The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal [2 ed.] 9783110213171, 9783110197020

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

W. Hugh Woodin

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

Second revised edition

De Gruyter

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

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

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

Contents

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

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1 2 6 10 13 15 18

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

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

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184 184 187 192 199 221 232 238 274

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

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287 288 306 306 334

vi

Contents

6.3

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

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370 377 383 403 421

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

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609 610 617 618 622 633 637 651 667 683

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

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

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11 Questions

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Bibliography

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Index

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Chapter 1

Introduction

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

2

1 Introduction

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

1.1

The nonstationary ideal on !1

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

1.1 The nonstationary ideal on !1

3

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

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

12

1 Introduction

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

1.4 Extensions of inner models beyond L.R/

13

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

1.4

Extensions of inner models beyond L.R/

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

14

1 Introduction

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

1.5 Concluding remarks – the view from Berlin in 1999

15

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

1.5

Concluding remarks – the view from Berlin in 1999

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

t u

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

16

1 Introduction

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

1.5 Concluding remarks – the view from Berlin in 1999

is -consistent, then

Pmax

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

 :

17

t u

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

t u

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

18

1 Introduction

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

1.6

W. Hugh Woodin

The view from Heidelberg in 2010

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

1.5 The view from Heidelberg in 2010

19

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

20

1 Introduction

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

W. Hugh Woodin

Chapter 2

Preliminaries

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

2.1

Weakly homogeneous trees and scales

For any set X , X !1 then V  ZFC . Also, t u assuming ZFC, L.P .!1 //  ZFC as does the transitive set H.!2 /. The following lemma is a standard variation of Łos’ theorem. Lemma 3.2. Suppose M is a transitive model of ZFC and that U is an ultrafilter on P .!1M / \ M . Let hN; Ei be the model obtained from the M -ultrapower, .M !1 /M =U where

® ¯ .M !1 /M D f W !1M ! M j f 2 M :

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

t u

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

3.1 The nonstationary ideal and ı12

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

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

(i) !1M D !1M , 

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

3.1 The nonstationary ideal and ı12

55

Suppose that either 

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

of M  such that for all ˛ < ˇ <  : Mˇ

(1) !1

M

D !1 ˇ ; 

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

Proof. This is immediate by induction on  .

t u

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

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(1.3) there is an iteration, of V0

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

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

3.1 The nonstationary ideal and ı12

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

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

N˛ D collapse.X˛ /

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

t u

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

3.1 The nonstationary ideal and ı12

59

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

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

3.1 The nonstationary ideal and ı12

61

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

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

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By (1.2) the definition of hP˛ W ˛ <  i for suitable  is absolute, more precisely suppose that N is a transitive model of ZFC,  < ı and that N D V : Suppose that jV j D  . Then hP˛ W ˛ <  i is the iteration of length  as defined in N . We note by (1.3), since ı is a Mahlo cardinal, the partial order P is ı-cc. Thus if GP is V -generic then H.!2 /V ŒG D Vı ŒG: Let ı  < ı be least such that ı  is a Woodin cardinal in L.Vı  / and such that Vı   Vı . Since Vı# exists it follows that ı  has cofinality !. Therefore we can construct in V an L.Vı  /-generic filter H  Q where Q is the poset for adding a generic subset of ı  . The point is that since ı  is a Woodin cardinal in L.Vı  / it follows that !. We shall assume the basic facts concerning Wadge reducibility in the context of AD, see the discussion after Definition 9.25. One such fact is that there exists a set B  ! ! of minimum Wadge rank such that B … . Fix B0 and let  D ¹B   ! ! j B  is a continuous preimage of B0 º: Let

O D ¹! ! n A j A 2 º

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

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

1 † 1

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

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

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

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

Therefore

A 2  \ O D :

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

1

ŒA0 . Therefore

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

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

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

f .˛/  g .˛/ < g.˛/:

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

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

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

1 † 1

then Z is bounded relative to y .

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

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

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

Fi .S / W !1 ! ı; Fi .S /jS D f jS;

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

Fi .S /jS ¤ f jS

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

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

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

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

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

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The second covering theorem is stronger. Again we prove a preliminary version. Theorem 3.45. Suppose that M is a transitive inner model such that M  ZF C DC C AD; and such that (i) R  M , (ii) Ord  M , (iii) for all A 2 M \ P .R/, the set ¹X  hH.!2 /; 2i j MX is A-iterable and X is countableº is stationary where MX is the transitive collapse of X . Suppose ı < ‚M , X  ı and jX j D !1 . Then there exists Y 2 M such that M  “jY j D !1 ” and such that X  Y . Proof. Fix ı < ‚M and let A 2 M be a prewellordering of the reals of length ı. Fix a set X  ı of cardinality !1 . As in the proof of the first covering theorem suppose that G  Coll.!; !2 , cof./ > !1 , and so L .H.!2 //  ZFC : Further by the Moschovakis Coding Lemma, for each ˛ < , P .˛/ \ L.R/ 2 L .R/: Let X  L .H.!2 // be a countable elementary substructure and let MX be the transitive collapse of X . We prove that MX is A-iterable for each A  R such that A 2 X \ L .R/: Fix A. Thus for some t 2 R, A is definable in L .R/ from t . The set A is 21 .t / in L.R/ and so by the Martin–Steel theorem, Theorem 2.3, there exist < , and trees T0 ; T1 on !  such that A D pŒT0 ; such that, R n A D pŒT1 ; and such that .T0 ; T1 / is †1 -definable in L.R/ from .t; R/. Since L .R/ †1 L.R/; it follows that L .R/ \ .HOD t /L.R/ D .HOD t /L .R/ ; and so .T0 ; T1 / 2 .HOD t /L .R/ : Let j W .HOD t /L.R/ ! N t be the elementary embedding computed in L.R/ where N t D .HOD!t 1 /L.R/ = and where is the club filter on !1 . Since DC holds in L.R/, this ultrapower is wellfounded and we identify it with its transitive collapse. It follows that j  .HOD t /L.R/ . Since L .R/ †1 L.R/ and since cof./ > !1 , j./ D . The structure   .HOD t /L .R/ ; j j.HOD t /L .R/ is naturally iterable and the iterates are wellfounded. The notion of iteration is the conventional (non-generic) one.

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

jO.B \ MX /  pŒjO.T1X /  pŒT1 :

Therefore

jO.A \ MX / D A \ MX :

This verifies that MX is A-iterable.

t u

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3.2

The nonstationary ideal and CH

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

t u

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

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

2@0 D 2@1 :

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109

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

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

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

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

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111

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

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3 The nonstationary ideal

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

is unbounded in !1 .

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

t u

We weaken (possibly) still further in the following definition. QQ There exists a sequence Definition 3.56. ˘: hAˇ j ˇ < !2 i of distinct subsets of !1 and a sequence hS˛ j ˛ < !1 i of countable sets such that for a stationary set of countable sets X  !2 , there exists ˛ < !1 such that X \ !1  ˛ and such that ¹ˇ j ˇ 2 X \ !2 and Aˇ \ ˛ 2 S˛ º is t u cofinal in X \ !2 . QQ the sequence Remark 3.57. (1) Suppose that 2@1 D @2 . Then in the definition of ˘, hAˇ j ˇ < !2 i can be taken to be any enumeration of P .!1 /. (2) If there is a Kurepa tree on !1 then ˘Q holds. We shall show in Section 6.2.5 that the existence of a weak Kurepa tree is consistent with the nonstationary ideal on !1 is !1 -dense. Therefore ˘QQ is not implied by the existence of a weak Kurepa tree. Recall that a tree T  ¹0; 1º .!1 /N0 and such that ı is a Silver indiscernible of LŒx for all x 2 [¹R \ Nk j k 2 !º. From this iterability follows by an argument essentially identical to that given in the proof of Theorem 3.16. There it is proved that assuming  ı 12 D !2 and that the nonstationary ideal is saturated then if X  H.!2 / is a countable elementary substructure, the transitive collapse of X is iterable. t u Remark 4.18. (1) It is important to note that the assumptions of Lemma 4.17 do not actually imply that any iterations exist; the only implication is that if iterates exist, they are wellfounded. It is easy to construct sequences which satisfy the conditions of Lemma 4.17 and for which no (nontrivial) iterations exist. Lemma 4.19 isolates a condition sufficient to prove the existence of nontrivial iterations. (2) The conditions (i) and (ii) of the hypothesis of Lemma 4.17 are equivalent to the assertions: a) if C 2 Nk is closed and unbounded in !1N0 then there exists x 2 NkC1 such that ¹˛ < !1N0 j L˛ Œx is admissibleº  C: b) V!C1 \ NkC1 †2 V!C1 .

t u

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

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

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127

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

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

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

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

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4 The Pmax -extension

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

t u

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

t u

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

rank.Nk / <  ı 12 ;

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

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129

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

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

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

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

t u

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

4.1 Iterable structures

131

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

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4 The Pmax -extension

Proof. For each ˛ < !2 let

˛ W !1 ! ˛

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

4.1 Iterable structures

133

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

t u

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

4 The Pmax -extension

134

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

4.1 Iterable structures

135

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

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4.2

The partial order Pmax

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

t u

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

4.2 The partial order Pmax

137

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

M  “!1 D !1L.a;x/ for some x 2 R”:

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

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

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

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4 The Pmax -extension

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

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

Let

ht˛ W ˛ < i D j1 .hs˛ W ˛ < !1M i/

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

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139

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

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

t u

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

t u

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

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

t u

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

4.2 The partial order Pmax

141

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

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

and

!1 D !1LŒa :

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

4.2 The partial order Pmax

143

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

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

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4 The Pmax -extension

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

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

4.2 The partial order Pmax

145

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

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4 The Pmax -extension

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

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

4.2 The partial order Pmax

147

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

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

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

t u

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

IG D [¹I  j h.M; I /; ai 2 Gº:

t u

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

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

4.2 The partial order Pmax

149

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

t u

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

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

t u

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4 The Pmax -extension

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

!1 n 5A 2 J

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

4.2 The partial order Pmax

151

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

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

h.M; I /; a i < q˛ < p0 :

152

4 The Pmax -extension

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

t u

The lemma follows.

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

4.2 The partial order Pmax

153

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

4.2 The partial order Pmax

155

To see this fix ˇ < !1 . Choose h.N ; J /; bi 2 G such that h.N ; J /; bi < h.M; I /; ag i and such that ˇ < !1N . Hence there is a unique iteration k W .M; I / ! .M ; I  / such that k.ag / D b. This iteration is an initial segment of the iteration which defines j , k.f /  F , and ˇ < dom.k.f //.  .M; I / is X -iterable and so it follows that for all ˛ < !1M , .p; k.f /jˇ/ 2 Y for some p 2 k.g/ and for some ˇ > ˛. Finally for all p 2 k.g/, h.N ; J /; bi < p; and so k.g/  G. Therefore we have that for all ˛ < !1N , k.f /j˛ 2  . Thus k.f /jˇ 2  and so F jˇ 2 . This proves that !1 -DC holds in L.R/Pmax . In fact we have proved something stronger:  Suppose that G  Pmax is L.R/-generic. Suppose that T 2 L.R/ŒG, T is an !-closed subtree of ¹0; 1º ˇ0 . For each ı 2 CkC1 if ˇ0 < ı then f .ı/ is definable in LŒykC1  from ı, finitely many elements of CkC1 below ı and finitely many of the !n ’s. Working in V we can find a stationary set S  CkC1 , a finite set of ordinals t and a definable Skolem function, , of LŒykC1  such that if ı 2 S then f .ı/ D  .ı; t /. Thus we have produced a function h W !1 ! !1 such that h 2 LŒykC1  and such that T D ¹˛ < !1 j f .˛/ D h.˛/º is stationary in !1 . Clearly T 2 LŒD; ykC1  and so T must contain a tail of CkC1 since it cannot be disjoint from a tail of CkC1 . Thus h is as desired. Let X  H.!2 / be a countable elementary substructure containing D and ¹yk j k < !º. Let Z D X \ .[¹L!2 ŒD; yk  j k < !º/: Define a †0 -elementary chain

hZ˛ W ˛ < !1 i

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

164

4 The Pmax -extension

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

4.2 The partial order Pmax

165

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

!1M

and d 2 M.

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

 ;x/

for some real x”.

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

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

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

166

4 The Pmax -extension

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

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

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

4.2 The partial order Pmax

167

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

h.N1 ; J1 /; b1 i

!1N1 D !1LŒx;t;d1  2 D0 .

t u

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

4 The Pmax -extension

168

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

Pmax

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

:

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

Pmax

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

.x/ is a …1 formula in the

 9x .x/:

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

hH.!2 /M1 ; 2; I1 i 

where b1 D j.b0 / and

Œb1 

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

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

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



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

The theorem now follows.

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

4.2 The partial order Pmax

169

Theorem 4.62. Assume AD holds in L.R/. Suppose .x/ is a …1 formula in the language for the structure hH.!2 /; 2; INS i: Suppose G  Pmax is L.R/-generic and that hH.!2 /; 2; INS iL.R/ŒG  ŒA where A  !1 and A 2 L.R/ŒG n L.R/. Let G   Pmax be the L.R/-generic filter such that A D AG  . Then there is a condition h.M; I /; ai 2 G  such that for all h.M ; I  /; a i 2 Pmax ;    if h.M ; I /; a i  h.M; I /; ai then  hH.!2 /M ; 2; I  i  Œa : Proof. By Theorem 4.60, A is L.R/-generic for Pmax and so the generic filter G  exists. As in the proof of Theorem 4.61,   H.!2 /L.R/ŒG  D [¹H.!2 /M j h.M; I /; ai 2 G  º; where for each h.M; I /; ai 2 G  let j W .M; I / ! .M ; I  / is the iteration such that j.a/ D AG  D A. t u The next theorem we prove gives the key absoluteness property of L.R/Pmax . Using its proof one can greatly strengthen the previous theorems. To prove this we use the following corollary of Theorem 2.61. This theorem is discussed in Section 2.4. An alternate proof is possible using the stationary tower forcing and the associated generic elementary embedding. The choice is simply a matter of taste, working with Theorem 2.61 is more in the spirit of Pmax . In Chapter 6 we shall consider various generalizations of Pmax and for some of the variations we shall prove the corresponding absoluteness theorems which are analogous to the absoluteness theorems proved here for Pmax . There we will have to use the stationary tower forcing cf. Theorem 6.85. Theorem 4.63. Suppose ı is a Woodin cardinal. Let Q D Coll.!1 ; i . Then: (1) For each i < !,  , a) Mi 2 MiC1 



b) .!1 /Mi D .!1 /Mi C1 , 



c) jMi jMi C1 D .!1 /M0 , M0

d) if C 2 Mk is closed and unbounded in !1 D2

then there exists

 MkC1

such that D  C , D is closed and unbounded in C and such that D 2 LŒx  . for some x 2 R \ MkC1

(2) For each i < ! let Qi be the partial order of Ii -positive sets computed in Mi . For each a 2 [¹Qi j i < !º there exists a sequence hgi W i < !i such that a) a 2 [¹gi j i < !º, b) for each i < !, gi  giC1 and gi is Mi -generic. (3) The sequence is iterable.

h.Mi ; Ii / W i < !i

5.4 The stationary tower and Pmax

211

Proof. For each i < j < ! let kij W .Mi ; Ii / ! .Mij ; Iij / be the iteration of .Mi ; Ii / in Xj . This iteration is the unique iteration k such that .h.Mi ; Ii /; Xi i; k/ 2 Xj : Mj

The iteration has length .!1 / . Suppose i < j1 < j2 . Then since h.Mj2 ; Ij2 /; Xj2 i < h.Mj1 ; Ij1 /; Xj1 i < h.Mi ; Ii /; Xi i it follows that the iteration corresponding to kij1 is an initial segment of the iteration corresponding to kij2 . Let ki be the embedding given by the induced iteration of .Mi ; Ii / of length Mj

M

sup¹kij .!1 i / j i < j < !º D sup¹!1 Thus

j i < j < !º:

ki W .Mi ; Ii / ! .Mi ; Ii /:

Suppose i < j < !. Then (T) h.Mj ; Ij /; Xj i 2 Pmax

and

.h.Mi ; Ii /; Xi i; ki / 2 Xj

where Xj D kj .Xj /. Thus h.Mj ; Ij /; Xj i < h.Mi ; Ii /; Xi i: (1) follows from Lemma 5.31 and the definitions. For each i < j < ! the iteration ki W .Mi ; Ii / ! .Mi ; Ii / is full in Mj . (2) follows from this and Lemma 5.26. (3) follows from (1) by Lemma 4.17. The relevant point is that by (1) and by Lemma 4.17 the sequence hMi W i < !i is iterable in the sense of Definition 4.15. Any iteration of h.Mi ; Ii / W i < !i defines in a unique fashion an iteration of hMi W i < !i and so is iterable.

h.Mi ; Ii / W i < !i t u

212

5 Applications

Remark 5.33. Lemma 5.32 has the following consequence which is really the key to (T) (cf. Theorem 5.39). establishing the relationship between Pmax and Pmax Suppose h.Mi ; Ii / W i < !i is as specified in Lemma 5.32. By Lemma 5.32 (1) and by Lemma 4.17, the sequence hMi W i < !i is iterable in the sense of Definition 4.15. This was noted in the proof of Lemma 5.32 and the observation is the basis for the reformulation of Pmax given in Section 5.5. By Lemma 5.32(2), for each i < !, 



.INS /Mi C1 \ Mi D .INS /Mi : Further it follows that iterations of hMi W i < !i correspond to iterations of and conversely, iterations of correspond to iterations of

h.Mi ; Ii / W i < !i h.Mi ; Ii / W i < !i hMi W i < !i:

t u

(T) can be carried in a fashion analogous to Using Lemma 5.32, the analysis of Pmax (T) that for Pmax provided Pmax is sufficiently nontrivial. The proof of Lemma 5.37 requires Theorem 5.34 and Theorem 5.35; these are proved in .Koellner and Woodin 2010/.

