < Θμ,

S ç a)¡ is stationary and f : S —> S. Suppose that g • ωι ->· δ

is a function such that g e M and such that /(a)


ωι(ωι)

by g (a) = min(D\ß), where β = a + 1. Thus g is as required.

•

The second covering theorem is an immediate corollary of Theorem 3.45. Theorem 3.47. Suppose that the nonstationary ideal on ω\ is a>2-saturated. Suppose that M is a transitive inner model such that M 1= ZF + DC + AD, and such that (i) R C M , (ii) Ord ç M, (iii) every set A e Μ Π η and he ¿ ( R ) then h e

106

3 The nonstationary ideal

Thus the hypothesis of the theorem holds in Ln(H{a>2)). Suppose that i : Μχ

Μχ

is an iteration of Μχ. Then, abusing notation slightly, j\N : (Ν, k)

(j(N),

j(k))

is an iteration of (Ν, k) and so Μχ is wellfounded. Let Β = R\A - p[T\]. Thus HA η Μχ) ç p[j(T0x)] ç ρ[τ0] and j{Br\Mx)\)/im, is &>2-complete; i· e. if X ç 3>{ (ωι)/1 Ν 5 . Theorem 3.50. Suppose that the quotient algebra ^(û>l)/i N S is cù\-generated (equivalently ω-generated) as an coj-complete boolean algebra. Then 2«o = 2^1.

•

We shall actually prove the following strengthening of Theorem 3.50. We fix some notation. Suppose A Qcoi. For each γ < cú2 such that ωι < γ, let bf e

Ρ(ω!)/Ins

be defined as follows. Fix a bijection π : a>\

γ.

3.2 The nonstationary ideal and CH

107

Let S = {η < ωι I ordertype(7r[?7]) e A}. Set by to be the element of Ρ(ω\)/XNS defined by S. It is easily checked that by is unambiguously defined. We let ΒΛ denote the a^-complete subalgebra of ! Ρ ( ω \ ) / g e n e r a t e d by {by I ω, < γ

(ttfi)/1NS is of cardinality Kj. Then there exists a set A ç ω\ such that Ζ Ç BA. Thus Theorem 3.50 is an immediate corollary of the next theorm. Theorem 3.51. Suppose that for some set A ç ω\ ®i4 = ¡P{0>x)/lm Then 2*° - 2*1. Proof. The key point is the following. Suppose Y < (Η(ω2), e) is a countable elementary substructure such that A € Y. Let Ν be the transitive collapse of Y and suppose that is a countable iteration such that N* is transitive. Then we claim that j is uniquely determined by j(Afi/) where Α ν = Α Π ω^. To see this let {Νβ, Ga, ja,ß I a < β < γ) be the iteration giving j. We first prove that Go is uniquely determined by This follows from the definitions noting that the property of Λ, Βλ = Ρ

j(Am)r\N.

(ωi)/lNS

is a first order property of A in Η {ω2). Therefore since Υ 2), 6) it follows that Ν h Ba = ¿P(ù)i)/1NS where a = Am. For each γ e Ν Π Ord with γ , let {b")N be as computed in Ν. Strictly speaking (by)N is not an element of N, instead it is a definable subset of N. Go is an Λ/-generic filter and so it follows since

108

3 The nonstationary ideal

that Go is uniquely determined by {γ εΝ

\ G0n(bayf

¿0}.

Finally [γ e Ν I Go η (b»)N

φ 0 } = (J(α) η

Ν)\ω?.

This verifies that Go is uniquely determined by j (AN) ON. It follows by induction that j is uniquely determined by y (Αλτ). Fix Β ç ω\ and fix a countable elementary substructure X -< H(a>2) with A e X and Β e X. Let {Χη : η < ω\) be the sequence of countable elementary substructures of Η (ω2) generated by X as follows. (1.1) x 0 = x . (1.2) For all η < ω\, Χη+l = Χη[Χη Π O)]] = {/(Χη

Π (tí\) \ f β Χη] .

(1.3) For all η < ω\, if η is a limit ordinal then Xr, = U [Xy I γ
\, M* is the transitive collapse of Χη. Let Χωι = U [Χη I η < ω\} and let Μ ω , be the transitive collapse of Χωι. For each γ < η < ω\ let y'y.i) : Μγ ->· Μη be the elementary embedding given by the image of the inclusion map XY ç Χη under the collapsing map. For each η < ω\, (ω\)Μι is the critical point of j,hV+\ and Μη+\ is the restricted ultrapower of Μη by Gn where Gn is the M^-ultrafilter on (oj\ )Mi given by }η,η+\· By Lemma 3.12, (Μη, Gy, jy,TJ : γ < η < ω\)

is an iteration of Mo. For each η < ω\ let Αη be the image of A under the collapsing map. Therefore Αη = Α Π (ω\)Μη and for each η < ω\, ϊη,η+\(Αη)

=

Αη+\.

Similarly for each η < α>ι let Βη be the image of Β under the collapsing map. Thus by Corollary 3.13, j0,Mi (Bo) = B.

3.2 The nonstationary ideal and CH

109

For all η < ω\, jn( ω ,)/1 Ν 5 )/Β Λ . is atomless. Thus if the nonstationary ideal on ωι is saturated and CH holds then Ρ(ωι)/ίΝ5 decomposes as Β * Τ where Γ is a Suslin tree in V B . We now define two weak forms of o. We shall see that if o holds in a transitive inner model which correctly computes ω2 then these forms of o hold in V. To motivate the definitions we recall the following equivalents of o, stating a theorem of Kunen. Theorem 3.53 (Kunen). The following are equivalent. (1)

o.

(2) There exists a sequence (Sa | a < ωι) of countable sets such that for each A Ç ω 1 the set {a | Α Π α e is stationary in ωι. (3) There exists a sequence {Sa | a < ωι) of countable sets such that for each A C. ωι the set {a > ω \ Α Π a e Sa] is nonempty. (4) There exists a sequence (Sa | a < ωι) of countable sets such that for each countable X ç p(α>ι ) the set {α > ω \ Α Π a e Sa for all A 6 X] is nonempty. Proof. (2) is commonly referred to as weak o. That (3) is also equivalent to o is perhaps at first glance surprising. We prove that (3) is equivalent to (2). Let {Sa \a < ωι ) be a sequence witnessing (3). For each a < ω\ let Ta = 3*(α) Π Ly({Sß

Iβ < a +

ω))

where γ < ω ι is the least ordinal such that Ly({Sß

I β < a + ω)) Ν ZF\Powerset.

