woodin - the axiom of determinacy, forcing axioms and the non-stationary ideal

7/23/2019 Woodin - The Axiom of Determinacy, Forcing Axioms and the Non-Stationary Ideal

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de Gruyter Series in Logic and Its Applications 1

Editors: Wilfrid A. Hodges (London)Steffen Lempp (Madison)

Menachem Magidor (Jerusalem)


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W. Hugh Woodin

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

Second revised edition

De Gruyter


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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 2010511.3dc22

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.

2010 Walter de Gruyter GmbH & Co. KG, Berlin/New York

Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.dePrinting and binding: Hubert & Co. GmbH & Co. KG, Gttingen Printed on acid-free paper

Printed in Germany

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Contents

1 Introduction 1

1.1 The nonstationary ideal on!1 . . . . . . . . . . . . . . . . . . . . . 2

1.2 The partial order Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Pmaxvariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Extensions of inner models beyondL.R/ . . . . . . . . . . . . . . . 13

1.5 Concluding remarks the view from Berlin in 1999 . . . . . . . . . . 15

1.6 The view from Heidelberg in 2010 . . . . . . . . . . . . . . . . . . . 18

2 Preliminaries 21

2.1 Weakly homogeneous trees and scales . . . . . . . . . . . . . . . . . 212.2 Generic absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 The stationary tower . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Forcing Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Reflection Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 Generic ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 The nonstationary ideal 51

3.1 The nonstationary ideal and

12 . . . . . . . . . . . . . . . . . . . . . 51

3.2 The nonstationary ideal and CH . . . . . . . . . . . . . . . . . . . . 108

4 The Pmax-extension 116

4.1 Iterable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.2 The partial order Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Applications 184

5.1 The sentenceAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.2 Martins Maximum andAC . . . . . . . . . . . . . . . . . . . . . . 187

5.3 The sentenceAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.4 The stationary tower and Pmax . . . . . . . . . . . . . . . . . . . . . 199

5.5 P

max

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.6 P0max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

5.7 The Axiom

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.8 Homogeneity properties ofP.!1/=INS . . . . . . . . . . . . . . . . . 274

6 Pmaxvariations 287

6.1 2Pmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6.2 Variations for obtaining!1-dense ideals . . . . . . . . . . . . . . . . 306

6.2.1 Qmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

6.2.2 Qmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334


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vi Contents

6.2.3 2Qmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

6.2.4 Weak Kurepa trees andQmax . . . . . . . . . . . . . . . . . . 377

6.2.5 KTQmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

6.2.6 Null sets and the nonstationary ideal . . . . . . . . . . . . . . 403

6.3 Nonregular ultrafilters on!1 . . . . . . . . . . . . . . . . . . . . . . 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

8.2 P|NS

max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

8.3 The principles, |C

NSand |

CC

NS . . . . . . . . . . . . . . . . . . . . . . 577

9 Extensions ofL.;R/ 609

9.1 ADC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

9.2 The Pmax-extension ofL.;R/ . . . . . . . . . . . . . . . . . . . . . 617

9.2.1 The basic analysis . . . . . . . . . . . . . . . . . . . . . . . 618

9.2.2 Martins Maximum CC.c/ . . . . . . . . . . . . . . . . . . . 622

9.3 The Qmax-extension ofL.;R/. . . . . . . . . . . . . . . . . . . . . 6339.4 Changs Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

9.5 Weak and Strong Reflection Principles . . . . . . . . . . . . . . . . . 651

9.6 Strong Changs Conjecture . . . . . . . . . . . . . . . . . . . . . . . 667

9.7 Ideals on!2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

10 Further results 694

10.1 Forcing notions and large cardinals . . . . . . . . . . . . . . . . . . . 694

10.2 Coding intoL.P.!1// . . . . . . . . . . . . . . . . . . . . . . . . . 701

10.2.1 Coding by sets, QS . . . . . . . . . . . . . . . . . . . . . . . . 703

10.2.2 Q.X/max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708

10.2.3 P.;/

max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

10.2.4 P.;;B/

max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768

10.3 Bounded forms of Martins Maximum . . . . . . . . . . . . . . . . . 784

10.4 -logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

10.5 -logic and the Continuum Hypothesis . . . . . . . . . . . . . . . . 813

10.6 The Axiom./C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82710.7 The Effective Singular Cardinals Hypothesis . . . . . . . . . . . . . . 835