Theorem 5.34 (ZF + AD). Suppose Z  Ord. For each x 2 R let HODLŒZ;x ¹Zº denote HOD as computed in LŒZ; x with Z as a parameter. Then there exists x0 2 R such that for all x 2 R, if x0 2 LŒZ; x then .!2 /LŒZ;x is a Woodin cardinal in HODLŒZ;x . ¹Zº

t u

Theorem 5.35 (ZF + DC + AD). Assume V D L.R/. Suppose a  !1 is a countable set. Then HOD¹aº D HODŒa: t u From Theorem 5.34 and Theorem 5.35 we obtain the following theorem which is quite useful in transferring theorems about weakly homogeneously Suslin sets to theorems about all sets of reals in L.R/ assuming ADL.R/ .

5.4 The stationary tower and Pmax

213

Theorem 5.36. Assume AD holds in L.R/. Suppose A  R and A 2 L.R/. Then for each n 2 ! there exist a countable transitive model M and an ordinal ı 2 M such that the following hold. (1) M  ZFC. (2) ı is the nth Woodin cardinal of M . (3) A \ M 2 M and hV!C1 \ M; A \ M; 2i  hV!C1 ; A; 2i: (4) A \ M is ı C -weakly homogeneously Suslin in M . Proof. We work in L.R/. Suppose the theorem fails. Then there exists A 2 L.R/ which is a counterexample and such that A is 21 in L.R/. Let B  R code the first order diagram of hV!C1 ; A; 2i: is 21

Thus B definable in L.R/. Therefore by the Martin–Steel theorem, Theorem 2.3, there exist (definable) trees S and T in L.R/ such that B D pŒS  and R n B D pŒT : Therefore if N  L.R/ is any transitive inner model of ZF such that ¹S; T º  N then A \ N 2 N , B \ N 2 N and hV!C1 \ N; A \ N; 2i  hV!C1 ; A; 2i: We claim that by Theorem 5.34, there exists a transitive inner model N  L.R/ and an increasing sequence hıi W i  n C 1i of countable ordinals such that (1.1) ¹S; T º  N , (1.2) N  ZFC, (1.3) for each i  n C 1, ıi is a Woodin cardinal in N . We indicate how to find N in the case that n D 0, in this case there are to be two Woodin cardinals in N . We work in L.R/. Since S; T are definable, ¹S; T º  HOD. Let Z0  Ord be such that HOD D LŒZ0 . Choose x0 such that .!2 /LŒZ0 ;x0  is a Woodin cardinal in

0 ;x0  : HODLŒZ ¹Z0 º

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5 Applications

Let ı0 D .!2 /LŒZ0 ;x0  : Choose a  ı0 such that and such that

0 ;x0  a 2 HODLŒZ ¹Z0 º 0 ;x0  \ Vı 0 : LŒa \ Vı0 D HODLŒZ ¹Z0 º

Let y0 2 R be such that for all y 2 R if y0 2 LŒZ0 ; aŒy then

0 ;aŒy0  0 ;aŒy D P .ı0 / \ HODLŒZ : P .ı0 / \ HODLŒZ ¹a;Z0 º ¹a;Z0 º

By Turing determinacy y0 exists and it follows that LŒZ0 ;aŒy0  P .ı0 / \ HOD¹a;Z  HOD¹aº : 0º

Therefore by Theorem 5.35, 0 ;aŒy0  P .ı0 / \ HODLŒZ  HODŒa ¹a;Z0 º

and so ı0 is a Woodin cardinal in LŒZ0 ;aŒy0  : HOD¹a;Z 0º

By Theorem 5.34, we may assume by increasing the Turing degree of y0 if necessary that .!2 /LŒZ0 ;a;y0  is a Woodin cardinal in

0 ;a;y0  : HODLŒZ ¹Z0 ;aº

Let ı1 D .!2 /LŒZ0 ;a;y0  and let

0 ;a;y0  : N D HODLŒZ ¹Z0 ;aº

N is as required. The general case for arbitrary n is similar. Let D .2ınC1 /N and let S  and T  be trees on !  such that .S  ; T  / 2 N and such that if g  Coll.!; ınC1 / is N -generic then in N Œg, pŒS  D pŒS   and

pŒT  D pŒT  :

The trees S  ; T  are easily constructed in N by an analysis of terms. Suppose g  Coll.!; ınC1 / is N -generic with g 2 L.R/. The generic filter g exists since !1 is strongly inaccessible in N .

5.4 The stationary tower and Pmax

Thus

215

pŒS   \ N Œg D .R n pŒT  / \ N Œg;

and so by Theorem 2.32, S  and T  are !1 and such that L .R/ †1 L.R/: Let X  L .R/ŒG be a countable elementary substructure such that ¹A; fG º  X . Let MX be the transitive collapse of X and let fX be the image of fG under the collapsing map. Let IX D .INS /MX which is the image of INS under the collapsing map. By Theorem 6.78, .MX ; IX / is A-iterable. Therefore h.MX ; IX /; fX i 2 Qmax : By Theorem 6.76, fg witnesses ˘++ .!1 0º,

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

t u

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

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

Q kQ ı jQ.f0 /.!1M0 / D g;

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

Q kQS ı jQ.f0 /.!1M0 / D g

and such that

Q !1M0 2 kQS .S /:

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

j.a0 / n 5Y1 2 I1

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

t u

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

b \ a 2 I1

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

t u

376

6 Pmax variations

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

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377

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

t u

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

t u

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

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6 Pmax variations

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

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

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

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

6.2 Variations for obtaining !1 -dense ideals

379

Since 1 LŒA0    2 -Determinacy;

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

.A;B/ D !2

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

t u

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

380

6 Pmax variations

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

ˇ >˛

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

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

KT

Qmax is L.R/-generic.

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

t u

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

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6 Pmax variations

Theorem 6.117. Assume AD holds in L.R/. Then for each set A  R with A 2 L.R/, there is a condition h.M; I /; f i 2 Qmax such that the following hold. (1) M  ZFC . (2) I D .INS /M . (3) A \ M 2 M. (4) hH.!1 /M ; A \ M; 2i  hH.!1 /; A; 2i. (5) .M; I / is A-iterable. (6) f witnesses ˘++ .!1 1=2 and such that .X \ O/ > 0 for all open sets O  Œ0;1 with X \ O ¤ ;. The latter condition serves to make A separative. The order on A is by set inclusion. Suppose G  A is V -generic and in V ŒG let X D \¹P

V ŒG

j P 2 Gº

V ŒG

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

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405

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

6 Pmax variations

406

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

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

.P˛ \ O/ > 0:

6.2 Variations for obtaining !1 -dense ideals

407

Let Q  Œ0;1 be a perfect (nowhere dense) set of positive measure. Since Q has positive measure, Q is of measure 1. Fix ˛ < !1 such that P˛  Q . Q is an F and so there exist closed (proper) intervals I  J  Œ0;1 with rational endpoints such that P˛ \ I ¤ ; and such that P˛ \ I D QJ \ P˛ \ I D Q \ P˛ \ I: This implies that J .P˛ \ I / D Q \ J .P˛ \ I / and so J .P˛ \ J / \ J .I /  Q and .J .P˛ \ J / \ J .I // > 0. There are only !1 many sets of the form J .P˛ \ J / \ J .I / where I  J  Œ0;1 are closed subintervals with rational endpoints, ˛ < !1 and I \ P˛ ¤ ;. . Therefore these sets collectively witness (2). Finally we show that (2) implies (3). Let hP˛ W ˛ < !1 i be a sequence of perfect subsets of Œ0;1 each of positive measure such that the sequence witnesses (2). Suppose Q  Œ0;1 is a perfect set of positive measure. For each ˇ < !1 let Xˇ D [¹P˛ j ˛ < ˇ and P˛  Qº: We claim that for all sufficiently large ˇ, .Q n Xˇ / D 0. This is immediate. Suppose ˇ < !1 and .Q n Xˇ / > 0. Then there exists ˛ < !1 such that P˛  Q n Xˇ and so .Xˇ / < .X / for some  < !1 . The claim follows. Let hQ˛ W ˛ < !1 i enumerate the perfect subsets of Œ0;1 which can be expressed as a finite union of the P˛ ’s. Thus hQ˛ W ˛ < !1 i witnesses (3). t u Lemma 6.127 (ZFC ). Assume ˘+ .!1 p for some p 2 f .˛/º: Thus on a club in !1 , F .˛/ is a perfect set of measure 1=2. It is straightforward to u t verify that .!1 ; Œ0;1/ 2 YA .F; I /. Definition 6.128. M Qmax consists of finite sequences h.M; I /; f; F; Y i such that: (1) h.M; I /; f i 2 Qmax ; (2) M  ZFC ; (3) f witnesses ˘++ .!1 1=2: Suppose that A 2 M,

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

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

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

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

.Q \ Pi / > 1=2 t u

and the lemma follows.

With this lemma the main iterations lemmas are easily proved. As usual it is really the proofs of these iteration lemmas which are the key to the analysis of M Qmax . Lemma 6.130 (ZFC C ˘+ .!1 1=2. Suppose A 2 MC1 and A is open dense in the partial order .P .!1 / n IC1 ; /M C1 : By Lemma 6.129, there exists .S  ; P  / 2 YC1 such that S   S , S  2 A and

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

6.2 Variations for obtaining !1 -dense ideals

411

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

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

M C1

/  XG

M C1

/  P :

jC1;C2 .FC1 /.!1 and so

jC1;C2 .FC1 /.!1

This verifies that condition (1.2) can be met at every relevant stage. We consider the effect of condition (1.1). Since g witnesses ˘+ .!1 1=2 then ¹˛ 2 S j j0;!1 .F /.˛/  Qº … J: The other requirements .S; P / must satisfy follow by absoluteness. We first prove (2.1). The key point is that there exists a club C0  !1 such that for all ˛ 2 C0 , D˛ D ¹P 2 AM˛ j P  Qk for some k < !º is dense in AM˛ . The existence of C0 follows from clause (1.2) in the construction of the iteration. Let X  H.!2 / be a countable elementary substructure such that h.M˛ ; I˛ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i and such that D 2 X where D D ¹P 2 AM!1 j P  Qk for some k < !º: Let ˛ D X \ !1 and let MX be the transitive collapse of X . g witnesses ˘+ .!1 1=2, (2.2) .Sk ; Pk / 2 Yk , (2.3) SkC1  Sk , (2.4) The set ¹S 2 .P .!1 //Mk j Si  S for some i 2 !º is Mk -generic for .P .!1 / n Ik ; /Mk .

6.2 Variations for obtaining !1 -dense ideals

415

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

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

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

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

t u

416

6 Pmax variations

The basic analysis of M Qmax follows from these lemmas in a by now familiar fashion. The results of this we give in the following theorem. The analysis requires that M Qmax is suitably nontrivial. More precisely one needs that for each set A  R with A 2 L.R/ there exists h.M; I /; f; F; Y i 2 M Qmax such that hH.!1 /M ; A \ H.!1 /M ; 2i  hH.!1 /; A; 2i and such that .M; I / is A-iterable. By Lemma 6.47 and Lemma 6.127, this follows from the existence of a huge cardinal. Theorem 6.132. Assume that for each set A  R with A 2 L.R/ there exists h.M; I /; f; F; Y i 2 M Qmax such that (i) hH.!1 /M ; A \ H.!1 /M ; 2i  hH.!1 /; A; 2i, (ii) .M; I / is A-iterable. Then M Qmax is !-closed and homogeneous. Suppose G  M Qmax is L.R/-generic. Then L.R/ŒG  !1 -DC and in L.R/ŒG: (1) P .!1 /G D P .!1 /; (2) IG is a normal !1 -dense ideal on !1 ; (3) IG is the nonstationary ideal; (4) YG D YA .FG ; INS /; (5) fG witnesses ˘++ .!1 1=2 then ¹a 2 S j F .a \ !1 /  Qº is stationary in S . (5) For all a 2 S , .Q/ D 1=2 where Q D F .a \ !1 /. The relationship between YA .F; INS / and YA .F; ı/ is summarized in the following lemma which is an immediate consequence of the definitions. Lemma 6.135. Suppose that ı is strongly inaccessible and that F W !1 ! P .Œ0;1/: Then YA .F; INS / D ¹.S; P / j .S; P / 2 YA .F; ı/ and S  !1 º:

t u

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

7.2 The Borel Conjecture

445

(3.3) p 2 Coll.!1 ; < C 1/, (3.4) the set ¹q 2 Coll.!1 ; < C 1/ j p < qº is X -generic, (3.5)

a) if  \ mc.Vı / has no maximum element then (2.2) is satisfied by X at  , otherwise, b) (2.3) is satisfied by X at  .

For the construction of this elementary chain we note that the requirements corresponding to the desired properties, (3.5(a)) and (3.5(b)), do not conflict. The requirement which yields (3.5(a)) is easily handled using the definition of a semiproper subset of P .!1 / n INS . The requirement for (3.5(b)) is handled using the following observation. Suppose Y M is a countable elementary substructure and that  2 Y \ ı is a measurable cardinal. Then there exists a closed unbounded set C  !1 such that for each ˛ 2 C there exists Y  M such that Y  Y  , such that Y \  is an initial segment of Y  \  and such that Y \ has ordertype ˛. Now suppose that 2 V Coll.!1 ;< C1/ is a term for a stationary, co-stationary subset of !1 and that q 2 Coll.!1 ; 1 and p.k/ D k m for all sufficiently large k < !; (3) for all i < n and for all j < !, hi .j / D hi .k/ where k < ! is such that p.k/  j < p.k C 1/.

t u

Remark 7.29. In Definition 7.27, condition (2) can be replaced by  for each m 2 ! there exists p 2 ! ! such that: (1) p.0/ D 0 and p.k/ < p.k C 1/ for all k < !; (2) p.k/ D k m for all sufficiently large k < !; (3)

.h.fi ; hi / W i < ni; S / 2 YBC .I /; where for all i < n and for all j < !, if k 2 ! and p.k/  j < p.k C 1/ then hi .j / D hi .k/:

The analogous remark applies to Definition 7.28. We record in the following lemma sufficient conditions for membership in YBC .I /. Lemma 7.30. Suppose I is a uniform normal ideal on !1 . Suppose that .h.fi ; hi / W i < ni; S / 2 ZBC .I / is such that (1) for all i < n, .h.fi ; hi /i; S / 2 YBC .I /,  (2) .h.fi ; hi / W i < ni; S / 2 YBC .I /.