110

3 The nonstationary ideal

We claim that (Ta | a < ω\) witnesses (2). To verify this fix A ç ω\ and fix a closed unbounded set C ç a>\. We may suppose that C contains only limit ordinals. It suffices to prove that for some ßeC,AP\ßeTß. Let B0 = {2 • a I a e A]. For each η e C U {0}, let χη ç ω be a set which codes Α Π η* where η* is the least element of C above η. Let Βι = {η + 2k + 1 I η e C and k e χη}. Let Β = Bo U Β]. Since (Sa : α < ω\) witnesses (3), there exists an infinite ordinal a such that Β Da € Sa. If α e C then set β = a. Thus β is as required since Sa ç r a . If a £ C let η be the largest element of C below a. Let η = Oif C H a = 0. Let η* be the least element of C above a. There are two cases. If η + ω < a then Α Π η* e Τη* since χη = {k < ω I (η + 2k + 1) € Β Π α]. Ιΐα2) of distinct subsets of ω\ and there exists a sequence {Sa | a < ω\) of countable sets such that {βχ I X ç ^Ρ(ωι) is countable and (Sa | a < ω\) guesses X] is stationary in ω*ι. Here βχ = sup [η + 1 | Αη € Χ}. We weaken (possibly) still further in the following definition. Definition 3.56. 5: There exists a sequence (Aß I β < W2) of distinct subsets of ω ι and a sequence (Sa I « < ω\) of countable sets such that for a stationary set of countable sets X Q ω2, there exists a < ω\ such that Χ Π ω ι < a and such that [β | β e Χ Π an and Aß Π a e 5 α } is cofinal in Χ Π &>2. •

•

3.2 The nonstationary ideal and CH

111

Remark 3.57. (1) Suppose that 2 N l = K2. Then in the definition of Ö, the sequence {Aß I β < ω2> can be taken to be any enumeration of Ρ(ω\ ). (2) If there is a Kurepa tree on ω\ then δ holds. We shall show in Section 6.2.5 that the existence of a weak Kurepa tree is consistent with the nonstationary ideal on ω ι is ω\ -dense. Therefore δ is not implied by the existence of a weak Kurepa tree. Recall that a tree Τ ç {0, 1 } < ω ι is a weak Kurepa tree if |Γ| = ω\ and Τ has cü2 branches of length ω\. • We do not know if CH actually implies Ö though this seems unlikely. Theorem 3.58. Assume that there is a transitive inner model o/ZFC in which o holds and which correctly computes an. Then Ô holds. Proof. Suppose M is a transitive inner model of ZFC such that o holds in M and such that Í02 - a>2 • Thus ω\ = ω™. Let (Sa : a < ω\) be a sequence in M which witnesses o in the sense of Theorem 3.53(4). Let {Aß : β < coi) be a sequence of distinct subsets of ω\ with (Aß : β < ύ>ι) e M. The key point is that the set Μ Π

(ci>2) is stationary in P W ] ( « 2 ) · To verify this,

let F : û>2 ω -* on be a function in V. We must prove that there exists a set σ e M η

(ft>2)

suchthat F [ a < û ) ] c σ . Let γ 2be an ordinal above ω\ such that Fly"™]

ç γ.

Let 7T e M be a bijection from ω\ to γ. For each a < ω\, π [ a ] e Μ Π Ρ ω ι (2) together witness Ô. Therefore there exists a countable elementary substructure X -< H(a>3) such that (1.1) (Sa : a < u>i) € X, (1.2) (Aß:ß2)€

X,

(1.3) for some a < ωι, Χ Π ω\ < a and [β

l^eXand^naeí«}

is cofinal in Χ Π cü2Fix a satisfying (1.3). Let (Χ γ : y < ω\) be the elementary chain where Xq = X and for all γ < ω\, (2.1) Χ κ + 1 =

{/(Χκηωι)|/εΧκ},

(2.2) if y is a limit ordinal, Xy = U { X „ I η
\ )) may be close to the inner model L(M). Perhaps the most important clue is given by Corollary 3.13; if the nonstationary ideal on ωι is saturated and there is a measurable cardinal, then every subset of ω\ appears in an iterate of a countable iterable model. Motivated by these considerations we shall define and analyze in Section 4.2 a partial order Pmax e L(M) for which the corresponding generic extension, L(M)[G], is an optimal version of L(P(a)i)) (assuming ADL(K^). First we generalize the notion of iterability slightly to accommodate the definition.

4.1 Iterable structures We formulate the obvious generalizations of the definitions of iterability from Chapter 3. Definition 4.1. Suppose M is a countable transitive model of ZFC*. Suppose 1 e M is a set of normal uniform ideals on ωj**. (1) A sequence {(Mß, iß), Ga, ja,ß

et < β < γ) is an iteration of (M, X) if:

a) Mo - M and lo = X. b) ja,β '• Ma

Mß is a commuting family of elementary embeddings.

c) Forali β < γ, iß = jo,ß(io)· d) For each η + 1 < y, Gn is

-generic for {3>(ω\)\Ι)Μ"

for some ideal

I e ίη, Μη+1 is the generic ultrapower of Μη by Οη and ϊη,η+\ '· Μη ~^ «Airç+l is the induced elementary embedding. e) For each β < γ if β is a (nonzero) limit ordinal then Μ β is the direct limit of {Ma I a < β] and for all α < β, ja,ß is the induced elementary embedding.

116

4 The Pmax -extension

(2) If y is a limit ordinal then γ is the length of the iteration, otherwise the length of the iteration is 5 where S + 1 = γ. (3) A pair (W, $) is an iterate of (M, 1) if it occurs in an iteration of (M, 1). (4) (M, 1) is ite rabie if every iterate is wellfounded.

•

Remark 4.2. ( 1 ) This is the natural definition for iterability relative to a set of ideals. We shall only use it in the case that the set of ideals is finite. (2) Suppose that M is a countable transitive model of ZFC* such that (Ρ(ωι))Μ

e M.

W

Then (M, {(1NS)' }) is iterable if and only if M is iterable in the sense of Definition 3.5. (3) We will often write (M, /) when referring to (M, {/}) in the case where only one ideal is designated. • We define the corresponding notion of X-iterability where X Ç 1 . Definition 4.3. Suppose M is a countable transitive model of ZFC*. Suppose I e M is a set of uniform normal ideals on ω"^. Suppose (Μ, 1) is iterable, Ï Ç R and that XDM e M. Then (M, i ) is X-iterable if for any iteration of (Μ, 1), j : ( M , i ) -> (Μ*, Γ) j(XDM)

= XnM*.

•

The next two lemmas are the generalizations of Lemma 3.8 and Lemma 3.10 respectively. The proofs are similar and we omit them. Lemma 4.4. Suppose that M and M* are countable models of ZFC* such that (i) Cûf = ( o f , (ii) Ρ 2 ( ω \ ) Μ = ¿Ρ2(ωι)Μ*, (iii) M e M*. Suppose 1 € M is a set of uniform, normal, ideals on ω^ and that ((Mß, iß), Ga, ja,β ι a < β < γ) is an iteration of (Μ, 1). Then there corresponds uniquely an iteration {(Mß, lß), G*, j*ß \ a < β < γ) of Μ* such that for alla < β < γ: (1)

Mß

Mtβ Μ

;

4.1 Iterable structures

117

(2) 3>{ωx)Me = 3>(ω (3) Ga = G*. Further for all β < γ there is an elementary embedding kß : (Μß, Ìβ) such that

jlß((M,

1))

ß\M = kß o j0 β.