Then .h.fi ; hi / W i < ni; S / 2 YBC .I /: Proof. Define for each ordinal ˛, a subset ˛ .I / ZBC

as follows. 0 ZBC .I / D ZBC .I / and if ˛ is a limit ordinal then ˛ ˇ .I / D \¹ZBC .I / j ˇ < ˛º: ZBC ˛C1 Finally for each ordinal ˛, ZBC .I / is the set of ˛ .h.fOi ; hO i / W i < ni; O SO / 2 ZBC .I /

t u

7.2 The Borel Conjecture

453

such that: ˛ .I / and such (1.1) for each i < nO there exists g 2 ! ! such that .h.fOi ; g/i; SO / 2 ZBC that for sufficiently large k 2 !,

g.j /  hO i .k/ for all j < 5k ; (1.2) for some p 2 ! ! ,

˛ O SO / 2 ZBC .I / .h.fOi ; hO i / W i < ni;

where: a) p.0/ D 0 and p.k/ < p.k C 1/ for all k < !; b) for some m 2 !, m > 1 and p.k/ D k m for all sufficiently large k < !; c) for all i < nO and for all j < !, hO i .j / D hO i .k/ where k < ! is such that p.k/  j < p.k C 1/. Thus for for sufficiently large ˛, ˛ .I / D YBC .I /: ZBC

Fix .h.fi ; hi / W i < ni; S / 2 ZBC .I / satisfying the conditions of the lemma and assume toward a contradiction that .h.fi ; hi / W i < ni; S / … YBC .I /: Thus for some ordinal ˛, ˛ .I /: .h.fi ; hi / W i < ni; S / … ZBC

We may suppose that the choice of .h.fi ; hi / W i < ni; S / minimizes ˛. Thus ˛ is a successor ordinal. Let ˛0 be such that ˛ D ˛0 C 1. Let p 2 ! ! be a function such that  .I / .h.fi ; hi / W i < ni; S / 2 YBC

where: (2.1) p.0/ D 0 and p.k/ < p.k C 1/ for all k < !; (2.2) for some m 2 !, m > 1 and for all sufficiently large k < !;

p.k/ D k m

454

7 Conditional variations

(2.3) for all i < n and for all j < !, hi .j / D hi .k/ where k < ! is such that p.k/  j < p.k C 1/. We claim that for all i < n, .h.fi ; hi /i; S / 2 YBC .I /: Fix i < n. Since

 .I / .h.fi ; hi /i; S / 2 YBC

it follows that the elements of YBC .I / which witness .h.fi ; hi /i; S / 2 YBC .I / can be used to witness that .h.fi ; hi /i; S / 2 YBC .I /: Thus .h.fi ; hi / W i < ni; S / satisfies the conditions of the lemma and so, ˛0 .h.fi ; hi / W i < ni; S / 2 ZBC .I /:

But this implies that ˛0 C1 .h.fi ; hi / W i < ni; S / 2 ZBC .I /;

t u

a contradiction.

Definition 7.31. Bmax consists of finite sequences h.M; I /; a; Y i such that the following hold. (1) M is a countable transitive model of ZFC. (2) M  CH. (3) I 2 M and M  “I is a normal uniform ideal on !1 ” . (4) M 

AC .I /.

(5) .M; I / is iterable. (6) Y 2 M is the set YBC .I / as computed in M. (7) a 2 M, a  !1M and M  “!1 D !1LŒaŒx for some real x”. The order on Bmax is defined as follows. Suppose that h.M1 ; I1 /; a1 ; Y1 i and h.M2 ; I2 /; a2 ; Y2 i are conditions in Bmax . Then h.M2 ; I2 /; a2 ; Y2 i < h.M1 ; I1 /; a1 ; Y1 i if

M1 2 H.!1 /M2

7.2 The Borel Conjecture

455

and there exists an iteration j W .M1 ; I1 / ! .M1 ; I1 / such that (1) j.a1 / D a2 , (2) I1 D I2 \ M1 , (3) j.Y1 / D Y2 \ M1 .

t u

Remark 7.32. (1) The only reason for requiring that the models occurring in the Bmax conditions actually be models of ZFC instead of ZFC is so that (6) in the definition of Bmax is unambiguous. The trivial point is that ZFC does not prove that YBC .I / exists. (2) There is actually a parameterized family of generalizations of Bmax . Fix a function h 2 ! ! . Let Bmax .h/ be the suborder of Bmax consisting of those conditions h.M; I /; a; Y i such that if g 2 ! ! and g occurs in Y then for sufficiently large i 2 !, h.i / < g.i /. t u We prove the basic iteration lemmas for Bmax . There are two iteration lemmas, one for models and one for sequences of models. The latter is necessary to show that Bmax is !-closed and its proof is an intrinsic part of the analysis of Bmax just as in the case of Pmax . We need several preliminary lemmas. For all m 2 ! and for all h 2 ! ! , let h.m/ be the function obtained by shifting h, h.m/ .k/ D h.m C k/ for k 2 !. Thus a set X  Œ0;1 is ŒhE -small if and only if X is ¹h.m/ j m < !º-small. We note that if .h.fi ; hi / W i < ni; S / 2 ZBC .I / then for all m 2 !,

.h.fi ; h.m/ i / W i < ni; S / 2 ZBC .I /:

This is immediate from the definition of ZBC .I /. We claim that if .h.fi ; hi / W i < ni; S / 2 YBC .I / then for all m 2 !,

.h.fi ; h.m/ i / W i < ni; S / 2 YBC .I /:

This is easily verified noting the following. Suppose g 2 ! ! and h 2 ! ! are such that for all k m0 , g.j /  h.k/

456

7 Conditional variations

for all j < 5k . Then for any m 2 !, g .m/ .j /  h.m/ .k/ for all j < 5k and for all k max¹m0 m; 0º. One reason for thinning ZBC .I / to obtain YBC .I / is the following. Suppose that .h.fi ; hi / W i < ni; S / 2 YBC .I / and fix i < n. Fix g 2 ! ! such that .h.fi ; g/i; S / 2 YBC .I / and such that for sufficiently large k 2 !, g.j /  hi .k/ for all j < 5k . Fix m0 such that for all k > m0 , g.j /  hi .k/ if j < 5k . For each m 2 ! let Hm 2 ! ! be such that for all j 2 !, Hm .j / D hi .k C m/ where k is such that 2  j C 1 < 2kC1 . Suppose that for each m 2 !, Om is Hm -small. For each m1 > m0 let k

Xm1 D [¹Om j m > m1 º: Then Xm1 is g .m1 / -small. This observation, which we prove in the next lemma, is the key to proving the subsequent lemmas. Lemma 7.33. Suppose g 2 ! ! , h 2 ! ! , and h.k/ g.j / for all j < 5k . For each m 2 ! let Hm 2 ! ! be such that for all j 2 !, Hm .j / D h.k C m/ where k is such that 2k  j C 1 < 2kC1 . Suppose that for each m 2 !, Om is Hm -small. For each n 2 ! let Xn D [¹Om j m > nº: Then for each n 2 !, Xn is g .n/ -small. Proof. For each m 2 ! let hIjm W j < !i be a sequence of open intervals which witnesses that Om is Hm -small, so that for each m 2 !,

.Ijm / < 1=.Hm .j / C 1/; for each j 2 !.

7.2 The Borel Conjecture

457

Fix n 2 !. It suffices to show that ¹Ijm j m > n; j 2 !º witnesses that Xn is g .n/ -small. We note that for each a 2 ! with a > n, X

2k D

kCmDa;am>n

a.nC1/ X

2k D 2an 1  2aC1 n

kD0

and so for each a 2 !, j¹Ijm j 2k  j C 1 < 2kC1 where k D a m and a m > nºj  2aC1 n: For each a 2 ! such that a > n let Ja D ¹Ijm j 2k  j C 1 < 2kC1 where k D a m and a m > nº: Thus [¹Ja j a 2 !; a > 0º D ¹Ijm j m > n; j 2 !º: Each interval I 2 Ja has length at most 1=.h.a/ C 1/. For each a 2 ! such that a > n, g .n/ .j /  h.a/ for all j such that 5a1  n C j < 5a . If a > n, j¹j j 5a1  n C j < 5a ºj 4 5a1 n and so jJa j  j¹j j 5a1  n C j < 5a ºj; noting that since a > n, jJa j  2aC1 n: Thus ¹Ijm j m > n; j 2 !º t u

witnesses that Xn is g .n/ -small. Lemma 7.34. Suppose I is a uniform normal ideal on !1 , g 2 ! ! , and .h.f; h/i; S / 2 ZBC .I /; where for all j 2 !, if k 2 ! and 2k  j C 1 < 2kC1 then h.j / D g.k/: Then

 .I /: .h.f; g/i; S / 2 YBC

Proof. This is immediate from the definitions.

t u

458

7 Conditional variations

Lemma 7.35. Suppose I is a uniform normal ideal on !1 . Suppose .h.f0 ; h0 /i; S / 2 YBC .I /; and that T  S is a set such that T … I . Then there exists m < ! such that .h.f0 ; h.m/ 0 /i; T / 2 YBC .I /: Proof. For each m 2 ! let Hm 2 ! ! be such that for all j 2 !, Hm .j / D h0 .k C m/ k where k is such that 2  j C 1 < 2kC1 . We first prove that for some m 2 !, .h.f0 ; Hm /i; T / 2 ZBC .I /: If this fails then there is a sequence hOm W m < !i of open sets such that (1.1) T n ¹˛ 2 T j f0 .˛/ 2 Om º 2 I , (1.2) for all m < !, Om is Hm -small. Let g 2 ! ! be such that (2.1) .h.f0 ; g/i; S / 2 ZBC .I /, (2.2) for sufficiently large k 2 !, g.j /  h0 .k/ if j < 5k . By Lemma 7.33, for sufficiently large k 2 !, [¹Om j m > kº .k/ is g -small. Let B D \¹[¹Om j m > kº j k 2 !º and so B is ŒgE -small. However by (1.1) ¹˛ 2 T j f0 .˛/ 2 Bº … I which is a contradiction since T  S . Fix m0 2 ! such that .h.f0 ; Hm0 /i; T / 2 ZBC .I /: By Lemma 7.34,  0/ .h.f0 ; h.m /i; T / 2 YBC .I /: 0 For each h 2 ! ! and for each s 2 ! kº is gi.k/ -small. For each i < n C 1 let Bi D \¹[¹Oim j m > kº j k 2 !º and so for each i < n C 1, Bi is Œgi E -small. Therefore, since for each i < n C 1, .h.fi ; gi /i; S / 2 ZBC .I /; there is a set T2  T1 such that T1 n T2 2 I and such that for all i < n C 1 and for all ˛ 2 T2 , fi .˛/ … Bi . Fix ˛ 2 T2 . Then ˛ 2 T1 and so for all m 2 !, fi .˛/ 2 Oim for some i < n C 1. Thus for some i < n C 1, the set ¹m 2 ! j fi .˛/ 2 Oim º is infinite and so fi .˛/ 2 Bi which is a contradiction. Therefore for some m 2 !, .h.fi ; h.m/ i / W i < n C 1i; T / 2 YBC .I /.

t u

Lemma 7.36 can be recast as follows. This reformulation is in essence what is required to prove the iteration lemmas. Lemma 7.37. Suppose that h.M; I /; a; Y i 2 Bmax , .h.fi ; hi / W i < ni; S / 2 Y , .h.fn ; hn /i; S / 2 Y , and that hOi W i < ni is a finite sequence of open sets such that each Oi is hi -small. For each i < n let hIki W k < !i be a sequence of open intervals in .0;1/ with rational endpoints such that the sequence witnesses that Oi is hi -small. Suppose A 2 M, A  .P .!1 / n I /M ; and A is dense below S in .P .!1 / n I; /M . Suppose m0 2 !. There exists m > m0 and there exists T 2 A such that (1) T  S , (2) .h.fi ; h.m/ i / W i < n C 1i; T / 2 Y , (3) fi .˛/ … Iki for all k < m, for all i < n, and for all ˛ 2 T . Proof. This follows from Lemma 7.36 by absoluteness. Fix m0 2 !. Let T be the tree of attempts to build the sequences hIki W k < !i to refute the lemma. So T is the set of hti W i < ni such that for some m > m0 :

462

7 Conditional variations

(1.1) For all i < n, ti D h.rki ; ski / W k < mi where for all k < m, a) 0  rki < ski  1, b) rki 2 Q, c) ski 2 Q, d) .ski rki / < 1=.hi .k/ C 1/. (1.2) For all T  S with T 2 A, either .h.fi ; h.m/ i / W i < n C 1i; T / … Y; or for some i < n and for some ˛ 2 T , fi .˛/ 2 [¹.rki ; ski / j k < mº: The ordering on T is by (pointwise) extension, hsi W i < ni  hti W i < ni if ti  si for all i < n. Clearly T 2 M. Suppose T has an infinite branch. Then by absoluteness, T has an infinite branch in M. We work in M and assume toward a contradiction that T has an infinite branch. Any such branch yields for each i < n a sequence h.rki ; ski / W k < !i of open intervals in .0;1/ with rational endpoints such that for all i < n and for all k < !, jski rki j < 1=.hi .k/ C 1/: These sequences have the additional property that for all T 2 A such that T  S and for all m < ! either .h.fi ; h.m/ i / W i < n C 1i; T / … Y; or for some i < n and for some ˛ 2 T , fi .˛/ 2 [¹.rki ; ski / j k < mº: For each i < n let OQ i D [¹.rki ; ski / j k < !º: Thus for each i < n, OQ i is hi -small. Let T0 D ¹˛ 2 S j fi .˛/ … OQ i for all i < nº: Since .h.fi ; hi / W i < ni; S / 2 Y ,

T0 … I:

A is dense below S and so there exists T 2 A such that T  T0 . By Lemma 7.36, there exists m < ! such that .h.fi ; h.m/ i / W i < n C 1i; T / 2 Y: This is a contradiction and so T is wellfounded in M. Hence T is wellfounded in V . t u

7.2 The Borel Conjecture

463

The iteration lemmas are proved using the following lemmas which in turn follow rather easily from the previous lemmas. Lemma 7.38. Suppose h.M; I /; a; Y i 2 Bmax . Suppose .h.fi ; hi / W i < ni; S / is an element of Y and suppose h.fi ; hi ; Si / W i < !i is a sequence extending h.fi ; hi ; S / W i < ni such that for each i < ! if i n then .h.fi ; hi /i; Si / 2 Y . Suppose hBi W i < !i is a sequence of borel sets such that each i < !, if i < n then Bi is hi -small and if i n then Bi is Œhi E -small. Then there is an iteration j W .M; I / ! .M ; I  / of length 1 such that (1) for all i < !, if !1M 2 j.Si / then j.fi /.!1M / … Bi , (2) !1M 2 j.S /. Proof. Let hAi W n  i < !i enumerate the sets in M which are dense in .P .!1 / n I /M . Using Lemma 7.37 it is straightforward to build sequences hTi W i < !i;

hIji W i; j < !i;

and

hNi W i < !i

such that for all i < ! the following hold. Let Z D ¹i < ! j Si \ Ti ¤ ;º: (1.1) Ni D 0 and Ti D S for i < n. (1.2) If i n then Ti 2 Ai and either Ti  Si or Ti \ Si D ;. (1.3) If i n then TiC1  Ti  S , Ni 2 ! and Ni < NiC1 . (1.4) Iji is an open interval with rational endpoints and

.Iji / < 1=.hi .Ni C j / C 1/: (1.5) Bi  [¹Iji j j < !º. .Ni /

(1.6) .h.fj ; hj

/ W j 2 Z \ i i; Ti / 2 Y .

(1.7) If i < n then for all ˛ 2 Tn , for all j < i , fj .˛/ … [¹Ikj j k < Nn º: (1.8) If i n then for all ˛ 2 TiC1 , for all j < i , if j 2 Z then fj .˛/ … [¹Ikj j k < NiC1 º:

464

7 Conditional variations

We first construct hTi W i  ni;

hIji W i < n; j < !i;

and

hNi W i  ni:

For this we need only specify hIji W i < n; j < !i; Tn and Nn . For each i < n let hIji W j < !i be a sequence of open intervals with rational endpoints such that Bi  [¹Iji j j < !º and such that for all j < !,

.Iji / < 1=.hi C 1/: By Lemma 7.37, there exist L0 2 ! and T 0 2 An such that (2.1) T 0  Sn or T 0 \ Sn D ;, (2.2) T 0  S , 0

(2.3) .h.fi ; hi.L / / W i < ni; T 0 / 2 Y , (2.4) for all k < L0 , for all i < n, fi .˛/ … Iki for all ˛ 2 T 0 . Let Tn D T 0 and let Nn D L0 . We next suppose m n and that hTi W i  mi;

hIji W i < m; j < !i;

and

hNi W i  mi

i are given. For each i < m and k < ! let Jki D IkCN . Therefore m m/ / W i 2 Z \ mi; Tm / 2 Y .h.fi ; h.N i

and for each i < m, the sequence hJki W k < !i witnesses that Oi is .hi.Nm / /-small where Oi D [¹Jki j k < !º: By Lemma 7.37, there exist L0 2 ! and T 0 2 AmC1 such that (3.1) T 0  SmC1 or T 0 \ SmC1 D ;, (3.2) T 0  Tm , 0

m / .L / / / W i 2 Z \ mi; T 0 / 2 Y , (3.3) .h.fi ; .h.N i

(3.4) for all k < L0 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 0 .

7.2 The Borel Conjecture

465

By Lemma 7.37 again, there exist L00 2 ! and T 00 2 AmC1 such that (4.1) T 00  T 0 , 00

m / .L / / / W i 2 Z \ m C 1i; T 00 / 2 Y , (4.2) .h.fi ; .h.N i

(4.3) for all k < L00 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 00 . Of course if T 0 \ SmC1 D ; then one can simply let T 00 D T 0 and L00 D L0 . Set TmC1 D T 00 and NmC1 D Nm C L00 . Choose a sequence hJk W k < !i such .N / that hJk W k < !i witnesses that Bm is hm mC1 -small. The sequence exists since Bm is Œhm E -small. For each k < ! set Ikm D Jk . Therefore by induction the sequences exist. Let G be the filter generated by ¹Ti j i < !º. Thus G is M-generic. Let j W .M; I / ! .M ; I  / be the associated iteration of length 1. It follows from (1.5), (1.7), and (1.8) that for all t u i < !, if !1M 2 j.Si / then j.fi /.!1M / … Bi . There is an analogous version of the previous lemma for sequences of models. We shall apply this lemma only to sequences which are iterable. However the lemma holds for sequences which are not necessarily iterable and it is this more general version which we shall prove, (for no particular reason). Lemma 7.39. Suppose that h.Mk ; Ik / W k < !i is a sequence such that for each k < !, Mk is a countable transitive model of ZFC, Ik 2 Mk and such that in Mk , Ik M is a uniform normal ideal on !1 k . For each k < ! let Yk D .YBC .Ik //Mk : Suppose that for all k < !, (i) Mk 2 MkC1 , (ii) jMk jMkC1 D .!1 /MkC1 , Mk

(iii) !1

MkC1

D !1

,

(iv) IkC1 \ Mk D Ik , (v) Yk D YkC1 \ Mk , M

(vi) for each A 2 Mk such that A  P .!1 k / \ Mk n Ik , if A is predense in .P .!1 / n Ik /Mk then A is predense in .P .!1 / n IkC1 /MkC1 .

466

7 Conditional variations

Suppose .h.fi ; hi / W i < ni; S / is an element of Y0 and suppose h.fi ; hi ; Si / W i < !i is a sequence extending h.fi ; hi ; S / W i < ni such that for each i < ! if i n then .h.fi ; hi /i; Si / 2 Yi . Suppose hBi W i < !i is a sequence of borel sets such that each i < !, if i < n then Bi is hi -small and if i n then Bi is Œhi E -small. Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i of length 1 such that (1) for all i < !, if !1M0 2 j.Si / then j.fi /.!1M0 / … Bi , (2) !1M0 2 j.S /. Proof. Let hAi W n  i < !i enumerate the sets A 2 [¹Mk j k 2 !º such that if A 2 Mk then

A  .P .!1 / n Ik /Mk

and A is predense in

.P .!1 / n Ik ; /Mk :

By (vi) in the hypothesis of the lemma we can suppose that for each i < !, Ai 2 Mi : Following the proof of Lemma 7.38 it is straightforward, using Lemma 7.37 and (v), to build sequences hTi W i < !i, hIji W i; j < !i and hNi W i < !i such that for all i < ! the following hold. Let Z D ¹i < ! j Si \ Ti ¤ ;º: (1.1) Ni D 0 and Ti D S for i < n. (1.2) If i n then Ti 2 Ai and either Ti  Si or Ti \ Si D ;. (1.3) If i n then TiC1  Ti  S , Ni 2 ! and Ni < NiC1 . (1.4) Iji is an open interval with rational endpoints and

.Iji / < 1=.hi .Ni C j / C 1/: (1.5) Bi  [¹Iji j j < !º. .Ni /

(1.6) .h.fj ; hj

/ W j 2 Z \ i i; Ti / 2 Yi .

7.2 The Borel Conjecture

467

(1.7) If i < n then for all ˛ 2 Tn , for all j < i , fj .˛/ … [¹Ikj j k < Nn º: (1.8) If i n then for all ˛ 2 TiC1 , for all j < i , if j 2 Z then fj .˛/ … [¹Ikj j k < NiC1 º: We first construct hTi W i  ni;

hIji W i < n; j < !i;

and hNi W i  ni:

For this we need only specify hIji W i < n; j < !i; Tn and Nn . For each i < n let hIji W j < !i be a sequence of open intervals with rational endpoints such that Bi  [¹Iji j j < !º and such that for all j < !,

.Iji / < 1=.hi C 1/: By Lemma 7.37, there exist L0 2 ! and T 0 2 An such that (2.1) T 0  Sn or T 0 \ Sn D ;, (2.2) T 0  S , 0

(2.3) .h.fi ; hi.L / / W i < ni; T 0 / 2 Yn , (2.4) for all k < L0 , for all i < n, fi .˛/ … Iki for all ˛ 2 T 0 . Let Tn D T 0 and let Nn D L0 . We next suppose m n and that hTi W i  mi;

hIji W i < m; j < !i;

and

hNi W i  mi

i are given. For each i < m and k < ! let Jki D IkCN . m Therefore m/ .h.fi ; h.N / W i 2 Z \ mi; Tm / 2 Ym i

and for each i < m, the sequence hJki W k < !i witnesses that Oi is .hi.Nm / /-small where Oi D [¹Jki j k < !º: By (v), Ym D YmC1 \ Mm and so

m/ .h.fi ; h.N / W i 2 Z \ mi; Sm / 2 YmC1 : i

468

7 Conditional variations

By Lemma 7.37, there exist L0 2 ! and T 0 2 MmC1 such that (3.1) T 0  !1M0 and T 0  a for some a 2 AmC1 , (3.2) T 0  SmC1 or T 0 \ SmC1 D ;, (3.3) T 0  Tm , 0

m / .L / (3.4) .h.fi ; .h.N / / W i 2 Z \ mi; T 0 / 2 Ym , i

(3.5) for all k < L0 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 0 . By Lemma 7.37 once more, there exist L00 2 ! and T 00 2 MmC1 such that (4.1) T 00  T 0 , 00

m / .L / / / W i 2 Z \ m C 1i; T 00 / 2 YmC1 , (4.2) .h.fi ; .h.N i

(4.3) for all k < L00 , for all i 2 Z \ m, fi .˛/ … Jki for all ˛ 2 T 00 . Of course, as in the proof of Lemma 7.38, if T 0 \ SmC1 D ; then one can simply let T 00 D T 0 and L00 D L0 . Set TmC1 D T 00 and NmC1 D Nm C L00 . Choose a sequence hJk W k < !i such .N / that hJk W k < !i witnesses that Bm is hm mC1 -small. The sequence exists since Bm is Œhm E -small. For each k < ! set Ikm D Jk . Therefore by induction the sequences exist. Let G be the filter generated by ¹Ti j i < !º. Thus G is [¹Mi j i < !º-generic. Let j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i be the associated iteration of length 1. It follows from (1.5), (1.7) and (1.8) that for all t u i < !, if !1M0 2 j.Si / then j.fi /.!1M0 / … Bi . With these lemmas the main iterations lemmas are easily proved. As usual it is really the proofs of these iteration lemmas which are the key to the analysis of Bmax . Lemma 7.40 (CH). Suppose h.M; I /; a; Y i 2 Bmax and that J is a normal uniform ideal on !1 . Then there is an iteration j W .M; I / ! .M ; I  / such that (1) J \ M D I  , (2) j.Y / D YBC .J / \ M .

7.2 The Borel Conjecture

469

Proof. Let hSk;˛ W k < !; ˛ < !1 i be a sequence of pairwise disjoint J -positive sets such that !1 D [¹Sk;˛ j k < !; ˛ < !1 º: Let hs˛ W ˛ < !1 i be an enumeration (with repetition) of all finite sequences of open subsets of .0;1/ such that for each finite sequence s of open subsets of .0;1/, and for each .k; ˛/ 2 !  !1 , ¹ı 2 Sk;˛ j s D sı º is a set which is J -positive. Let hB˛ W ˛ < !1 i be an enumeration of all the borel subsets of .0;1/. Let x be a real which codes M and let C  !1 be a closed unbounded set of ordinals which are admissible relative to x. Fix a function F W !  !1M ! M such that (1.1) F is onto, (1.2) for all k < !, F jk  !1M 2 M, (1.3) for all A 2 M if A has cardinality !1M in M then A  ran.F jk  !1M / for some k < !. The function F is simply used to anticipate elements in the final model. Our situation is similar to that in the proof of Lemma 7.7. Suppose j W .M; I / ! .M ; I  / is an iteration. Then we define j.F / D [¹j.F jk  !1M / j k < !º and it is easily verified that M is the range of j.F /. This follows from (1.3). Implicit in what follows is that for ˇ 2 C if j W .M; I / ! .M ; I  / is an iteration of length ˇ then j.!1M / D ˇ. This is a consequence of Lemma 4.6(1). We construct an iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i of M of length !1 using the function F to provide a book-keeping device for dealing with elements of j0;!1 ..P .!1 / n I /M / and for dealing with elements of j0;!1 .Y /.

470

7 Conditional variations

More precisely construct by induction, an iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i as follows. Suppose ı < !1 and that h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  ıi: Fix .k; / 2 !  !1 such that ı 2 Sk; . If ı … C or if  ı then choose Gı to be any Mı -generic filter. If ı 2 C and if  < ı there are three cases. We first suppose that j0;ı .F /.k; / D .h.fi ; hi / W i < ni; S / and that .h.fi ; hi / W i < ni; S / 2 j0;ı .Y /. Suppose sı D hOi W i < ni is a sequence of length n such that for each i < n, Oi is hi -small. Let h.fi ; hi ; Si / W i < !i be a sequence extending the sequence h.fi ; hi ; S / W i < ni such that for all i < !, .h.fi ; hi /i; Si / 2 j0;ı .Y / and such that for all

.h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y /;

.h.f 0 ; h0 /i; S 0 / D .h.fi ; hi /i; Si / for infinitely many i < !. Let hBi0 W i < !i be a sequence of borel sets extending hOi W i < ni such that for all i n, Bi0 is Œhi E -small and such that for all ˛ < ı if .h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y / and if B˛ is h0 -small then for some j > n, B˛ D Bj0 and .h.f 0 ; h0 /i; S 0 / D .h.fj ; hj /i; Sj /: By Lemma 7.38, there exists an iteration j W .Mı ; Iı / ! .MıC1 ; IıC1 / of length 1 such that Mı

(2.1) for all i < !, if !1 Mı

(2.2) !1

M

2 j.Si / then j.fi /.!1 ı / … Bi0 ,

2 j.S /.

Let jı;ıC1 D j and let Gı be the associated Mı -generic filter. The remaining cases are similar. Choose j W .Mı ; Iı / ! .MıC1 ; IıC1 /

7.2 The Borel Conjecture

471

of length 1 such that for all .h.f 0 ; h0 /i; S 0 / 2 j0;ı .Y / if

Mı

!1 then

2 j.S 0 / M

j.f 0 /.!1 ı / … B˛

for all ˛ < ı such that B˛ is h0 -small. Let jı;ıC1 D j and let Gı be the associated Mı -generic filter. If j0;ı .F /.k; / 2 .P .!1 / n Iı /Mı ; then choose j such that in addition to the requirement above, Mı

!1

2 j.S /

where S D j0;ı .F /.k; /. In each of these last two cases j exists by Lemma 7.38. This completes the inductive construction of the iteration h.Mˇ ; Iˇ /; G˛ ; j˛;ˇ W ˛ < ˇ  !1 i: It is straightforward to verify that this iteration is as required. The first case of the construction at the inductive step guarantees that j0;!1 .Y /  ZBC .J / and this implies that j0;!1 .Y /  YBC .J /: The second case guarantees J \ M!1 D I!1 and so j0;!1 .Y / D YBC .J / \ M!1 :

t u

The analysis of the Bmax -extension requires the generalization of Lemma 7.40 to sequences of models. We state this lemma only for the sequences that arise, specifically those sequences of structures coming from descending sequences of conditions in Bmax . Suppose that hpk W k < !i is a sequence of conditions in Bmax such that for all k < !, pkC1 < pk : We let hpk W k < !i be the associated sequence of conditions which is defined as follows. For each k < ! let h.Mk ; Ik /; ak ; Yk i D pk and let

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

472

7 Conditional variations

be the iteration obtained by combining the iterations given by the conditions pi for i > k. Thus jk is uniquely specified by the requirement that jk .ak / D [¹ai j i < !º: For each k < !, pk D h.M ; Ik /; jk .ak /; jk .Yk /i: We note that by Corollary 4.20, the sequence h.Mk ; Ik / W k < !i is iterable (in the sense of Definition 4.8). Lemma 7.41 (CH). Suppose hpk W k < !i is a sequence of conditions in Bmax such that for each k < ! pkC1 < pk : Let hpk W k < !i be the associated sequence of Bmax conditions and for each k < ! let h.Mk ; Ik /; ak ; Yk i D pk : Suppose that J is a normal uniform ideal on !1 . Then there is an iteration j W h.Mk ; Ik / W k < !i ! h.Mk ; Ik / W k < !i such that for all k < !; (1) J \ Mk D Ik , (2) YBC .J / \ Mk D j.Yk /. Proof. By Corollary 4.20, the sequence h.Mk ; Ik / W k < !i is iterable. The lemma follows by a routine modification of the proof of Lemma 7.40 using Lemma 7.39 in place of Lemma 7.38. t u Theorem 7.43 establishes the nontriviality of Bmax in the sense required for the analysis of L.R/Bmax . The proof requires Theorem 7.18, Theorem 7.42 and the transfer principle supplied by Theorem 5.36.

7.2 The Borel Conjecture

473

Theorem 7.42. Suppose that ı is a Woodin cardinal. Suppose A  R and that every set of reals which is projective in A is ı C -weakly homogeneously Suslin. Then there is an iterable structure .M; I / such that M  ZFC C CH and such that (1) M 

AC .I /,

(2) A \ M 2 M, (3) hH.!1 /M ; A \ Mi  hH.!1 /; Ai, (4) .M; I / is A-iterable. Proof. Suppose that G  Coll.!1 ; 0, if necessary, we may suppose that for all k < !, Nk < NkC1 : For each m 2 ! let m 2 U be such that .t; m /  1 .k/  H t .k/ for all t  m C 1 and for all k  NmC1 . The ultrafilter U is selective and so there exists  2 U such that for all k 2 !, j  \ ŒNk ; NkC1 /j  1; and such that for all s 2 Œ   mº  m where m D [s. For each k 2 ! let ak be the least element of  n Nk . Let J be the set of intervals of the form Hs .j / such that for some k 2 !, Nk  j < NkC1 and such that

s  .NkC1 \  / [ ¹akC1 º:

7.2 The Borel Conjecture

Let W D [J. We claim that W is h -small and that .;;  /  0  WG : We first prove that Suppose

.;;  /  0  WG : .s; /  .;;  /

and that j < [s. Let k 2 ! be such that Nk  j < NkC1 : We may suppose that

 ¹k 2  j k > [sº:

There are two cases. First, suppose that akC1 2 s: Then .s; /  1 .j /  H t .j / where t D s \ .akC1 C 1/. However H t .j / 2 J and so .s; /  1 .j /  WG : Second, suppose that akC1 … s: Then .s; /  1 .j /  H t .j / where t D s \ akC1 . Again H t .j / 2 J and so again .s; /  1 .j /  WG : Thus

.;;  /  0  WG :

We finish by proving that W is h -small. We note that since for all k 2 !, j  \ ŒNk ; NkC1 /j  1; it follows that for all k 2 !, jP .  \ NkC1 /j  2kC1 : Suppose m 2 ! and let k 2 ! be such that Nk  m < NkC1 : Let Jm be the set of intervals of the form Hs .m/ such that s  .NkC1 \  / [ ¹akC1 º: Thus J D [¹Jm j m 2 !º:

479

480

7 Conditional variations

Further for each m 2 !,

jJm j  2kC2

and each interval in Jm has length at most 2=.h.m/ C 1/ where k 2 ! is such that

Nk  m < NkC1 :

 be the collection of intervals For each m 2 !, let Jm  interval .a; b/ 2 Jm , Jm contains the intervals,

.a; .a C b/=2/; 

Let J D

 [¹Jm

obtained as follows. For each

.a C .b a/=4; b .b a/=4/;

and

..a C b/=2; b/:

j m < !º. Thus W D [J  :

Suppose m 2 ! and let k 2 ! be such that Nk  m < NkC1 : Each interval in

 Jm

has length at most 1=.h.m/ C 1/ and  j  3 2kC2 : jJm

For each j 2 ! such that

p.m/  j < p.m C 1/;

we have that h .j / D h.m/. Further since m Nk , p.m C 1/ p.m/ 3 2kC2 : It follows that W is h -small.

t u

Suppose G  PU is V -generic. Let aG D [¹s j .s; / 2 Gº and let hG W ! ! ! be the enumeration function of aG . We note the following. Suppose that I is a normal, uniform, ideal on !1 and that P is ccc. Suppose that G  P is V -generic. Then in V ŒG the ideal I defines three ideals, (1) I0 which is the ideal generated by I , I0 D ¹A  !1 j A  B for some B 2 I º; (2) I1 which is the -ideal generated by I0 , (3) I2 which is the normal ideal generated by I0 . Under certain circumstances, these three ideals can coincide.