•

Lemma 4.5. Suppose M is a countable transitive model of ZFC and that le M is a set of normal precipitous ideals on ((Mß,lß),Ga,ja,ß

Suppose I a < β < γ)

is an iteration of (M, 1) of length γ where γ < Μ Π Ord. Then Μβ is wellfounded for all β < γ. a We shall need boundedness for iterable structures. Lemma 4.6(1) is proved by an argument analogous to the proof of Lemma 3.15 and Lemma 4.6(2) follows easily from Lemma 4.6(1). Lemma 4.6 (ZFC*). Suppose that χ 6 M codes a countable iterable structure, (Μ, 1). (1) Suppose that j : (Μ, 1) is an iteration of length η. Then

(Μ*,

Γ)

rank(M*) < η*

where η* is the least ordinal such that η < η* and such that Ln* [Λ] is admissible. (2) Suppose that j : (M, 1)

(AT,

Γ)

is an iteration of length a>\. Let D = [η < ω ι I Lv[x] is

admissible}.

Then for each closed set C Q ω ι such that C € M*, D\C is countable.

•

As an immediate corollary to Lemma 4.6 we obtain the following boundedness lemma. Lemma 4.7 (ZFC*). Assume that for all χ € 1, exists. Suppose ( M , i) is a countable iterable structure and that j : (M, 1) (M*, 1*) is an iteration of length ω\. Then rank(M*) < «5^.

•

118

4 The ?max -extension

We extend Definition 4.1 to sequences of models. Definition 4.8. Suppose f° then there exists χ e J^k+i such that { a < o)f° I La[x\ is admissible} ç C. b) ν ω + ι n«Ak+1 < ς 2 Kh-i·

•

Lemma 4.19. Suppose that (Jfk : k < ω) is a sequence of countable transitive sets such that for all k < ω,

e -Nk+\,

Λrk Ν ZFC*, and *

η ( i N S ) ^ + l = ifk η ( i N S ) · ^ .

Suppose that k e ω and that

Then there exists

G ç υ{(^ > (α>ι)) Λί ' I i < ω}

such that a e G and such that for all i < ω, G Π Jf¡ is a uniform J/i-normal ultrafilter. Proof. Fix by replacing [JJ, : i < ω) with {JS¡+k '• i < ω), we may suppose that a e J^oLet ( f i : i < ω) enumerate all functions .-No /f .. ,ω,

«Vo ω,

such that /

and such that for all a
,

(1.2) α, is cofinal in a>f°, (1.3) a¡ e (1.4) f i \üí is constant, (1.5) ai+1 C

ai.

The sequence is easily constructed by induction on i. Suppose a¡ is given. By (1.2) it follows that "i i

Uns)^

for all j > i. This is the key point. Thus a¡ is a stationary subset of a>'f° in M¡+2 and so since /,+1 is regressive there exists β
77

be a suqection such that {(*,>

k(H(o>i)M*)

is a countable iteration. By Theorem 3.34, Η(o>i)Mx is A-iterable. Therefore k(A η Νγ) = ΑΠ Ν*

and so since Β is Δ[ (Λ) in parameters from Νγ, k(Br\NY)

=

ΒΠΝ*.

By elementarily, it follows that k(jiY)

: ΜΠΛ^*

Ηηγ)

is a suijection and that Β Γ\ N* = {(*, y) I k(nY)(x)


a. Finally for all ρ e k(g), {(«y, J), b) < p, and so k(g) ç G. Therefore we have that for all a < cof, k(f)\a e τ. Thus k(f)\ß e τ and so F\β€τ. This proves that ÚJ\ -DC holds in L(K)p™*. In fact we have proved something stronger: • Suppose that G c P m a x is ¿(M)-generic. Suppose that Τ e L(R)[G], Τ is an ω-closed subtree of {0, 1}\ =

for some real χ ".

(ii) Let £>\ be the set of {(J/, J), b) e P m a x such that a) ((Jf,J),b)

< ((M,l),a),

b) «V Ν "d* is constructible from a real" . Then DQ U ¿Di is open, dense in P m a x below ((M, I), a). Here d* denotes the image ofd under the iteration of (M, I) which sends a to b. Proof Fix a condition ρ e P m a x with ρ < ((M, I), a). There are two cases. First suppose there is a sequence ((Pk,Xk) : k < co) such that for all k < ω; (1.1) pk e Pmax and pk+\ < pk < p, (1.2) Xk e Κ Π Mk and xf is recursive in xk+\, (1.3) xo codes ρ and {(M, I), a), (1.4) every subset of

which belongs to L[dk+1, Xk\ either contains oris disjoint

from a tail of the indiscernibles of L[xk+1] below

.

where for each k < ω, pk = ((Mk, 4), a^), dk — jk(d) and jk is the elementary embedding from the unique iteration of ( Μ , I) such that jkifl) = ak. Implicit in (1.4) is the fact that if A ç ω^* and if Λ 6 Mk then every subset of which is in L[A] belongs to La[A] where a = Mk Π Ord. This is because A# € Mk which in turn follows from the iterability of (Mk, Jk)· We use this frequently. Choose a condition ((J/, J), b) e P m a x such that for all k < ω, ((Λr,J),b)

< ((Mk, h), ak).

For each k < ω let jk : (Mk, Ik)

(M¡, J¿)

be the unique iteration such that jk(ak) = b. Let d* = jk(dk)· This is unambiguously defined and we may apply Lemma 4.57 in M to obtain that there is a real t e ,N such that d* e ¿[f]. The condition ((M, J), b) e £)\ and ((M, J), b) < p. The second case is that no such sequence ((Pk,Xk) : k < ω) exists. Notice that if ((Mx,Jx),b\)

< ((Mo, Jo), bo)

4.2 The partial order Pmax

165

in Pmax and if j : (Jfo,

is the unique iteration such that j(bo)

Jo)

(^0'

Jo)

= b\ then for every D e J* a tail of indiscernibles

of L[x] below off' is disjoint from D where χ is any real in ,N\ which codes «Afo. Therefore since the sequence ((pk, Xk) exist a condition {{Mo,

a real XQ € J^o, and a set D ç (2.1)

and a>f°\D

D

where do = =

Jo),

bo)


| ( ( J J , J ) , b ) e g } ; i. e. that A is the set "Ac" computed from g. Therefore A is L(R)-generic for P m a x .

•

The next theorem is the key for actually verifying that specific Π2 sentences hold in

4.2 The partial order P m a x

Theorem 4.61. Assume language for the

AD holds in L ( R ) . Suppose

ψ(χ)

is α

Πι formula

167 in the

structure (Η(ω2),

e,

ÍNS)

and that

Then there is a condition such that forali

((M\,

((Mo,

/o), ao) e P m a x and a set bo ç

with bo e

Mo

I\), a\) e P m ax, if ((Muh),

ai)


(ωι)

= Ό{Ρ(ωχ)Μ'

G}

and so H((02)LmG]

= U { H ( a > 2 ) M * I ((M,

The theorem now follows.

I), a) e

G}.