7.2 The Borel Conjecture

481

Lemma 7.49. Suppose U is a selective ultrafilter on ! and that for all X  U , if jX j D !1 then there exists 2 U such that n  is finite for all  2 X . Suppose I is a normal uniform ideal on !1 . Suppose G  PU is V -generic. Let I.G/ be the ideal generated by I in V ŒG. Then in V ŒG: (1) I.G/ is a normal uniform ideal on !1 ; (2) suppose f W !1 ! .0;1/ is an injective function such that f 2 V , then V ŒG .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

for some n 2 !. Proof. Suppose F 2 V ŒG is a function, F W !1 ! !1 such that F .˛/ < ˛ for all ˛ > 0. Suppose A  !1 , A 2 V ŒG and A … I.G/ . We may suppose 0 … A. We must show that there exists B such that B  A, B … I.G/ and such that F jB is constant. Let F 2 V PU be a term for the function F and let A 2 V P be a term for the set A. Fix a condition .s0 ; 0 / 2 G. We may suppose that .s0 ; 0 /  A … I.G/ : 

We work in V . Let A be the set of ˛ < !1 such that there exists a condition .s; / < .s0 ; 0 / with the property that .s; /  ˛ 2 A : Since .s0 ; 0 /  A … I.G/ ; 

it follows that A … I . For each ˛ 2 !1 choose a condition .s˛ ; ˛ / < .s0 ; 0 / and an ordinal ˛ < ˛ such that .s˛ ; ˛ /   .˛/ D ˛ ; and such that if ˛ 2 A then .s˛ ; ˛ /  ˛ 2 A ; 

and if ˛ … A then

.s˛ ; ˛ /  ˛ … A :

Let 2 U be such that for all ˛ 2 A, \ .! n ˛ / is finite.

482

7 Conditional variations

For each ˛ 2 A let n˛ 2 ! be such that n n˛  ˛ . The ideal I is normal. Therefore there exists a set B  A such that B … I and there exists .s; n; / 2 Œ! 1 and p0 .k/ D k m for all sufficiently large k < !;

484

7 Conditional variations

(1.3) for all i < n and for all j < !, hi .j / D hi .k/ where k < ! is such that p0 .k/  j < p0 .k C 1/. Suppose O 2 V ŒG is an open set such that O is h-small. By Lemma 7.48, there exists an open set W 2 V such that W is h -small and such that O  WG : In V , ¹˛ 2 S j f .˛/ … W º is I -positive. Hence in V ŒG, ¹˛ 2 S j f .˛/ … Oº is I.G/ positive since by (1) I.G/ \ V D I: Therefore .h.f; h/i; S / 2 .ZBC .I.G/ //V ŒG : The general case is similar. Finally we prove (3). Fix f W !1 ! .0;1/ such that f is injective and such that f 2 V . Suppose V ŒG0  is a ccc extension of V such that V ŒG0   MA C “.2@0 /V < 2@0 ”: Let U0 2 V ŒG0  be a selective ultrafilter such that U  U0 and such that in V ŒG0 , for all X  U0 , if jX j D !1 then there exists 2 U0 such that \ .! n  / is finite for all  2 X. Suppose G1  PU0 is V ŒG0 -generic. By Lemma 7.47(2), G1 \ PU is V -generic. Therefore without loss of generality we may suppose that G1 \ PU D G. Let I.G0 / be the normal ideal generated by I in V ŒG0  and let I.G0 ;G1 / be the ideal generated by I.G0 / in V ŒG0 ŒG1 . By Lemma 7.49, there exists n 2 ! such that V ŒG0 ;G1  : .h.f; h.n/ G /i; !1 / 2 .YBC .I.G0 ;G1 / //

Therefore V ŒG .h.f; h.n/ G /i; !1 / 2 .YBC .I.G/ //

since I.G/  I.G0 ;G1 / . Combining Theorem 7.44 and Lemma 7.50 we obtain the following corollary.

t u

7.2 The Borel Conjecture

485

Theorem 7.51. Assume ADL.R/ . Then L.R/Bmax  ZFC C Borel Conjecture: Proof. By Theorem 7.44 it suffices to prove the following. Suppose h.M0 ; I0 /; a0 ; Y0 i 2 Bmax and that

f0 W !1M0 ! .0;1/

is an injective function such that f0 2 M0 . Then there exists a condition h.M1 ; I1 /; a1 ; Y1 i 2 Bmax such that h.M1 ; I1 /; a1 ; Y1 i < h.M0 ; I0 /; a0 ; Y0 i and such that for some h 2 M1 , .h.j.f0 /; h/i; !1M1 / 2 Y1 ; where j is the iteration of .M0 ; I0 / such that j.a0 / D a1 . Fix f0 and h.M0 ; I0 /; a0 ; Y0 i. Let z 2 R code M0 and let N be a transitive inner model of ZFC C CH such that Ord  N , z 2 N and such that for some ı < !1 , N  “ı is a Woodin cardinal”: We also require that !1 is strongly inaccessible in N . Since AD holds in L.R/, N exists by Theorem 5.34. By Lemma 7.40, there exists an iteration j W .M0 ; I0 / ! .M0 ; I0 / such that j 2 N and such that (1.1) .INS /N \ M0 D I0 , (1.2) j.Y / D .YBC .INS //N \ M0 . Let U 2 N be a selective ultrafilter on ! and let G0  .PU /N be N -generic. Let G1  .Coll.!1 ; !1 , G 2 M and in M , G satisfies all of the requirements of the lemma except possibly (4) or (5). If either (4) or (5) fail let 0 2 S be least such that M0

!1

D !1

and such that in M 0 , (4) or (5) fails to be satisfied by G. We prove that G satisfies (4) and (5) in M 0 . Assume otherwise. Let ˛0 be least such that h.˛0 / 2 .YG /M0 and such that h.˛0 / witnesses in M 0 the failure of either (4) or (5) for G. Set U D h.˛0 /. Arguing as in the proof of Lemma 8.40 using countable elementary substructures of hM 0 ; hj0 ; 2i it follows that .IU;FG /M0  INS and so (4) must hold for .U; G/ in M 0 . Therefore (5) fails for .U; G/ in M 0 . Let ˛1 be least such that h.˛1 / 2 G  ! and such that either 

 ¤ .IW;FG /M.p;k/ , or (9.1) .IU;FG /M0 \ M.p;k/ 

 ¤ .RW;FG /M.p;k/ , (9.2) .RU;FG /M0 \ M.p;k/

where .p; k/ D h.˛1 / and where  jp;G W hM.p;k/ W k < !i ! hM.p;k/ W k < !i

is the iteration given by G. We first assume that (9.1) holds. Let 0 be least such that 

 h.0 / 2 .IU;FG /M0 \ M.p;k/ n .IW;FG /M.p;k/ :

550

8 | principles for !1

Using countable elementary substructures of hM 0 ; hj0 ; 2i it follows that !1 n h.0 / 2 INS ; cf. the proof of Lemma 8.40. This is a contradiction since .IU;FG /M0  INS . Finally we assume that (9.2) holds. Let 0 be least such that h.0 / D .S0 ; .s0 ; f0 /; .s1 ; f1 // where  .S0 ; .s0 ; f0 // 2 .RW;F /M.p;k/ ; .s1 ; f1 / 2 .PU /M0 ; M0 .s1 ; f1 / < .s0 ; f0 / in .PU / , and S0 \ .Z.s1 ;f1 /;F /M0 2 .IU;F /M0 : The existence of 0 follows from Lemma 8.29. Again using countable elementary substructures of hM 0 ; hj0 ; 2i it follows that !1 n S0 2 INS and that !1 n .Z.s1 ;f1 /;F /M0 2 INS : This contradicts .IU;FG /M0  INS . Thus G satisfies (4) and (5) in M 0 and so the semi-generic filter G satisfies the requirements of the lemma. t u 1 Lemma 8.56 ( 2 -Determinacy). Suppose that |

NS G  Pmax

is a semi-generic filter and that Y0  YG is a set such that the following hold where for each p 2 G,  W k < !i jp;G W hM.p;k/ W k < !i ! hM.p;k/ is the iteration given by G. (i) For each .p; k/ 2 G  !, and for each W 2 jp;G .Y.p;k/ /, there exists U 2 Y0 such that  D W: U \ M.p;k/ (ii) For each U 2 Y0 , a) the ideal IU;FG is proper, b) for each .p; k/ 2 G  !,   .IW;FG /M.p;k/ D IU;FG \ M.p;k/ and   ; .RW;FG /M.p;k/ D RU;FG \ M.p;k/  where W D M.p;k/ .

|

NS 8.2 Pmax

551

(iii) Suppose that U0 2 YG , U1 2 Y0 and that U0 \ P .!1 /G D U1 \ P .!1 /G : Then U0 2 Y0 . Then there exists a set Y  Y0 such that: (1) for each U 2 Y0 there exists U  2 Y such that for all .p; k/ 2 G  !,   D U  \ M.p;k/ I U \ M.p;k/

(2) let I be the ideal dual to the filter F D \¹U j U 2 Y º; then \¹IU;FG j U 2 Y º  I: Proof. The proof is essentially identical to that of Lemma 8.50. We define by induction on ˛ a normal ideal J˛ as follows: J0 D \¹IU;F j U 2 Y0 º and for all ˛ > 0, J˛ D \¹IU;F j U 2 Y0 and for all  < ˛, J \ U D ;º: It follows easily by induction that if ˛1 < ˛2 then J˛1  J˛2 . Thus for each ˛, J˛ is unambiguously defined as the intersection of a nonempty set of uniform normal ideals on !1 . The sequence of ideals is necessarily eventually constant. Let ˛ be least such that J˛ D J˛C1 and let J D J˛ . Thus J is a uniform normal ideal on !1 . Let Y be the set of U 2 Y0 such that U \ J D ; and let I be the ideal dual to the filter F D \¹U j U 2 Y º: Then \¹IU;F j U 2 Y º  I; and so Y satisfies the second requirement. Finally it follows by induction that for each ˛, the ideal J˛ has the property: For each U0 2 Y0 there exists U1 2 Y0 such that U1 \ J˛ D ;; and such that U0 \ P .!1 /G D U1 \ P .!1 /G : Therefore the set Y satisfies the first requirement.

t u

As a corollary to Lemma 8.55 and Lemma 8.56 we obtain the following lemma with |NS which the basic analysis of the Pmax -extension is easily accomplished. Lemma 8.57 is analogous to Lemma 4.46, though this formulation is more efficient.

8 | principles for !1

552

Lemma 8.57 (ADL.R/ ). Suppose that A  R and that A 2 L.R/. Then for each |NS |NS there exists p0 2 Pmax such that p0 < q0 and such that: q0 2 Pmax (1) hM.p0 ;k/ W k < !i is A-iterable; (2) hV!C1 \ M.p0 ;0/ ; A \ M.p0 ;0/ ; 2i  hV!C1 ; A; 2i; |

NS (3) Suppose that D  Pmax is a dense set which is definable in the structure hH.!1 /; A; 2i M.p0 ;0/ from parameters in H.!1 / . Then D \ ¹p > p0 j p 2 M.p0 ;0/ º ¤ ;:

Proof. Fix A and let B be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; A; 2i: Let B  be the set of x 2 R such that x codes an element of the first order diagram of the structure hV!C1 ; B; ¹q0 º; 2i: Thus B  2 L.R/. By Theorem 8.42 applied to B  , there exist a countable transitive model M and an ordinal ı 2 M such that the following hold. (1.1) M  ZFC. (1.2) ı is a Woodin cardinal in M . (1.3) B  \ M 2 M and hV!C1 \ M; B  \ M; 2i  hV!C1 ; B  ; 2i. (1.4) B  \ M is ı C -weakly homogeneously Suslin in M . (1.5) Suppose  is the least inaccessible cardinal of M . Then strong condensation holds for M in M . |

NS M / . (1.6) q0 2 .Pmax

Let hD˛ W ˛ < !1M i |

NS M enumerate all the dense subsets of .Pmax / which are first order definable in the structure hH.!1 /M ; A \ M; 2i:

By Lemma 8.55 there exists a filter |

NS M g  .Pmax / such that the following hold in M where for each p 2 g,  jp;g W hM.p;k/ W k < !i ! hM.p;k/ W k < !i is the iteration given by p.

|

NS 8.2 Pmax

553

(2.1) q0 2 g. (2.2) For each ˛ < !1M ,

g \ D˛ ¤ ;:

(2.3) Suppose that p 2 g. For each W 2 jp;g .Y.p;k/ /, there exists U 2 Yg such that  U \ M.p;k/ D W:

(2.4) For each U 2 Yg , the normal ideal IU;Fg is proper. (2.5) Suppose that .p; k/ 2 g  !. For each U 2 Yg , 

 a) IU;Fg \ M.p;k/ D .IW;Fg /M.p;k/ , 

 D .RW;Fg /M.p;k/ , b) RU;Fg \ M.p;k/  . where W D U \ M.p;k/

By Lemma 8.56, there exists Y 2 M such that Y  Yg and such that in M : (3.1) For each U 2 Yg there exists U  2 Y such that for all .p; k/ 2 g  !,   U \ M.p;k/ D U  \ M.p;k/ I

(3.2) Let I be the ideal dual to the filter F D \¹U j U 2 Y º; then \¹IU;Fg j U 2 Y º  I: Let a D .aI /M where I D ¹.IU;Fg /M j U 2 Y º: Let be the least strongly inaccessible cardinal of M above ı. By .1:3/, B  \ M 1 is not † 1 in M and so by (1.4), exists. and let X0  M

be an elementary substructure such that X0 2 M , X0 is countable in M and such that ¹B \ M; Y; gº 2 X0 : Let M0 be the transitive collapse of X0 . Let .a0 ; Y0 ; F0 ; g0 / be the image of .a; Y; Fg ; g/ under the collapsing map and let I0 be the image of .I p0 j p 2 M.p0 ;0/ º ¤ ;: By genericity we may suppose that p0 2 G. Let B  !1 be the interpretation of  by G, The key point is that |

NS \ M.p0 ;0/ j p0 < pº 2 M.p0 ;1/ ¹p 2 Pmax

and so by (1.2) and (1.3),

M.p0 ;0/

B \ !1 Let

2 M.p0 ;1/ :

 W k < !i jp0 ;G W hM.p0 ;k/ W k < !i ! hM.p 0 ;k/

be the iteration given by G. By (1.1)–(1.3), again using the fact that |

NS ¹p 2 Pmax \ M.p0 ;0/ j p0 < pº 2 M.p0 ;1/

it follows that B D jp0 ;G .b/ M.p

;0/

where b D B \ !1 0 . This proves (1). A similar argument shows that L.R/ŒG  !1 -DC: The remaining claims, (2) and (3), are immediate consequences of (1) and the defini|NS tion of the order on Pmax . t u

556

8 | principles for !1 |

NS We now begin the analysis of the nonstationary ideal on !1 in the Pmax -extension. Our goal is to show that the ideal is saturated. We begin with the following lemma |NS -extension. which is the analog of Lemma 6.77 for the Pmax

Lemma 8.59. Assume ADL.R/ and suppose |

NS G  Pmax

is L.R/-generic. Then in L.R/ŒG, for every set A 2 P .R/ \ L.R/ the set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Proof. The proof is identical to that of Lemma 6.77 using the basic analysis provided by Theorem 8.58 (i. e. using Theorem 8.58 in place of Theorem 6.74) and using the |NS is !-closed (Corollary 8.52 in place of Theorem 6.73). t u fact that Pmax Remark 8.60. An immediate corollary of Lemma 8.59 is the following. Assume AD |NS is L.R/-generic. Then in L.R/ŒG, INS is semiholds in L.R/ and that G  Pmax saturated. The verification is a routine application of Lemma 4.24. t u Assume AD holds in L.R/ and that |

NS G  Pmax

is L.R/-generic. Then it is not difficult to show that in L.R/ŒG, the set YG is empty. However one can force over L.R/ŒG to make YG nonempty. The resulting model is |NS . We shall define and briefly itself a generic extension of L.R/ for a variant of Pmax |NS analyze this variant which we denote Umax . |NS The basic property of Umax is the following. Suppose that AD holds in L.R/ and that |NS G  Umax is L.R/-generic. Then L.R/ŒG D L.R/ŒgŒY  where in L.R/ŒG; |

NS is L.R/-generic, (1) g  Pmax

(2) L.R/Œg is closed under !1 sequences in L.R/ŒG, (3) Y D Yg , (4) INS D \¹IU;Fg j U 2 Y º, (5) for all U 2 Y , INS \ U D ;. |

NS . We now define Umax

|

NS 8.2 Pmax

557

|

NS is the set of pairs .p; f / such that Definition 8.61. Umax

|

NS , (1) p 2 Pmax

(2) f 2 M.p;0/ and f W .!3 /M.p;0/ ! Y.p;0/ is a surjection. |

NS The ordering on Umax is defined as follows:

.p1 ; f1 / < .p0 ; f0 / |

NS if p1 < p0 in Pmax and for all ˛ 2 dom.f0 /,

 j.f0 .˛// D f1 .j.˛// \ M.p 0 ;0/

where j W hM.p0 ;k/ W k < !i ! hM.p0 ;k/ W k < !i is the unique iteration such that j.F.p0 / / D F.p1 / .

t u

|

|

NS NS Suppose that G  Umax is a filter. Then G projects to define a filter FG  Pmax . |NS The filter G is semi-generic if the projection FG is a semi-generic filter in Pmax . We |NS is a semi-generic filter. Let fix some more notation. Suppose that G  Umax

fG D ¹jp;FG .f / j .p; f / 2 Gº: Thus fG is a function with domain, dom.fG / D sup¹!2LŒA j A 2 P .!1 /FG º: For each ˛ 2 dom.fG /,

fG .˛/  P .!1 /FG

and fG .˛/ is an ultrafilter in P .!1 /FG . |

NS -extension of L.R/ is a routine generalization of the analThe analysis of the Umax |NS ysis of the Pmax -extension of L.R/. We summarize the basic results in the next theorem the proof of which we leave as an exercise for the dedicated reader.

|

NS is L.R/-generic. Theorem 8.62. Assume AD L.R/ . Suppose G  Umax F D FFG and let Y D .YFG /L.R/ŒG :

Then in L.R/ŒG: |

NS ; (1) FG is L.R/-generic for Pmax

(2) P .!1 /FG D P .!1 /; (3) dom.fG / D !2 ; (4) for all ˇ < !2 , f .ˇ/ 2 Y ;

Let

558

8 | principles for !1

(5) for each U 2 Y , IU;F is a proper ideal and .!1 ; p/ 2 RU;F where p D 1PU ; (6) INS D \¹IU;F j U 2 Y º; (7) for each U 2 Y , INS \ U D ;.

t u |

NS Suppose that for each x 2 R there exists p 2 Pmax such that

x 2 M.p;0/ : We fix some more notation. Suppose that |

NS G  Pmax

is a semi-generic filter. Then 

 IG D [¹M.p;0/ \ .INS /M.p;1/ j p 2 Gº

where for each p 2 G,  jp;G W hM.p;k/ W k < !i ! hM.p;k/ W k < !i

is the iteration given by G. A set |

NS   Pmax  H.!1 /

defines a term for a dense subset of .P .!1 /G n IG ; / if the following conditions are satisfied. (1)  is a set of pairs .p; b/ such that M.p;0/

b  !1 and such that

b 2 M.p;0/ n .INS /M.p;1/ : |

NS (2) For each .p0 ; b0 / 2 Pmax  H.!1 / such that

M.p0 ;0/

b0  !1 and such that

b0 2 M.p0 ;0/ n .INS /M.p0 ;1/ ;

there exists .p1 ; b1 / 2  such that p1 < p0 and such that b1  j.b0 / where  j W hM.p0 ;k/ W k < !i ! hM.p W k < !i 0 ;k/

is the (unique) iteration such that j.F.p0 / / D F.p1 / . |

NS is a semi-generic filter. Then Suppose G  Pmax

G D ¹jp;G .b/ j p 2 G and .p; b/ 2 º: If the filter G is sufficiently generic then G is dense in the partial order, .P .!1 /G n IG ; /:

|

NS 8.2 Pmax

559

|

NS 1 Lemma 8.63 ( 2 -Determinacy). Suppose that   Pmax  H.!1 / defines a term

|

NS for a dense subset of .P .!1 /G n IG ; / and that q0 2 Pmax . Suppose that strong condensation holds for H.!3 /. Then there is a semi-generic filter

|

NS G  Pmax

and a set Y0  YG such that the following hold where for each p 2 G,  W k < !i jp;G W hM.p;k/ W k < !i ! hM.p;k/

is the iteration given by G. (1) q0 2 G. (2) For each U 2 Y0 ,

G n IU;FG

is dense in .P .!1 /G n IU;FG ; /. (3) Suppose that .p; k/ 2 G  !. For each W 2 jp;G .Y.p;k/ /, there exists U 2 Y0 such that  D W: U \ M.p;k/ (4) For each U 2 Y0 , the normal ideal IU;FG is proper. (5) Suppose that .p; k/ 2 G  !. For each U 2 Y0 , 

 D .IW;FG /M.p;k/ , a) IU;FG \ M.p;k/ 

 b) RU;FG \ M.p;k/ D .RW;FG /M.p;k/ ,  . where W D U \ M.p;k/

(6) Suppose that U0 2 YG , U1 2 Y0 and that U0 \ P .!1 /G D U1 \ P .!1 /G : Then U0 2 Y0 . Proof. Fix a function f W !3 ! H.!3 / which witnesses strong condensation for H.!3 /. Define a function h W !3 ! H.!3 / as follows. Let hD˛ W ˛ < !i

8 | principles for !1

560

|

NS enumerate all the dense subsets of Pmax which are first order definable in the structure

hH.!1 /; ; 2i: We require that for each limit ordinal ˛, ¹Dˇ j ˇ < ˛º contains all the dense sets which are definable with parameters from ¹f .ˇ/ j ˇ < ˛º: Let X be the set of t  ! such that t codes a pair .ˇ; p/ where ˇ < !1 and |NS . p 2 Pmax For each ˛ < !3 let ˛ D ! ˛. Thus h˛ W ˛ < !3 i is the increasing enumeration of the limit ordinals (with 0) less than !3 . Suppose ˛ < !3 then for each k < ! h.˛ C k C 2/ D f .˛/: Suppose ˛ < !3 and f .˛/ … X . Then h.˛ / D f .˛/: Suppose ˛ < !3 and f .˛/ 2 X . Let .ˇ; p/ be the pair coded by f .˛/. Then h.˛ / D f .˛  / where ˛  is least such that f .˛  / 2 Dˇ and such that f .˛  /  p. Finally for each ˛ < !3 , ² 1 if f .˛/ 2  ; h.˛ C 1/ D 0 otherwise. Just as in the proof of Lemma 8.55, h witnesses strong condensation for H.!3 /. The additional feature we have obtained here is that (using the notation from the proof of Lemma 8.55) for each  2 S,  \ M 2 M : Let hp˛ W ˛ < !1 i be as constructed in the proof of Lemma 8.55 using the function h and the sequence hD˛ W ˛ < !1 i: Let G 

|NS Pmax

be the filter, |

NS j for some ˛ < !1 ; p˛ < pº: G D ¹p 2 Pmax

Thus G is a semi-generic filter, YG ¤ ;, and the following hold. (1.1) Suppose that .p; k/ 2 G  !. For each W 2 jp;G .Y.p;k/ /, there exists U 2 YG such that  D W: U \ M.p;k/

|

NS 8.2 Pmax

(1.2) For each U 2 YG , the normal ideal IU;FG is proper. (1.3) Suppose that .p; k/ 2 G  !. For each U 2 YG , 

 D .IW;FG /M.p;k/ , a) IU;FG \ M.p;k/ 

 D .RW;FG /M.p;k/ , b) RU;FG \ M.p;k/  . where W D U \ M.p;k/

Let Y0 be the set of U 2 YG such that G n IU;FG is dense in .P .!1 /G n IU;FG ; /. Y0 satisfies the requirements of the lemma provided for each p 2 G,  jp;G .Y.p;0/ / D ¹U \ M.p;0/ j U 2 Y0 º:

For each  < !3 let

M D ¹h.ˇ/ j ˇ < º

and let h D hj: Let S be the set of  < !3 such that (2.1) M is transitive, (2.2) hˇ 2 M for all ˇ < , (2.3) hM ; h ; 2i  ZFC n Powerset, M

(2.4) !2

M

exists and !2

2 M ,

(2.5) q0 2 H.!1 /M , M

(2.6) H.!1 /M D ¹h.ˇ/ j ˇ < !1  º. Let Q D [¹jp;G .Y.p;k/ / j .p; k/ 2 G  !º: Define a partial order on Q by W1 < W2 if W2  W1 . Suppose that U 2 YG . Then ¹W 2 Q j W  U º is a maximal filter in Q. Suppose that  2 S and !1 <  < !2 . Then Q 2 M :

561

562

8 | principles for !1

Suppose that F Q is a filter which is M -generic and let U D UF . We shall prove that G n IU;FG is dense in .P .!1 /G n IU;FG ; /. We first prove that the relevant filters exists. More precisely suppose that D  P .Q/ is set of dense subsets of Q such that jDj  !1 . We prove that there exists a filter F Q such that W 2 F and such that F is D-generic. The proof is essentially the same as the proof that YG ¤ ;. Fix D and let X  hH.!3 /; h; 2i be the elementary substructure of elements which are definable in the structure with parameters from !1 [ ¹Dº. For each ˛ < !1 let X˛ D ¹f .s/ j f 2 X and s 2 ˛ 0, J˛ D \¹IU;F j U 2 Z and for all  < ˛, J \ U D ;º: It follows easily by induction that if ˛1 < ˛2 then J˛1  J˛2 : Thus for each ˛, J˛ is unambiguously defined as the intersection of a nonempty set of uniform normal ideals on !1 . The sequence of ideals is necessarily eventually constant. Let ˛ be least such that J˛ D J˛C1 and let J D J˛ : Thus J is a uniform normal ideal on !1 . Let Y be the set of U 2 Z such that U \ J D ; and let I be the ideal dual to the filter F D \¹U j U 2 Y º: Then it follows that \¹IU;F j U 2 Y º  I: Similarly and

j0 .Y0 / D ¹U \ M0 j U 2 Y º j1 .Y1 / D ¹U \ M1 j U 2 Y º: |

t u

NS is an immediate corollary. We isolate the relevant fact The homogeneity of Pmax in the following lemma.

Lemma 8.91. Suppose that for each x 2 R, there exists |

h.M; I/; g; Y; F i 2 M0 NS |

|

NS NS such that x 2 M. Suppose that p0 2 Pmax and p1 2 Pmax . There exist

|

NS .h.Mk ; Yk / W k < !i; F / 2 Pmax

and functions F0 , F1 such that |

NS and .h.Mk ; Yk / W k < !i; F0 / < p0 , (1) .h.Mk ; Yk / W k < !i; F0 / 2 Pmax

|

NS (2) .h.Mk ; Yk / W k < !i; F1 / 2 Pmax and .h.Mk ; Yk / W k < !i; F1 / < p1 ,

(3) ¹˛ < !1M0 j F0 .˛/ ¤ F1 .˛/º 2 .INS /M0 .

594

8 | principles for !1

Proof. Let x 2 R code the pair .p0 ; p0 / and let |

h.M; I/; g; Y; F i 2 M0 NS be such that x 2 M. Thus

|

NS M / : ¹p0 ; p1 º  .Pmax

Let N D .L.R//M and for i 2 ¹0;1º let |

NS N / gi  .Umax

be N -generic with pi 2 Fgi where |

NS N / Fgi  .Pmax

|

NS N is the induced N -generic filter on .Pmax / . Let hYi ; Fi i D hYgi ; Fgi iN Œgi 

and let Ii D ¹.IU;Fi /N Œgi  j U 2 Yi º: A key point is that since .M; I/ is iterable it follows by Lemma 8.66 and Theorem 3.46, that for each i 2 ¹0;1º, the structure .N Œgi ; Ii / is also iterable and so |

h.Mi ; Ii /; gi ; Yi ; Fi i 2 M0 NS ; where Mi D N Œgi . Strictly speaking Lemma 8.66 and Theorem 3.46 cannot be applied in N Œg since we have only N Œg  ZFC C ZC C †1 -Replacement; but both are easily seen to hold in this case. Let y 2 R code .N ; g0 ; g1 / and let O I; O ı; / 2 H.!1 / .M; be such that O (1.1) x 2 M, (1.2) MO is transitive and MO  ZFC C “ı is a Woodin cardinal”, O (1.3) IO D .I ˛00 . Let Z D [¹P .!/ \ Mk j k < !º and let

Z  D [¹P .!/ \ Mk j k < !º:

Since .hMk W k < !i; j.f // is an iterate of .hMk W k < !i; f / and since .hMk W k < !i; f / 2 Q max ; it follows that the Z  -uniform indiscernibles above !1N0 coincide with the Z-uniform indiscernibles above !1N0 . Since .hMk W k < !i; j.h/; x/ 2 Q.X/ max it follows that the Y˛0 0 -uniform indiscernibles above !1N0 coincide with the Z  -uniform indiscernibles above !1N0 . Further these coincide with the Y˛0 -uniform indiscernibles above !1N0 .

10.2 Coding into L.P .!1 //

715

Finally j.h/.˛/ D g.˛/ for all ˛ < !1N0 such that ˛ is a Z-uniform indiscernible and such that ˛ > !1M0 . Therefore by induction on  it follows that if ˛0 C  < ı then 0 Y.˛0 C / n Y˛0 D Y.˛ n Y˛0 0 0 C / 0

and

0

0 n X˛0 0 : X.˛0 C / n X˛0 D X.˛ 0 C / 0

0

t u

(2) follows.  Remark 10.31. There is an important difference between Q.X/ max and Qmax . Suppose

.hMk W k < !i; f; z/ 2 Q.X/ max ; and h 2 M0 is a function such that ¹˛ < !1M0 j h.˛/ ¤ f .˛/º 2 .INS /M1 : Then in general .hMk W k < !i; h; x/ … Q.X/ max for any choice of x. This will cause problems in the analysis that follows. This diffit u culty does not arise in the case of Qmax . Lemma 10.32. Suppose that X  P .!/. Suppose that .hNk W k < !i; g; x/ 2 Q.X/ max ; .hMk W k < !i; f; z/ 2 Q.X/ max ; and t  ! codes hMk W k < !i. Suppose t 2 LŒx and G is LŒx-generic where G  Coll.!;  and such that q 2 j.f /./: The iteration exists since the critical sequence of any iteration of hNk W k < !i is an initial segment of the Z-uniform indiscernibles above !1N0 . Let hi W i < !i be O

an increasing sequence of elements of Z n , cofinal in !1N0 . For each i < ! let ˛ ˛ ki W hM i W k < !i ! hMO i W k < !i k

k

be the (unique) iteration such that ki .f˛i / D j.f /ji , and let ˛ pO˛i D .hMO k i W k < !i; fO˛i ; z/

be the corresponding condition in Q.X/ max . Note that for all ˛ < ˛i < !1M , pO˛i < p˛ : Now choose a condition p 2

Q.X/ max

\ M Œg such that for all i < !, p < pO˛i

and such that hAk W k < !i 2

M0.p/

where we let

.hMk.p/ W k < !i; f.p/ ; z/ D p: Let

˛ Z  D [¹P .!/ \ hMO 0 i j i < !º:

724

10 Further results

Let IZ be the class of Z-uniform indiscernibles and let IZ  be the class of Z  -uniform indiscernibles. Thus N IZ  D IZ n !1 0 : Further Z  2 M0.p/ and Z  is countable in M0.p/ . The key point is the following. Let M

.p/

h˛ W ˛ < !1 0 i be the increasing enumeration of IZ  and let I D ¹˛Cˇ j ˛ D ˛ and 0 < ˇ < !1M º: Suppose that fO 2 M.p/ is a function such that 0

.p/

M (1.1) dom.fO/ D !1 0 , .p/

M0

(1.2) for all ˛ < !1

,

fO.˛/  Coll.!; ˛/

and fO.˛/ is a filter, .p/

M0

(1.3) for all ˛ 2 !1 Then

n I,

f.p/ .˛/ D fO.˛/:

.hMk.p/ W k < !i; fO; z/ 2 Q.X/ max

and for each i < !,

.hMk.p/ W k < !i; fO; z/ < pO˛i :

Since hAk W k < !i 2 M0.p/ , we can choose fO so that requirement (4) of the lemma is satisfied by the condition .hMk.p/ W k < !i; fO; z/ by modifying fOjI if necessary. But this implies that requirement (4) is satisfied by any condition q 2 Q.X/ max such that .p/ O q < .hM W k < !i; f ; z/: k

Let pO D .hMk.p/ W k < !i; fO; z/. Finally by (vi), hM Œg \ V!C1 ; pŒT  \ M Œg; 2i  hV!C1 ; Y; 2i: and so M Œg \ D! M 1

is dense in

Q.X/ max

\ M Œg. Let .hMk W k < !i; h; z/ 2 D! M \ M Œg 1

be a condition such that O .hMk W k < !i; h; z/ < p: The condition .hMk W k < !i; h; z/ is as required.

t u

10.2 Coding into L.P .!1 //

725

Lemma 10.38. Suppose that X  P .!/. Suppose that hD˛ W ˛ < !1 i is a sequence of dense subsets of Q.X/ max . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ and suppose that .M; I/ 2 H.!1 / is such that .M; I/ is strongly Y -iterable. Let ı 2 M be the Woodin cardinal associated to I. Suppose t  !, t codes M and .hNk W k < !i; f; z/ 2 Q.X/ max ; is a condition such that t 2 LŒz. Let 2 Mı be a normal .uniform/ measure and let .M  ;  / be the !1N0 -th iterate of .M; /. Suppose that Gf is LŒz-generic where Gf  Coll.!; be an ordinal such that L Œt   ZFC and suppose X  L Œt  is a countable elementary substructure containing t and F . Let  D X \ !1N0 . Then  2 E and F is the image of F under the collapsing map. For each ˇ < ! N0 let Tˇ D ¹ 2 E j ¹.0; ˇ/º 2 f . /º: Thus

hTˇ j ˇ < !1N0 i 2 LŒzŒGf 

and for each ˇ < !1N0 , Tˇ … .INS /N1 . We modify G0 to obtain G as follows. For each ˛ < !1N0 , GjColl.!; ˛/ D G0 jColl.!; ˛/; unless ˛ D . C /

M

for some  2 E. In this case GjColl.!; ˛/ D ¹p _ q j q 2 G0 jColl.!; ˛/º:

where p D ¹.0; F .ˇ//º and  2 Tˇ . For each  2 C such that C \  is bounded in  , GjColl.!; ˛/ D G0 jColl.!; ˛/ for all but at most one ˛ in the interval Œ  ;   where   is the largest element of C \ . Further for this one possible exception, GjColl.!; ˛/ D ¹p _ q j q 2 G0 jColl.!; ˛/º for some condition p 2 Coll.!; ˛/. Finally G0 j0 D Gj0 where 0 is the least element of C . Thus by induction on  2 C it follows that for all  2 C , Gj is M  -generic for Coll.!; < /. Therefore G is M  -generic for Coll.!; 0 . By Theorem 10.40 and Lemma 10.41, L.A; R/ŒG  ZFC and further the following hold in L.A; R/ŒG. (1.1) X(Code) .SfG ; zG / D ¹;º. (1.2) Y(Code) .SfG ; zG / D P .!/. (1.3)

AC

holds.