•

The next theorem is simply a reformulation. This theorem strongly suggests that if AD holds in L(K) and if G Ç Pmax

is ¿(K)-generic then in L(M)[G] one should be able to analyze all subsets of Ρ ( ω \ ) which are definable in the structure (H(ù>2), € , l N S > ¿ ( R ) t G |

by a Πι formula. Thus while a Π2 sentence may fail in L(R)[G] one can analyze completely the counterexamples.

168

4 The '•max -extension

Theorem 4.62. Assume AD holds in L(R). Suppose ψ(χ) is α Πι formula in the language for the structure (H(Û)2), €, 1NS). Suppose G Ç P m a x is L(M)-generic and that (H(a>2), e, im)LmG] ^ nA] where A 2)M\e,I*)i=f[a*l Proof. By Theorem 4.60, A is L(R)-generic for P m a x and so the generic filter G* exists. As in the proof of Theorem 4.61, //(w2)¿(1R)[g*]

= υ{Η(ω2)Μ"

I ((M, I), a) e G*},

where for each ({M, I), a) e G* let j : (M, I)

(M*, I*)

is the iteration such that j(a) = Ac* — A.

•

The next theorem we prove gives the key absoluteness property of L (R)Pmax. Using its proof one can greatly strengthen the previous theorems. To prove this we use the following corollary of Theorem 2.61. This theorem is discussed in Section 2.4. An alternate proof is possible using the stationary tower forcing and the associated generic elementary embedding. The choice is simply a matter of taste, working with Theorem 2.61 is more in the spirit of Pmax· In Chapter 6 we shall consider various generalizations of Pmax and for some of the variations we shall prove the corresponding absoluteness theorems which are analogous to the absoluteness theorems proved here for Pmax· There we will have to use the stationary tower forcing cf. Theorem 6.85. Theorem 4.63. Suppose S is a Woodin cardinal. Let Q = C o l l i i , < 8) * Ρ be an iteration defined in V such that F is ccc in y Coll («i,«5) Then the nonstationary ideal on ω\ is precipitous in V1®. Proof. If the nonstationary ideal is precipitous in V then in any ccc forcing extension of V, the nonstationary ideal is precipitous. This is a relatively standard fact. Using this, the theorem follows from Theorem 2.61 •

4.2 The partial order Pmax

169

Theorem 4.64. Assume AD L ( R ) and that there is a Woodin cardinal with a measurable above. Suppose φ is α Π 2 sentence in the language for the structure {Η(ω2),

6,1NS)

and that {Η(ω2), €, 1NS)

φ.

Then (Η(ω2),β,1Ν5)^™ϊφ.

a

There is a stronger absoluteness theorem that is true and this is the version which we prove. Theorem 4.65. Assume A D ¿ ® and that there is a Woodin cardinal with a measurable above. Suppose that J is a normal uniform ideal on ω\, φ is α Π 2 sentence in the language for the structure (H(a>2),e,J), and that (Η(ω2),

J) 1= φ.

Then ( / / ( ^ . e J J ^ N f Proof. Let ψ(χ, y) be a Σο formula such that φ = Vx3y->\¡/(x, y) (up to logical equivalence). Assume towards a contradiction that

Then by Theorem 4.61, there is a condition ((Mo, Io), ao) e

Pmax

and a set

Mo

b0 e H(a>2) such that if {(Μ,Ι),α)


2)M,e,l)

^Sy^m

where b = j(bo) and j : (Mo, 10) (M^, /Q) is the iteration such that j(ao) = a. By Lemma 4.36, there is an iteration j : (Mo, Io) ^ such that: (1.1) (1.2) jnM*0

= I*.

(M*0,

φ

170

4 The Pmax-extension

Let Β = j(bo). The sentence φ holds in V and so there exists a set D e H (an) such that

(Η(ω2), e, J) Ν ->ψ[Β, D]. Let δ be a Woodin cardinal and κ be a measurable cardinal above S. Let g be a Vgeneric enumeration of J of length ω\. The poset is simply J y

^^x¡f[a*,b].

The structure (MQ, /Q ) is an iterate of (Mo, Io) and the iteration is uniquely determined by a*. Therefore M* e Y. Let Ζ ç cof be such that Ζ e Μ and Y e L{Z]. Let .

4.2 The partial order P m ax

175

Suppose that A ç ω\, χ e R, and that A e L[x\ Suppose that M is a countable transitive model, M 1= ZFC + "There is a Woodin cardinal with a measurable above", χ € M and that M

3

V.

Then if and only if (H(œ2),e,im)M

ϊφ[ΑΜ]

where AM = ΑΓ\ a>f.

•

As a corollary to Theorem 4.67 one obtains the following technical strengthening of Theorem 4.64. Generalizations of this theorem are the subject of Section 10.3. Theorem 4.69. Assume ADL(M) and that for each partial order P, V¥ 1= Aj-Determinacy. Suppose φ is α Π2 sentence in the language for the structure {Η(αν), €, 1NS) and that for some partial order P, (H{a>2),e,lm)v

ϊφ.

Then Proof The theorem follows by a simple absoluteness argument, noting that from the hypothesis that for every partial order P, Vp Ν A\-Determinacy, it follows that for every partial order P, V -(ωχ))Μ* ç b) œ f = ι w f ; c) 03(5) c «Λ/\ for each S e M such that 5 ç tuf ; d) If 5 ç cûf, S e M* and if S i I* then 5 is a stationary set in J/. Then ( ί / ω , ε , Ι , / Ν φ ! Now assume toward a contradiction that Ρ is a partial order such that [5] and Y = p[T]. Suppose Ρ e VK and G ç Ρ « V-generic. Let Xq = p[S] and let Yg — p[T], each computed in V[G\ Then in V[G], (H(a>l)v,e,X)l)vlG],e,XG).

•

4.2 The partial order Pmax

177

Theorem 4.71. Assume AD holds in L(R). Suppose ψ(χ) is α Πι formula in the language for the structure {H(a>2), X, iNS) where X OR is a set in L(K). Suppose that (H(co2),X,e,lNS)L^™ï3xir(x). Then there is a condition ((M, I), a) e Pmax and a set b e H(a>2)'M such that if ((M*,I*),a*) < {(M,I),a) and if(M*, /*) is X-iterable then {H(a>2)M\ Χ Π M*, e, /*) 1= f\b*ì where b* = j(b) and j : (M, /)

(M*, /*) is the iteration such that j(a) = a*.

Proof The proof is identical to the proof of Theorem 4.61.

•

We now prove the strong form of the absoluteness theorem. Theorem 4.72. Assume S is a Woodin cardinal and that every set X e ί'(Μ) Π L(R) +

is δ weakly homogeneously Suslin. Suppose φ is α Π2 sentence in the language for the structure {Η(ω2), e, 1 NS , X; X 6 LÇR), X Ç K ) and that {Η(ω2), €, 1NS, Χ-, X e L(R), X ç Then

R}ϊ=φ.

(H(a>2), e, 1NS, Χ; X e L(R), X ç R>LWPmax |= φ.

Proof We sketch the argument which is really just a minor modification of the proof of Theorem 4.65. Let ψ(χ, y) be a Σο formula such that φ = Vx3y->i¡r(x, y) (up to logical equivalence). Clearly we may assume that ψ contains only 1 unary predicate from those additional predicates for the sets of reals we have added to the structure {Η(ω2), e, 1 NS ). Let X be the corresponding set of reals. Assume toward a contradiction that

Then by Theorem 4.71, there is a condition ((Μ, I), a) e P max and a set b e H(a>2)M

178

4 The Umax "Extension

such that: (1.1) Forali ((M*, I*), a*) < ( ( M , I ) , a > ,

if (M*, /*) is X-iterable then {Η(ω2)M\ e, /*, Χ Π M*) 1= Vy^ß*] where b* = j(b) and j : (M, / ) - » ( M * , I * ) is the iteration such that j(a) = a*. By Theorem 4.41, we may assume by refining ((Μ, I), a) if necessary, that (M, I) is X-iterable and that X Ft M e M. By Lemma 4.36, there is an iteration j : (M, I)

(M*, I*)

such that: (2.1) y ( < ) = ωι; (2.2) J D M * = 1*; (2.3) ΧΠΜ*

=

j(XDM).

Let Β = j(b) and let A = j{a). The sentence φ holds in V and so there exists a set D e H (un) such that (H(cü2),€,X n s ,X) Let δ be the least Woodin cardinal and let Γ be a ,)) = L ( R ) [ G ] for some G ç P m a x which is Z/(R)-generic.

•

182

4 The -max -extension

Remark 4.77. We shall show in Chapter 10 that it is essential for Theorem 4.76 that 1NS be a predicate of the structure even if one assumes in addition Martin's Maximum for partial orders of cardinality c, cf. Theorem 10.70. We shall also show that "cofinally" many sets in ^ ( K ) η L(M) must also be added, (Theorem 10.90). • If one assumes in addition that R # exists then Theorem 4.76 can be reformulated so as to refer only to a structure of countable signature; i. e. the structure of a countable language. For each η e ω let Un be a set which Σι definable in the structure 2),€, •iNS> U„;n < ω) if (H(CO2),

e, XNS, Un,n
2),E)VP

(=2), €)V'P t= φ, then (Η(αν), e) Ν φ. Assume there is an inaccessible cardinal. Must AD L ( R ) hold?

•

Chapter 5

Applications

We give some applications of the axiom: Definition 5.1. Axiom ( * ) : A D holds in L ( M ) and L(JP(a)i)) sion of L(R).

is a P max -generic exten•

We begin by proving that ( * ) implies that 1= AC.

L(P(CÛ]))

We actually give two proofs, the first involves a sentence 0AC which is the subject of Section 5.1. The second proof works through a variant of 0AC » this is the sentence VAC which is discussed in Section 5.3. In fact the latter approach is much simpler, however the sentence 0AC introduces concepts which we shall use in Chapter 10. Martin's Maximum implies both 0AC and VAC and so Martin's Maximum also implies that Ν AC.

L(P(fi>ι))

The proof that Martin's Maximum implies 0AC adapts to show that Martin's Maximum implies that o holds at ωι on the ordinals of cofinality ω, this is Theorem 5.11. The main work of the chapter is in Section 5.7 where we give a reformulation of ( * ) which does not involve the definition of P m a x or the notion of iterable structures.

5.1

The sentence 0\c

We now prove Theorem 4.54; i. e. that L(R)IPmax

|= Z F C _

As we have noted, a second (simpler) proof is given in Section 5.3. First we fix some notation. Definition 5.2. Suppose 5 ç a>\. Then S is the set of all a < a>2 such that a>\ < a and such that if R is a wellordering of ω \ of length a then {γ I ordert y pe(fl|y) e 5} contains a club in ω\.

•

Thus 5 is the set of a < a>2 such that ω\ < a and 1 lhB « e

j(S)

5.1 The sentence 0AC

185

where Β = RO(¿P(Ú>I)\! ns ) and j :V

(M, E) C V B

is the corresponding generic elementary embedding. Note that ω% is always contained in the wellfounded part of the generic ultrapower (Μ, E). Definition 5.3.

i = U {7; I ι < ω]. Then there exists y < α>2 and a continuous increasing function F : ω\ cofinal range such that F[T¡] ç Si for each i < ω.

y with

•

Clearly AC is Π2 in the structure {H(o>2), 6). The next lemma is immediate. The idea for using subsets of ari to define a wellordering of the reals in this fashion originates in (Foreman, Magidor, and Shelah 1988). They use sets S ç {α < cù2 I cof(a) = ω} which are stationary in ω2- The additional ingredient here is using subsets of ω] to generate these sets. This yields a wellordering which is simpler to define. Lemma 5.4 (ZF + DC). Assume 2), 1 N S , e) from {S¡ : i < ω). Proof. Let (5, : i < ω) be a sequence of pairwise disjoint stationary sets. An immediate consequence ofAC is that for every set λ: ç ω with χ φ 0 there exists an ordinal y < α>2 such that cof(y) = ω\ and such that χ = {i I Si Π y is stationary in γ}. Let γχ be the least such ordinal. The wellordering of ίΡ{ω)\ {0} is given by χ < y if γχ < y v . This wellordering is Δ ι definable in {H(a>2), 1NS, €) from {Si : i < ω).

•

186

5 Applications

Lemma 5.5.

Suppose that for each set {(M, I ) , a) € P m a x such that

X ç M with

X

e

L(R),

there is a

condition

(i) Χ η M e M, (ii) (H(fi>\)M, (iii) (M, / ) is Suppose

G

Χ Π M) < (Η(ωι),

X),

X-iterable.

ç Pmax

is L(K)-generic.

Then

L(R)[G] t= ΦΑΟ. Proof. We work in L(M)[G], Necessarily, 3>(ωχ) ç L ( R ) [ G ] .

Suppose (S¡ : i < ω) and (T¡ : i < ω) are sequences of pairwise disjoint subsets of a>\. Suppose the 5,· are stationary and suppose that ω\ = U {T¡ | i < ω}. Let {(M, I), a) e G be such that (5, : i < ω), {Ti : i < ω) e M* where j : (M, I )

(M*,

I*)

is the iteration such that j(a) = Ac- Let (s¡ : i < ω), (ί, : i < ω) in Μ be such that j(({si

: i < ω), (t¡ : i < ω))) = ({S,· : i < ω), {T¡ : i < ω)).

Thus in Μ, (Si : i < ω) and (ί, : i < ω) are sequences of pairwise disjoint subsets of tt)·^, the Si are not in / , and of

= υ{t¡

I i < ω}.

Let D be the set of conditions ((JS, J), b) < ((Μ, I), a) such that in M there exist γ < ω'2 and a continuous increasing function F : wf —• γ with cofinal range such that F{tf)ç

sf

for each i < ω where tf = k(t¡), sf = k{s¡) and k is the embedding of the iteration of (M, I) which sends a to b. For each i < ω, sf denotes the set λ as computed in M where A = s f . It suffices to show that B> is dense below {(M, I), a). We show something slightly stronger. Suppose (ΟΛΑ, J), b) < (GM), Jo), bo) < ((M, / ) , a).