(1.4) The set ¹X  hH.!2 /; A; 2i j MX is A-iterable and X is countableº contains a club, where MX is the transitive collapse of X . (1.5) The set ¹X  hH.!2 /; B; 2i j MX is B-iterable and X is countableº contains a club, where MX is the transitive collapse of X . Let SfG D hSi W i < !i and let AG D ¹˛ C i C 1 j ˛ is a limit ordinal and ˛ 2 Si º: Thus SAG D hSi \ C W i < !i where C is the set of countable limit ordinals and so by (1.1), in L.A; R/ŒG, X(Code) .SAG ; zG / D ¹;º: By (1.4) and Lemma 4.24, the set of ¹Y  L 0 .A; R/ŒG j Y is countable and MY is strongly iterableº contains a club in P!1 .M /. Here MY is the transitive collapse of Y . Thus, by (1.5), there exists a countable elementary substructure, X  L 0 .A; R/ŒG; such that .;/ hM; a; zi 2 Pmax

and satisfies the requirements of the lemma, where (2.1) z D zG , (2.2) M is the transitive collapse of X , (2.3) a D AG \ X \ !1 D AG \ .!1 /M .

t u

742

10 Further results

.;/ It is convenient to organize the analysis of Pmax following closely that of Q.X/ max . The reason is simply that most of the proofs adapt easily to the new context. The next four lemmas summarize the basic iteration facts that one needs. These lemmas are direct analogs of the lemmas we proved as part of the analysis of Q.X/ max . We leave the details to the reader. .;/ , such as the !-closure The first two easily yield elementary consequences for Pmax .;/ and homogeneity of Pmax , the latter two allow one to complete the basic analysis.

Lemma 10.48. Suppose hM1 ; a1 ; z1 i < hM0 ; a0 ; z0 i in

.;/ Pmax

and let

j W M0 ! M0

be the .unique/ iteration such that j.a0 / D a1 . (1) X(Code) .M0 ; Sa0 ; z0 /  X(Code) .M1 ; Sa1 ; z1 /. (2) Suppose that b0 2 M0 is such that for each i < !, Sia0 M Sib0 2 .INS /M0 ; where hSia0 W i < !i D .Sa0 /M0 and hSib0 W i < !i D .Sb0 /M0 . Suppose that x0 2 M0 is a subset of ! such that .;/ : hM0 ; b0 ; x0 i 2 Pmax .;/ Then hM1 ; j.b0 /; x0 i 2 Pmax and hM1 ; j.b0 /; x0 i < hM0 ; b0 ; x0 i.

t u

.;/ As we have indicated, the iteration lemmas required for the analysis of Pmax are .;/ .;/ routine generalizations of those for Qmax . The situation for Pmax is actually quite a bit .;/ conditions are simpler and there is more freedom in less complicated since the Pmax constructing iterations.

Lemma 10.49. Suppose that .;/ ; hM1 ; a1 ; z1 i 2 Pmax .;/ hM0 ; a0 ; z0 i 2 Pmax ;

t  ! codes M0 , and that t 2 LŒz1 . Let hSia0 W i < !i D .Sa0 /M0 ; hSia1 W i < !i D .Sa1 /M1 ; and let C be the set of  < !1M1 such that  is an indiscernible of LŒt . Then there exists an iteration j W M0 ! M0 such that j 2 M1 and such that: (1) for each i < !, C \ j.Sia0 / D C \ Sia1 ; .;/ (2) hM1 ; j.a0 /; z0 i 2 Pmax ;

(3) hM1 ; j.a0 /; z0 i < hM0 ; a0 ; z0 i.

t u

10.2 Coding into L.P .!1 //

743

Lemma 10.50. Suppose that hD˛ W ˛ < !1 i .;/ is a sequence of dense subsets of Pmax . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ . Suppose that .M; T; ı/ 2 H.!1 / is such that:

(i) M is transitive and M  ZFC. (ii) ı 2 M \ Ord, and ı is strongly inaccessible in M . (iii) T 2 M and T is a tree on !  ı. (iv) Suppose P 2 Mı is a partial order and that g  P is an M -generic filter with g 2 H.!1 /. Then hM Œg \ V!C1 ; pŒT  \ M Œg; 2i  hV!C1 ; Y; 2i: .;/ Suppose that hp˛ W ˛ < !1M i is a sequence of conditions in Pmax such that

(v) hp˛ W ˛ < !1M i 2 M , (vi) for all ˛ < !1M , p˛ 2 D˛ , (vii) for all ˛ < ˇ < !1M , pˇ < p˛ : Suppose g  Coll.!; !1M / is M -generic and let Z D [¹Z˛ j ˛ < !1M º where for each ˛ < !1M , Z˛ D P .!/ \ M˛ and hM˛ ; a˛ ; z0 i D p˛ . Suppose  is a Z-uniform indiscernible, !1M <  < !2M and that hAi W i < !i 2 M Œg is a sequence of subsets of !1M . Then for each m < !, there exists a condition hN ; a; zi 2 D! M 1

such that the following hold where hSi W i < !i D .Sa /N ; and where for each ˛ < !1M , ˛ is the ˛ th Z-uniform indiscernible above !1N . (1) hN ; a; zi 2 M Œg. (2)  < !1N and  2 Sm . (3) For each i < ! and for each ˛ < !1M , ˛ 2 Ai if and only if ˛ 2 .SQi /N . (4) For all ˛ < !1M , hN ; a; zi < p˛ . (5) hp˛ W ˛ < !1M i 2 N .

t u

744

10 Further results

Lemma 10.51. Suppose that hD˛ W ˛ < !1 i .;/ is a sequence of dense subsets of Pmax . Let Y  R be the set of reals x such that x codes a pair .p; ˛/ with p 2 D˛ and suppose that

.M; I/ 2 H.!1 / is such that .M; I/ is strongly Y -iterable. Let ı 2 M be the Woodin cardinal associated to I. Suppose t  !, t codes M and .;/ hN ; a; zi 2 Pmax ;

is a condition such that t 2 LŒz. Let 2 Mı be a normal .uniform/ measure and let .M  ;  / be the !1N -th iterate of .M; /. Then there exists a sequence hp˛ W ˛ < !1N i 2 N and there exists .b; x/ 2 N such that .;/ (1) hN ; b; xi 2 Pmax ,

(2) for all ˛ < !1N , p˛ 2 D˛ and hN ; b; xi < p˛ ; (3) there exists an M  -generic filter g  Coll.!; !:

Again by the chain condition of P , it suffices to prove that .P!1 .˛//V is cofinal in

P

.P!1 .˛//V ; where ˛ D !2V . But this is immediate.

t u

Theorem 9.134, Theorem 9.135, and Theorem 9.136 (these are the theorems of Section 9.7 concerning ideals on !2 ) are immediate corollaries of the boundedness theorems, Theorem 10.62 and Theorem 10.63, together with the next lemma. Lemma 10.65. Suppose that I  P .!2 / is a normal uniform ideal such that ¹˛ j cof.˛/ D !º 2 I: Let P D hP .!2 / n I; i. Suppose that either (1) I is !3 -saturated, or (2) I is !-presaturated and that P is @! -cc, or (3) 2@2 D @3 and

P

.!1 /V D .!1 /V : Then P is weakly proper. Proof. (1) is an immediate corollary of Theorem 10.64. The proof of (2) is straightforward. The relevant observation is that since the ideal I is !-presaturated and since ¹˛ < !2 j cof.˛/ D !º 2 I; it follows that for each k < !, V //V .cof.!kC1

P

> !:

Since P is @! -cc, every countable set of ordinals in V P is covered by a set in V of cardinality (in V ) less than @! . This combined with the observation above, yields (2). The proof of (3) is a straightforward adaptation of the proof of Theorem 10.64, again one shows that for each k < !, V .cof.!kC1 //V

P

> !;

and of course one is only concerned with those values of k < ! such that P is not !kC1 -cc; i. e. with cardinals below the chain condition satisfied by P .

758

10 Further results

Since 2@2 D @3 , P is !4 -cc in V and so all cardinals above !3V are preserved. Therefore we need consider only the cases k  2. For k D 0 this is immediate and the case k D 1 follows by appealing to the generic ultrapower associated to the V -generic filter G  P . This leaves the case k D 2; i. e. !3V . But this case now follows by Lemma 9.120. t u Lemma 10.68, below, isolates the application of Lemma 10.59 within the proof of Theorem 10.69. This lemma in turn requires the following two lemmas. Lemma 10.66. Suppose that hS˛ W ˛ < !1 i is a sequence of stationary subsets of !1 and that h ˛ W ˛  !1 i is a closed increasing sequence of cardinals such that for each ˛ < !1 , ˛C1 is measurable and such that !1 < . Suppose that S  !1 is stationary and let Z be the set of X 2 P!1 . / such that (1) X \ !1 2 S , (2) For each ˛  X \ !1 , ordertype.X \ ˛ / 2 S˛ : Then Z is stationary in P!1 . /. Proof. Suppose T  !1 is stationary. Let GT be the game played on !1 for T :  The players alternate choosing countable ordinals, i , for i < ! with Player I choosing i for i even. Player I wins if sup¹i j i < !º 2 T: Since T is stationary, Player II cannot have a winning strategy. For each ˛ < !1 let G˛ be the game of length ! .1 C ˛/ defined as follows. A play of the game is an increasing sequence h W  < ! .1 C ˛/i of countable ordinals. Player I chooses  for  even and Player II chooses  for  odd. Player II wins if for some ˇ  ˛, sup¹ j  < ! .1 C ˇ/º … Sˇ : We claim that for each ˛, Player II cannot have a winning strategy in G˛ . This is easily proved by induction on ˛. Let ˛0 be least such that Player II has a winning strategy in G˛0 and let  W !1 ı and is inaccessible then there exists a countable elementary substructure X  V Q Aº such that hM; EQM i has a weakly A-good iteration scheme containing ¹ı; E; 1 which is -homogeneously Suslin. Here M is the transitive collapse of X and t u EQM is the image of EQ under the collapsing map. The following theorem shows that in many cases, WHIH implies WHIHC . Recall that if there exists a proper class of Woodin cardinals then a set A  R is universally Baire if and only if it is 1 -homogeneously Suslin. Theorem 10.165. Suppose that there exists a proper class of Woodin cardinals. Suppose that Q ı/ 2 H.!1 / .M; E; and that (i) M  ZFC, (ii) EQ 2 M is a weakly coherent Doddage (in M) witnessing ı is a Woodin cardinal in M, (iii) there exist 2 Ord and X  V such that M is the transitive collapse of X . Q ı/ which is universally Baire. Suppose that I is an iteration scheme for .M; E; Suppose that A  R is universally Baire and that B  R is  21 -definable in L.A; R/ with parameters from M. Then the iteration scheme, I , is weakly B-good. t u Having made the requisite definitions we can now state the first theorem regarding Coll.!1 ; R/ as an analog of Pmax in the context of CH.

10.5 -logic and the Continuum Hypothesis

817

Theorem 10.166. Suppose that there exists a proper class of Woodin cardinals. Let  be the set of A  R such that A is universally Baire. Suppose that 0   is a pointclass such that: (i) L.0 ; R/ \ P .R/ D 0 . (ii) For each A 2 0 there exists

Q ı/ 2 H.!1 / .M; E;

such that a) M  ZFC, b) EQ 2 M is a weakly coherent Doddage (in M) witnessing ı is a Woodin cardinal in M, c) in M there is a measurable Woodin cardinal above ı, d) M is A-closed, Q has an iteration scheme in M which is weakly A-good. e) .M; E/ 0 Suppose that  is a …2 sentence in the language for the structure hH.!2 /; INS ; 2; A W A 2 0 i and that Coll.!1 ;R/  ” ZFC C “hH.!2 /; INS ; 2; A W A 2 0 iV is -consistent. Suppose that G  Coll.!1 ; R/ is V -generic. Then hH.!2 /; INS ; 2; A W A 2 0 iL.0 ;R/ŒG  :

t u

Remark 10.167.

(1) In Theorem 10.166, if in addition one requires L.0 ; R/  ADR C “‚ is regular”; then L.0 ; R/ŒG  !1 -DC, and so one can further force over L.0 ; R/ŒG to obtain ZFC without adding new subsets of !1 .

(2) It seems quite likely that if there exists a proper class of measurable Woodin cardinals then the pointclass of the universally Baire sets necessarily satisfies the requirement (ii) of Theorem 10.166. t u We note the following theorem. Theorem 10.168. Suppose that there exists a proper class of Woodin cardinals and Q ı/ 2 H.!1 / such that that A  R is universally Baire. Then there exists .M; E; (1) M  ZFC and M is A-closed, (2) EQ 2 M is a weakly coherent Doddage (in M) witnessing ı is a Woodin cardinal in M, Q has a weakly A-good iteration scheme in M where   P .R/ is the (3) .M; E/ pointclass of all universally Baire sets. u t

818

10 Further results

Another, though weaker, version of Theorem 10.166 is: Theorem 10.169. Suppose that there exists a proper class of measurable Woodin cardinals and that for each partial order P , V P  WHIHC : Let  be the set of A  R such that A is universally Baire and suppose that L.; R/ \ P .R/ D : Suppose that  is a …2 sentence in the language for the structure hH.!2 /; INS ; 2; A W A 2 i and that ZFC C “hH.!2 /; INS ; 2; A W A 2 iV

Coll.!1 ;R/

 ”

is -consistent. Suppose that G  Coll.!1 ; R/ is V -generic. Then hH.!2 /; INS ; 2; A W A 2 iL.;R/ŒG  :

t u

The requirement in Theorem 10.169 that L.; R/ \ P .R/ D  where  is the pointclass of all universally Baire sets is not difficult to achieve. With additional (substantial) large cardinal assumptions one can also require, L.; R/  ADR C “‚ is regular”; see Remark 10.167. Theorem 10.170. Suppose that there exists a proper class of Woodin cardinals and that is an inaccessible limit of strong cardinals. Suppose that G  Coll.!; < / 1 be the pointclass of all universally Baire sets as defined in V ŒG. is V -generic. Let G Then in V ŒG:

(1) There is a proper class of Woodin cardinals. 1 1 ; RG / \ P .RG / D G . (2) L.G

(3) Suppose that in V , is the critical point of an elementary embedding j W VC1 ! VC1 : Then (in V ŒG) 1 ; RG /  ADR C “‚ is regular”: L.G

We note the following theorem which is a variation of Theorem 4.79.

t u

10.5 -logic and the Continuum Hypothesis

819

Theorem 10.171. Suppose that there exists a model hM; Ei such that hM; Ei  ZFC C CH; and such that for each …2 sentence  if there exists a partial order P such that hH.!2 /; 2iV VP

where Q D .Coll.!1 ; R//

P Q

 ;

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

Assume there exists a proper class of inaccessible cardinals. Then for all partial orders P , V P  ADL.R/ : u t Remark 10.172. The proof of Theorem 10.171 uses the core model induction, this is the machinery used to prove Theorem 5.104 and Theorem 6.149; and the conclusion can be strengthened. A plausible upper bound in the consistency strength of the hypothesis is an inaccessible limit of Woodin cardinals which are limits of Woodin cardinals. Of course without the assumption that hM; Ei  CH; the hypothesis is relatively weak, the upper bound, eliminating the inaccessibles, being the consistency strength of 1 ZFC C ./ C “For all P , V P   2 -Determinacy”; by Theorem 4.69.

t u

We generalize the notion of a mixed iteration scheme to weakly A-good mixed iteration schemes, and similarly we formulate the analogous mixed iteration hypothesis. These definitions allow us to state Theorem 10.175 which is the corresponding generalization of Theorem 10.12. With the version of Theorem 10.12 given here, it is possible to prove various generalizations of Theorem 10.128. But this we shall not do. Definition 10.173. Suppose that Q ı1 / 2 H.!1 / .M; ı0 ; E; and that (1) M  ZFC, (2) ı0 < ı1 and each is a Woodin cardinal in M, (3) EQ 2 M is a weakly coherent Doddage (in M) witnessing ı1 is a Woodin cardinal in M. Suppose that A is universally Baire and that M is A-closed. Q ı1 / is weakly A-good if every iterate A mixed iteration scheme, I , for .M; ı0 ; E; of M constructed according to I , is A-closed. t u

820

10 Further results

Definition 10.174 (MIHC ).