Then for some c e Ν, ((f. Let C be the set of indiscernibles of L{x] less than cof. Let D ç C be the set of η e C such that C Π η has ordertype η. Thus Ζ) is a closed unbounded subset of C. Let o) · Thus {(J/, J), c) e P m a x and 2).

188

5 Applications

Lemma 5.8.

Assume

Martin's Maximum.

Suppose

{α I a e S and cofia) is stationary

that

S ç 2 I

χ

e

f} | =

ωι.

The order on Ρ is defined in the natural fashion: (s*,t*)

Wi(a>l)

be the interpretation of τ. By a straightforward fusion argument there exists a condition (si, t\) < (ίο, to) and a function / : ii ->· a>2 such that: (3.1) Suppose that g ç Ρ is V-generic with (ii, ii) e g. Let ν π g : ω —»· ω2

be the function such that for each k < ω, there exists t such that (ng Then {f{ng\k)\k

i, ι {β < ω2

ι Χ~β

e i l a n d f(x~ß)

=

a] \ =

un-

t) e g.

5.2 Martin's Maximum, 0AC

οω(ω2)

189

It suffices to find a condition (S2, H) < (si, ?i) and a pair (αο, α ϊ ) € So x Si such that: (4.1) ao < a i . (4.2) Suppose π e fo] and let X = {f(n\k)

\ k < ω].

Then a) Χ Π ωχ = αο, b) ordertype(X) = a \ . To see this suppose that g c Ρ is V-generic with («2, h) e G. Then in V[g], ng e fo]. Let Thus Xg € Cg. By absoluteness it follows from (4.2(a)) and (4.2(b)) that Xg Γ\ω\ = αο and that ordertype(λ^) = αι. Therefore Xg e Zg[So, S|] and so cg η ζ,[So, Si] Φ 0. To find (ao, a i ) and (s2, ¿2) we associate to each pair (κο, η ) e SO

Χ

SI

with Ko < Kl a game, $(yo, Ki)> a s follows: Player / plays to construct a sequence ((W,ß!):i

)

of pairs such that (77,, ßf ) e a>2 χ κι · Player II plays to construct a sequence {(bi, m, ßl1) : i < ω)

of triples (b¡, n¡, β'') e t\ χ ω χ γι. Let (β, : i < ω) be the sequence such that for all i < ω, β2,+\ = ß! and ß2i = ßj'. The requirements are as follows: For each i < j < ω, (5.1) bi ç bi+i and dom (£>,-) = i, (5.2) if ίι ç b¡ then b¡+\ = bi

for some δ >

,

(5.3) /(¿>2i+i) < ωι if and only if β{ < γ0, (5.4) η, is odd and

f(bi)

=

f(bn¡),

(5.5) /(¿>2,+i) < f(b2j+\) if and only if ßi < β,.

190

5 Applications

The first player to violate the requirements loses otherwise Player II wins. Thus the game is determined. The key property of the game is the following. Suppose that ((77,·, ßf) : i < ω) and ((bi,rii, ß f ) : i < ω) define an infinite run of the game which satisfies (5.1)—(5.5) (and so represents a win for Player II). Let (βι : i < ω) be the sequence such that for all i < ω, βϊί = β! and ßh+\ — ßf. Suppose that Kl = Let Χ -- {f(bi)

{ßi

Ii


i)\iNS,ç) be V-generic with So s G and let >0 : V - *

M0

ç V[Go]

be the associated generic elementary embedding. Let Gi ç (J»(tui)\l N S , ç ) M ° be Mo-generic with 70(Si) e Gi and let 71 : Mo ->- Mi ç Mo[Gi] be the generic elementary embedding given by G1. Thus 71 ο 7ο(ω 2 ) = sup {71 o 70(a) | a < o*i\. Further, since

= jo(co^), K , 0)2 ) e 71 o 70(So) χ 71 0 yo(^i)·

It follows by absoluteness, using the property (3.2) of / , that in Mi, Player II has a winning strategy in the game $( ù>2 and a function F : γωχ (γ) and that F[X] ç χ. Then ordertype(X) e S. Thus γ e C Π 5.

•

From Lemma 5.8 and the results of (Foreman, Magidor, and Shelah 1988) we obtain the following corollary. Theorem 5.9. Assume Martin's Maximum. Then Η {ω2) 1= 0AC· Proof. The relevant result of (Foreman, Magidor, and Shelah 1988) is the following. Assume Martin's Maximum. Suppose (T, : i < ω) are pairwise disjoint subsets of OJ\ and that ω\ = U {T¿ \ i < ω}. Suppose (S, : i < ω) are pairwise disjoint stationary subsets of ü>2 such that for all i < ω, S¡ ç CM where ϋω = {a < c&i I cof a = ω]. Then there exists an ordinal γ < (&ι and a continuous (strictly) increasing function F : ω\ —>• γ with cofinal range such that f [7i] £ Si for each i < ω. This together with the previous lemma yields that Martin's Maximum implies 2). Thus Martin's Maximum(c) implies οωχ (&>2)· Theorem 5.11. Assume Martin's Maximum Then οω{ωι) holds.

5.2 Martin's Maximum, \c a n d

(W2)

193

Proof. Fix an enumeration (xa : a < ω2) of R. For each limit ordinal γ of cofinality ω let Cy be the set of ordinals η < u>2 such that (1.1) Υ < η , (1.2) for each a < γ, η is a Silver indiscernible of L[xa\. For each k < ω let ηζ be the klii element of Cy. Fix a stationary set S ç ω\ such that ω\\S is stationary and fix a sequence (τα : a < ω\) of pairwise almost disjoint infinite subsets of ω. Let ScoU»*i) = [γ < ω2 I cof(y) = ω]. For each γ e Sw(a>2) let ay = {k < ω \ ηζ e S] and let Ay = {a < ü)\ I \σγ Π r a | = ω}. We prove that for each set Β ç û>2, the set {y e Suím)

I ΒΠγ

e

L[AYì)

is stationary in α>ι. For each set A ç ω\, A* exists and so un is strongly inaccessible in L[A]. Thus οω(α>2) will follow from this by the general form of Kunen's theorem; i. e. the generalization of Theorem 3.53 to a>2. Suppose that : ι < ω) is an increasing sequence of ordinals in the interval (α>ι, ω2> and that σ ç ω. Let Q[S, σ, (Ç, : i < ω)] be the following partial order. Conditions are partial functions Ρ:

^ ξω,

with countable domain, such that: (2.1) ξω = sup({£, I i < ω}). (2.2) If X ç ξω is a countable set such that X ç dom(p) and such that ρ[Χ2)

195

(4.4) For each i < ω, ordertype({/(jr|fc) | k < ω}) Π £,· = g(7r|i). To show that (S2, ti), «0 and g exist one associates to each «o e Γ a game o) as follows. The game $(ao) is the natural analog of the game $(αο, «ι) defined in the proof of Lemma 5.8: Player I plays to construct a sequence ((m, β!)

:ί

,

(5.6) f(b2i+i) < ωι if and only if βι < ao, (5.7) for a l U < i, f(b2l+i)

< ξΐι if and only if βι
2/+i) < fibij+i)

f(bnt), if and only if A < ßj.

The first player to violate the requirements loses otherwise Player II wins. Thus the game is determined. The key property of the game is the following. Suppose that ((rç,, β!) : i < ω) and ((bi, n¡, Yi, ßj1) : i < ω) define an infinite run of the game which satisfies (5.1)—(5.9) (and so is winning for Player II). Let {ßi : i < ω) be the sequence such that for each / < ω, ßii = ßj and ßj1 = /Ö2/+1. Let γ — sup ({y, | i < ω}) and suppose that γ = {ßi I i < ω}. Let Χ = [f(b¡) I i < ω}. Then Χ Π ωι = ao and for each i < ω, ordertype(X Π ξι) = γ, . We prove that for some αο £ T, Player II has a winning strategy in the game $(ao). The proof is similar to the proof of the corresponding claim in the proof of Lemma 5.8 except that here we use a generic iteration of V of length ω.

196

5 Applications

Let G ç

(,P(Ù>I)\!NS, Ç)

be V-generic with Γ 6 G and let j : y -» M c V[G]

be the associated generic elementary embedding. We claim that in M, Player II has a winning strategy in the game To prove this assume toward a contradiction that Σ e M is a winning strategy for Player I. Let H ç Coll (ω, be V [G J-generic. We work in V[G][H], The key point is that there exists an iteration (M¡, G,, : i < m < ω) of M and a sequence ((¿>,,/i,,#'):i • o>2 defines a cofinal branch of ¡2 then the conditions (4.1)-(4.4) are satisfied. This proves that the iteration Ρ * Q[5, σ, (ft : i < ω)] preserves stationary subsets of ω\ for any choice of σ. It is now a simple application of Martin's Maximum which shows that the indicated sequence yields οω((ϋ2). Fix a set Β ç ω^ and a closed unbounded set C ç co2- We must produce y € C Π SM{üy¿) such that Β Π γ e L[Ay], Let G ç Ρ be V-generic. The key point is that in V[G] there exists a set σ ç ω such that Β e L[A] where A = {α < ω\ | |σ Π τα| = ω}. This follows from the fact that in V[G], ω^ has cofinality ω and from the fact that Martin's Axiom holds in V. Fix such a set σ and let H Ç Q [ S , er, (ft : i < ω)] be V[G]-generic. Then (1 NS

)V

= V Π (l N S ) V [ C ] [ / / ] and in V[G][H], σ = {/ < ω I ξί e 5}

and ω\σ = {i < ω | ft e (twi\S)~}. Applying Martin's Maximum to the iteration Ρ * Q[S, σ, (ft : i < ω)] yields γ € C η 5 ω (ω2) such that Β Π γ e L[Ay].

5.3

The sentence ^AC

We prove that (*) implies a variant of AC· This sentence implies • L(P(Û) i))t=AC, and in addition it implies

•

198

5 Applications

Further this sentence can be used in place of ΜΑ ω , in defining P m a x , an alternate approach which will be useful in defining some of the P m a x variations, cf. Definition 6.91. We will also consider, in Section 7.2, versions of this sentence relativized to a normal ideal on ω\. Definition 5.12. Ψα€· Suppose S ç ωι and Τ ç ωι are stationary, co-stationary, sets. Then there exist a < o>2, a bijection 7Γ I ù){ —> Ci, and a closed unbounded set C Q ω ι such that {η < ωι I ordertype(7r[rç]) e Γ} Π C = S Π C.

D

Thus ψAC asserts that for each pair (S, Τ) of stationary, co-stationary, subsets of α>ι, there exists an ordinal a < ω2 such that [S\s=[[aej(T)]]

in V®

where Β = RO (¿Ρ(ωι)/1 Ν5 ) and ; :V

(M, E) C Ve

is the corresponding generic elementary embedding. This implies (in ZF) that the boolean algebra 3>((ûX)/im can be wellordered (in length at most i&i). Lemma 5.13 (ZF + DC). Assume I/^AC holds in (H(C02), e). Suppose (Sa : a < ω\) is a partition ofa)\ into ωι many stationary sets. Then there is a surjection ρ :ω2 ->· Ρ{ωι) which is Δι definable in (Η(ω2),lNS,e) from (Sa : a < ωι). Proof. For each set A ç ω] let SA = U{S a + i I α e A} and let Sa = So if A = 0. The key point is that if A ç ωι, Β ç ωι, and if Αφ Α Δ Β i 1 NS . Define

ρ : on ^ ¡Ρ{ωO

Β then

5.3 The sentence i/^ac

199

by ρ (a) = A if there is a suijection π : ω\ —»· a and a closed set C C w i such that {η < ω\ I ordertype(7T[jj]) e So} η C = SA Π C. If no such set A exists then p(a) = 0. Since ψAC holds, ρ is a suijection. It is easily verified that ρ is is Δ ι definable in (H(C02),1NS,€) from (SA '• A < Ω\).

•

The proof that Martin's Maximum implies ^ a c is actually much simpler then the proof we have given that Martin's Maximum implies ac· The reason is that our approach to proving ac from Martin's Maximum was through Lemma 5.8 which established quite a bit more than is necessary. Here we take a more direct approach which only requires a special case of the reflection principle, SRP, an observation due independently to P. Larson. The special case is SRP for subsets of {ori), which can be proved from just Martin's Maximum(c). This special case is discussed in Section 9.5. Theorem 5.14. Assume Martin's Maximum(c). Then H(C02) 1= ΨACProof. Fix stationary sets Sç, c ω\ and To ΩΙ (AN) I Χ Π Ω\ e Sq if and only if ordertype(X) e 7b}. It suffices to prove that for each stationary set S ç Ω Ι, the set Zs = {X e Ζ I Χ Π ω\ € 5} is stationary in Ρ ω ] (ωζ). Let 5 ç ω\ be stationary. The claim that Z s is stationary in ¡Ρωχ (üy¿) follows by an absoluteness argument using the fact that 1 N S is (^-saturated. Fix a function Η : a>2W —> u>2· We must prove that ZS

Π {X

6 S>M (AN) I Η[Χ2 ],

the image of ω^ under 70,2· Thus (2.1) ordertype(X) < j o ¿ ( ( ú \ ) , (2.2) > 0 , 2 ( H ) [ X < n ç X ,

(2.3) Χ η jo,2i· ao be a suijection. It follows that (πο, αο) witnesses that ^ a c holds for (So, 7b).