(1) There is a proper class of Woodin cardinals.

(2) There exist a Woodin cardinals ı0 < ı1 and a weakly coherent Doddage EQ D hE˛ W ˛ < ı1 i which witnesses ı1 is a Woodin cardinal such that if ı1 < and if is inaccessible then there exists a countable elementary substructure, X  V containing ı0 ; ı1 and EQ such that hM; ı0M ; EQM ; ı1M i has a mixed iteration scheme which is 1 -homogeneously Suslin and weakly A-good for each universally Baire set A 2 X . Here M is the transitive collapse Q ı1 / under the collapsing map. u t of X and .ı0M ; EQM ; ı1M / is the image of .ı0 ; E; Theorem 10.175. Assume there is a proper class of measurable cardinals which are limits of Woodin cardinals. Then for each ordinal ˛ there exists a transitive inner model containing the ordinals such that (1) V˛  N , (2) N  ZFC C MIHC , WH N WH /  1 . (3) .1

t u

There are two natural candidates for canonical models of the form, L.P .!1 //, in the context of CH. (I) Suppose that 0 is 1 -huge. Suppose that G0  Coll.!; < 0 / is V -generic and that G1  .Coll.!1 ; < 1 //V ŒG0  is V ŒG0 -generic. The first candidate is L.P .!1 //V ŒG1  : (II) Suppose that   P .R/ is a pointclass, closed under continuous preimages, such that L.; R/  ZF C ADR C “‚ is regular”: Suppose that G  Coll.!1 ; R/ is L.; R/-generic. The second candidate is L.P .!1 //L.;R/ŒG : The first class of models, or at least the background models V ŒG1 , have two interesting features. These models were the subject of Theorem 6.28, which shows that a much stronger version of (1) below actually holds.

10.5 -logic and the Continuum Hypothesis

821

(1) In V ŒG1  there is a normal, uniform, !1 -dense ideal on !1 . (2) There is a stationary set S  !1V ŒG1  such that if C  !1V ŒG1  is a V ŒG1 -generic club contained in S then hH.!2 /; 2iV ŒG1 ŒC    where  is any †2 -sentence such that ZFC C CH C “hH.!2 /; 2i  ” is -satisfiable in V1 . A very interesting question is:  Can ODR -Determinacy hold in L.P .!1 //V ŒG1  ? The second class of models have strong homogeneity properties and also a plethora of saturated ideals. (1) Suppose that H  Coll.!2 ; /L.;R/ŒG is L.; R/ŒG-generic. Then in L.; R/ŒGŒH , there is a normal, uniform, !1 dense ideal on !1 . (2) .INS /L.;R/ŒG is quasi-homogeneous in L.; R/ŒG. (3) Suppose that P 2 L.; R/ŒG is an .!; 1/-distributive partial order, in L.; R/ŒG, of cardinality !1 . Suppose that g  P is L.; R/ŒG-generic. Then .L.P .!1 ///L.;R/ŒG .L.P .!1 ///L.;R/ŒGŒg : An appealing conjecture is that these models, with the proper choices of the underlying ground models, do yield generalizations of ./ to the context of CH. Theorem 10.166 and Theorem 10.169 offer some evidence for this in the case of the second class of models. We end this section by stating several theorems which impose a rather fundamental limit on possible generalizations of Pmax to the context of CH. These theorems are conditioned on the following conjecture from .Woodin 2010b/. Definition 10.176 (ADC Conjecture). Suppose that L.A; R/ and L.B; R/ each satisfy ADC . Suppose that every set X 2 .L.A; R/ [ L.B; R// \ P .R/ is !1 -universally Baire. 2 L.A;R/ 2 L.B;R/ 2 L.B;R/ 2 L.A;R/ Then either .  . or .  . . 1 / 1 / 1 / 1 / There is a stronger version of this conjecture.

t u

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10 Further results

Definition 10.177 (Strong ADC Conjecture). Suppose that L.A; R/ and L.B; R/ each satisfy ADC . Suppose that every set X 2 .L.A; R/ [ L.B; R// \ P .R/ is !1 -universally Baire. Then either A 2 L.B; R/ or B 2 L.A; R/.

t u

Theorem 10.178. Assume there exists a proper class of Woodin cardinals. Let  1 be the pointclass of all A  R such that A is universally Baire and let T D Th.H.!2 // Then following are equivalent. (1) There exists a sentence ‰ such that V  ZFC C ‰ for some , and such that for each sentence , either a) ZFC C ‰ ` “H.!2 /  ”, or b) ZFC C ‰ ` “H.!2 /  :”. (2) T is is †1 definable (equivalently 1 -definable) in the structure hM 1 ; 2; ¹Rºi:

t u

Remark 10.179. (1) First order logic is definable in V! and as a result the theory of V! cannot be finitely axiomatized over ZFC in first order logic. This of course is the essence of the incompleteness theorems of G¨odel. The key question raised by Theorem 10.178 concerns the intrinsic complexity of -logic; i. e. of the set: 0 D ¹ j ZFC ` º for this places a limit on how large a fragment of V one can consistently assert has a theory which is finitely axiomatized over ZFC in -logic. The only immediate upper bound is Vı where ı is the second Woodin cardinal, noting that Wadge determinacy holds in this case for the sets A  R which are homogeneously Suslin for each < ı. Neeman has proved that if there is a Woodin cardinal then all universally Baire sets are determined and using this result, the set 0 is definable in Vı0 C1 where ı0 is the least Woodin cardinal. The set 0 cannot be defined in H.!1 / and assuming ./ it cannot be defined in H.!2 /. (2) Assume there exists a proper class of Woodin cardinals. Let  1 be the pointclass of all A  R such that A is universally Baire. Then 0 has the same Turing degree as the †1 -theory of the structure hM 1 ; 2; ¹RºiI in fact each is recursively reducible to the other. Thus the complexity of -logic t u is the same as that of the complete †21 . 1 / subset of !.

10.5 -logic and the Continuum Hypothesis

823

Theorem 10.180 (CH C ADC Conjecture). Assume there exists a proper class of Woodin cardinals. Let  1 be the pointclass of all A  R such that A is universally Baire. Let T be the †1 theory of hM 1 ; 2; ¹Rºi. Then either (1) T is †2 definable in the structure; hH.!2 /; INS ; 2i; or (2) T is …2 definable in the structure; hH.!2 /; INS ; 2i:

t u

There is a version of Theorem 10.180 which is not dependent on the ADC Conjecture. This theorem is proved using the core model induction. Theorem 10.181. Assume there exists a proper class of Woodin cardinals and that either Martin’s Maximum.c/ holds or there is an !1 -dense ideal on !1 . Let  1 be the pointclass of all A  R such that A is universally Baire and let T t u be the †1 theory of hM 1 ; 2; ¹Rºi. Then T is definable H.c C /. As a corollary of Theorem 10.180, using Tarski’s theorem on the undefinability of truth, one obtains the first theorem regarding CH. This theorem shows that the most optimistic possibility of a version of Pmax for CH must fail. Theorem 10.182 (ADC Conjecture). Suppose that there exist a proper class of Woodin cardinals and that ‰ is a sentence such that V  ‰ for some strongly inaccessible cardinal, . Suppose that for each sentence , either (i) ZFC C ‰ ` “H.!2 /  ”, or (ii) ZFC C ‰ ` “H.!2 /  :”. t u

Then CH is false.

The axiom ./ is a natural example of an axiom which axiomatizes the theory of H.!2 / in -logic. An immediate consequence of ./ is that there exists a surjection  W R ! !2 such that the induced prewellordering, ¹.x; y/ j .x/  .y/º is 13 . Let ˆ express: There exists a surjection  W R ! !2 such that   is 2 -definable in the structure hH.!2 /; INS ; 2i; (without parameters),

824

10 Further results

 if there exists a proper class of Woodin cardinals then – the relation R D ¹.x; y/ j .x/ < .y/º is universally Baire, moreover – there exists a universally Baire set A such that R is .21 /L.A;R/ . We require the following generalization of Theorem 10.180. Under a variety of additional assumptions, alternative (2) can be eliminated. For example if either of the following hold:  There is a normal, uniform, !2 -saturated ideal on !1 ;  Chang’s Conjecture; then it can be eliminated. Theorem 10.183 (ADC Conjecture). Assume there exists a proper class of Woodin cardinals. Let  1 be the pointclass of all A  R such that A is universally Baire. Let T be the †1 theory of hM 1 ; 2; ¹Rºi. Then either (1) T is †2 definable in the structure, hH.!2 /; INS ; 2iI or (2) T is …2 definable in the structure, hH.!2 /; INS ; 2iI or (3) there exists a universally Baire set A  R and a surjection  W R ! !2V such that  is 2 -definable in the structure hH.!2 /; INS ; 2i and such that the prewellordering R D ¹.x; y/ j .x/  .y/º t u

is .21 /L.A;R/ .

The second theorem regarding CH generalizes the fact that ./ implies  ı 12 D !2 . It is a corollary of Theorem 10.183. Theorem 10.184 (ADC Conjecture). Suppose that there exist a proper class of Woodin cardinals and that ‰ is a sentence such that V  ‰ for some strongly inaccessible cardinal, . Suppose that for each sentence , either (i) ZFC C ‰ ` “H.!2 /  ”, or (ii) ZFC C ‰ ` “H.!2 /  :”. Then ˆ holds.

t u

10.5 -logic and the Continuum Hypothesis

825

We note the following theorem which shows that Theorem 10.184 is essentially the strongest possible. This theorem was independently proved by Neeman. Theorem 10.185. Assume there exists a proper class of Woodin cardinals. Suppose that A  R is universally Baire, 0 is sentence and (i) L.A; R/  0 , (ii) for all B 2 L.A; R/ \ P .R/, either L.B; R/ D L.A; R/, or L.B; R/  :0 . Let ‚A D .‚/L.A;R/ . Then there exists a sentence ‰ such that: (1) For each sentence , either a) ZFC C ‰ ` “H.!2 /  ”, or b) ZFC C ‰ ` “H.!2 /  :”. (2) ZFC C ‰ is -consistent. t u

(3) ZFC C ‰ ` ‚A < !2 :

In the fall of 2009, Aspero, Larson, and Moore showed that there are …2 -sentences 1 and 2 such that both 1 and 2 can be forced to hold with CH but .1 ^2 / implies the :CH. The main question which remains is: Question. Can there exist a sentence ‰ such that for all †2 sentences, , either (1) ZFC C CH C ‰ ` “H.!2 /  ”, or (2) ZFC C CH C ‰ ` “H.!2 /  :”; and such that ZFC C CH C ‰ is -consistent?

t u

Remark 10.186. (1) A natural conjecture is that if the answer to the question above is yes, then under suitable large cardinal hypotheses, or suitable determinacy hypotheses, the witness for ‰, is simply the sentence:  H.!2 / †2 H.!2 /V

Coll.!1 ;R/

;

which is a generic form of ˘, see Theorem 10.198. (2) As indicated in Remark 10.194 of the next section, one of the statements (Version III) preceding Remark 10.194 gives an example of a …2 sentence in the language for the structure hH.!2 /; INS ; 2i which looks quite difficult to obtain (together with CH) except by forcing over a suitable model of ZF C ADC .

826

10 Further results

(3) We note the following corollary of Theorem 10.171. Suppose that CH holds in V and that if  is a …2 sentence for which there exists a partial order P such that hH.!2 /; 2iV

P

 CH C ;

then H.!2 /  . Assume there exists a proper class of inaccessible cardinals. Then for all partial orders P , V P  ADL.R/ :

t u

Finally if the  Conjecture is false (and there is a proper class of Woodin cardinals) then a very interesting question is the following. Question. Can there exist a sentence ‰ such that for all  either (1) ZFC C CH C ‰  “H.!2 /  ”, or (2) ZFC C CH C ‰  “H.!2 /  :”; and such that ZFC C CH C ‰ is -satisfiable?

t u

The next theorem, in conjunction with Theorem 10.184, shows that this question must have a negative answer in any sufficiently iterable model provided that ZFC C CH C ‰ is -consistent in V . Theorem 10.187. Suppose that there exists a proper class of Woodin cardinals and that Q ı/ 2 H.!1 / .M; E; is such that: (i) M is transitive and M  ZFC C “There exists a proper class of Woodin cardinals”: (ii) M  “ı is a Woodin cardinal”. (iii) E 2 M and in M is a weakly coherent Doddage witnessing that ı is a Woodin cardinal. Q has an iteration scheme which is universally Baire. (iv) .M; E/ Suppose that T 2 M is a theory containing ZFC,  is a sentence and that M  “T  ”: Then T ` .

t u

10.6 The Axiom ./C

827

10.6 The Axiom ./C The results of Section 9.2 suggest the following variations of the axiom ./. Definition 10.188. Axiom ./C : For each set X  R there exists a set A  R such that (1) L.A; R/  ADC , (2) there is an L.A; R/-generic filter, g  Pmax , such that X 2 L.A; R/Œg:

t u

Definition 10.189. Axiom ./CC : There exists a pointclass   P .R/ and a filter g  Pmax such that (1) L.; R/  ADC , (2) g is L.; R/-generic, (3) P .R/  L.; R/Œg.

t u

The successful extension of the fine structural analysis of HODL.R/ to the analysis of HODL.A;R/ for all sets A  R such that L.A; R/  ADC should yield a proof of the following conjecture. Definition 10.190 (The Cofinality Conjecture). Suppose that L.A; R/ and L.B; R/ are Wadge incomparable inner models of ADC . Let ı D sup¹‚L.C;R/ j C 2 P .R/ \ L.A; R/ \ L.B; R/º: t u

Then cof.ı/ D !1 .

Assuming Cofinality Conjecture an argument using the core model induction and which is quite similar to the proof of Theorem 10.181 yields the following theorem and it is a corollary of this theorem that the two axioms, ./C and ./CC , are equivalent. The statement of the theorem involves the following notation – for each set A  R, ./A abbreviates: (1) L.A; R/  ADC , (2) L.P .!1 /; A/ D L.A; R/ŒG, for some L.A; R/-generic filter G  Pmax . Theorem 10.191 (Cofinality Conjecture). Let  be the pointclass of sets A  R such that ./A holds and suppose that A and B are in . Then L.A; B; R/  ADC :

t u

828

10 Further results

Related to the problem of Martin’s Maximum vs. ./ is the following question: Is ZFC C Martin’s Maximum C ./CC consistent? A simpler question concerns the value of  ı 12 in V Œg where g is V -generic for Namba forcing. Note that if ı 1 D !2 2 then necessarily, .ı12 /V < .ı12 /V Œg : A bound for .ı12 /V Œg is provided by the following theorem. Theorem 10.192. Assume that for all A  !2 , A exists. Suppose that g is V -generic for Namba forcing. Then in V Œg: (1) For all x 2 R, x # exists; (2) ı12  !3V . Proof. We sketch the proof. Let P be the Namba partial order. The elements of P are pairs .s; t / such that (1.1) t  !2 . b) .ZF C DC/ For all x 2 R, ¹xº is OD if and only if for some A 2  1 , x is OD in L.A; R/. (25) Assume there exists an elementary embedding j W L.VC1 / ! L.VC1 / with critical point below . Define ‚L.VC1 / to be; sup¹˛ 2 Ord j there exists a surjection, f W P . / ! ˛, with f 2 L.VC1 /º: Must

‚L.VC1 / < CC ‹

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Index

A-Bounded Martin’s Maximum, 798 A-Bounded Martin’s MaximumCC , 798 A(Code) .S; z; B/, 769 A-closed structure, 808 ADC Conjecture, 821 ADC , 611 A-iterable model, M , 73 A-iterable structure, hMk W k < !i, 224 Axiom of Strong Condensation, 499 Axiom ./, 184 Axiom ./C , 827 Axiom ./ CC , 827 Axiom  , 241 Œˇ˛ , 493 Œˇ