•

Larson has also noted that the proof of Theorem 5.14 easily adapts to show that Martin's Maximum(c) implies AC· We note that Lemma 5.8 cannot be proved from just Martin's Maximum(c). Therefore, for the proof that we have given that Martin's Maximum implies AC> Martin's Maximum(c) does not suffice. Finally Larson has proved versions of Lemma 5.8 showing for example that Martin's Maximum{c) implies that for each stationary set S ç ω\, S is stationary in 2 and that S Π {α < ω2 I c o f ( a ) = ω} φ 0. The sentence, i/^ac, implies that for each stationary, co-stationary, set Τ c.a>\, the boolean algebra ¿Ρ(ωι)/1 Ν 5 is (trivially) generated by the term for j(T). This fact combined with Theorem 3.51 yields the following lemma as an immediate corollary. Lemma 5.15. Suppose that i/^ac holds. Then 2*o = 2*1 - κ 2 . Proof. By Theorem 3.51, 2*° = 2 s 1 . By Lemma 5.13, 2 K l < K2-

•

The next lemma shows that i/^ac serves successfully in place of ΜΑ ω , in the definition of P m a x . This lemma is really just a special case of the claim given at the beginning of the proof of Theorem 3.51.

5.3 The sentence i/^ac

201

Lemma 5.16. Suppose M is a countable transitive set such that M Ν ZFC* + TAac· Suppose a e M, a C a)f, and Μ Ν "a is a stationary, co-stationary, set in ω\ ". Suppose ji : M-+

M\

Ì2 : M

Mí

and are semi-iterations of M such that M\ is transitive, M2 is transitive and such that ji(a) =

j2(a).

Then M\ = Mi and j\ = ji. Proof. Fix a and suppose that (Mß, Ga, ίαφ I a < β < γ) is a semi-iteration of M such that Mß is transitive for all β < γ. We prove that Go, «Μι and 70,1 : M

M\

are uniquely specified by jo,y(a) Π ω^. We note that since Go is an M-normal ultrafilter, G 0 = [b ç of Therefore since

I b e M and of

e 70,1 (¿>>} -

M t= ψAC

it follows that Go is completely determined by 70,1 ( e ) n a i f . To see this fix b e M such that b ç ojf. We may suppose that b i and that a>f\bi(lm)M. Therefore there exist a < o j f , a bijection π : of and c ç of

such that

(1.1) π e M, c e M,

a

202

5 Applications

(1.2) c is closed and cofinal in ω^, (1.3) {η i} Ç C ' . Following the proof of Lemma 4.36 construct the iteration {(Mß, Iß), Ga, ja,β : a < β < ω\) of (Μ, I) to satisfy the additional requirement that for all y e C", Y e

jo,Y+i(s)

if and only if jo,ß(t) e Gß

where β = γ+. For each y e C if ((Mß, Iß), Ga, ja,β • a < β < γ} is any iteration of length γ then fay(o>f)

=Y

and so this additional requirement does not interfere with the original requirements indicated in the proof of Lemma 4.36. Thus jo,a> 1 : (M, I) ->• (Mm,

/ω|)

is as desired.

•

Lemma 5.18. Assume (*) holds. Then

Ì/TAC

holds.

Proof. Fix a filter G ç P m a x such that G is L(R)-generic. Necessarily, !P(u>\) ç L(K)[G]. Fix subsets S and Τ of a>i such that each are both stationary and co-stationary. Therefore there exist {(Mo, Io), ao) e G, s e Mo and t e Mo such that j(s) — S and j(t) = Τ where j : (Mo, Ιο)

(Λ[S] and R\B = p[T]. Therefore if Ν ç L(R) is any transitive inner model of ZF such that {5, T] ç Ν then ADN

e Ν, BHNeN

and

(VWi Π Ν , Α Π Ν , β ) < 0] {α,Ζo} By Theorem 5.35, we may assume by increasing the Turing degree of yo if necessary that (ù)2)LlZo'a-yo] is a Woodin cardinal in Let

8i = (w2) L [ Z o ^ y o i

and let Ν = HOD fyo]. (ZoÍ ,a) Ν isas required. The general case for arbitrary η is similar. Let κ = (2 Λ+1)

220

5 Applications

and let S* and Τ* be trees on ω χ κ such that (5*, Τ*) e Ν and such that if g c Coll (ω, [r*]) η tfb], and so by Theorem 2.32, S* and T* are < (ω\);

(2) IG is a normal saturated ideal; (3) IG is the nonstationary ideal.

•

Theorem 5.64. Assume AD holds in L(R). Suppose G ç p j ^ is L (IR) -generic. Then L(R)[G] = L(R)[g][A] where g ç P m a x is L(R)-generic and h ç is L(R)[g]-generic.

•

5.6

241

The generalization of Theorem 5.54 and Theorem 5.55 that we seek is the following. Theorem 5.65. The following are equivalent. (1) 2^° = 2 X | and there exists a countable elementary substructure X < Η (on) such that the transitive collapse of X is iterable. (2) There exists a semi-generic filter such that 3>(ω\)β =

Ρ(ω\).

Proof. Let κ = (2*') + · Fix a wellordering, (œi)F*. It is easily verified that this implies (2). We build !FQ as the filter generated by Ζζ = U {Zi : i < ω] where ((Ζ,, Y¡) : i < ω) is a sequence defined as follows. (2.1) Yo is the set of pairs (p, j ) such that ρ e Zo and j — jp,F0. (2.2) Y¡ ç Yl+{ and Yi+\\Y¡ is the set of pairs (({Mk:k

),X),j)

such that there exists a sequence (Xjt : k < ω) such that: a) For some ao € the structure

Xo is the set of è € Η (κ) such that b is definable in {H(K),(Zi,Yi), (M*0,

is an iteration then

Í j(hi)(a)

φ

1

JH*

I α , < ω\), and ρ II· a e σ ο } . Thus r ç ωχ χ Coll(û>, < ω ι ) and r e L [ f ] . We finish by proving that t and r are as desired. Suppose g ç Coll ( ω , < ω ι ) is L[/]-generic and that for each a < ω\, S® is a stationary subset of ω\. Let j : (M0,

Ιο) -> (M¡,

φ

be the iteration given by σ and g. Thus Ig(ωλ), Χ φ 0, and that X is definable from real and ordinal parameters. Then there exist ( e l and a term t ç ω\ χ Coll(i χ Coll(û), < ω\) such that

τ e ¿[r]

and such that for all filters g c Coll(ft), < ω ι ) if g is L[/]-generic and if for each a < ωι, Sa is stationary, then Ig{τ)

e Χ.

Since f # exists we may suppose, by replacing t if necessary, that τ is definable in L [ f ] from t and ω\. Let yo be the least Silver indiscernible of L [ f ] and let go Ç Coll (ω, < yo) be an L[i]-generic filter. Fix t0 e L [ í ] [ g 0 ] Π E such that ¿[ί][£θ] =

L[to]

and define το Ç: ωι χ Coll(ft), < &>i) as follows. (a, ρ ) e το if there exists q e Coll(û), < ω ι ) such that (1.1) q\(cox [0,K) + 1])

€g0,

(1.2) q\(co χ (γο + 1, ω ι ) ) = ρ\(ω χ (yo + 1, ωΟ), (1.3) ((o + a , q ) e τ . Now suppose h ç Coll(ft>, < ωι ) is an L[fo]-generic filter such that for all α < ωι, S¡¡¡ is stationary. Define g ç Coll(o>, < ω ι ) by g = {q e Coll (ω, < ω ι ) |