11 Questions 840

Bibliography 845

Index 849


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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

Godels constructible universeLand 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 thecontext 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

completionofP.!1/, thepowersetof!1.

This model is a forcing extension ofL.R/ and the method can be varied to pro-

duce 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, obtain-

ing 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 ofdefinable scales which is ultimately the driving force behind the ar-

guments. Boundedness arguments also play a key role. These results contribute to a

curious circle of relationships between large cardinals, determinacy, and forcing ax-

ioms. 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 becharacterized without reference to this. For example, as we have indicated above, our

canonical model is a generic extension ofL.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 ofP.!1/.


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2 1 Introduction

The original motivation for the definition of these models resulted from the dis-

covery that it is possible, in the presence of the appropriate large cardinals, to force

(quite by accident) theeffectivefailure 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 settlequestions that were otherwise unsolvable. This has been remarkably successful partic-

ularly 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

cardinalscould notbe used to settle theContinuum Hypothesis. This was first argued

by Levy and Solovay.1967/.

Nevertheless large cardinals do provide some insight to theContinuum 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 then21 sentences are absolute between all set generic extensions ofVwhich 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 familyS j < !2 of stationary subsets of!1 such thatS \ S is

nonstationary whenever ?

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 determi-

nacy.

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 letINS

denote the set of

subsets of!1 which are not stationary. ClearlyINS 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 !2chain 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 ob-

tained by Steel and VanWesep .1982/. They used the consistency of a very strong form

of theAxiom of Determinacy(AD), see.Kanamori 2008/and Moschovakis.1980/for

the history of these axioms.


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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 ADR C is regular:

ADRis the assertion that all real games of length ! are determined anddenotes 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 !2is 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 theAxiom of Choice

holds) and forces both that !2is the second uniform indiscernible and (by arguments of

VanWesep) that the nonstationary ideal on!1is !2-saturated. The method of.Woodin1983/ 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 RO.Coll.!;

We have proved that this in turn implies ADL.R/ and so the hypothesis used (the con-

sistency of AD) is the best possible.

The next progress on the problem of the saturation of the nonstationary ideal was

obtained in a series of results by Foreman, Magidor, and Shelah .1988/. They provedthat a generalization ofMartins Axiomwhich they termedMartins Maximumactually

implies that the nonstationary ideal is saturated. They also proved that if there is a

supercompact cardinal then Martins Maximum is true in a forcing extension ofV.

Later Shelah proved that if there exists a Woodin cardinal then in a forcing extension

ofVthe nonstationary ideal is saturated. This latter result is most likely optimal in the

sense that it seems very plausible that

ZFC C The nonstationary ideal on!1is!2-saturated

is equiconsistent with

ZFC C There exists a Woodin cardinal

see.Steel 1996/.

There was little apparent progress on obtaining a model in which !2 is the second

uniform indiscernible beyond the original results of.Steel and VanWesep 1982/ and

.Woodin 1983/. Recall that assuming that for every real x ,x # exists, the second uni-

form indiscernible is equal to

12, the supremum of the lengths of

12prewellorderings.

Thus the problem of the size of the second uniform indiscernible is an instance of the


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4 1 Introduction

more general problem of computing the effectivesize of the continuum. This problem

has a variety of formulations, two natural versions are combined in the following:

Is there a (consistent) large cardinal whose existence implies that the length of

any prewellordering arising in either of the following fashions, is less than the

least weakly inaccessible cardinal?

The prewellordering exists in a transitive inner model of AD containing all

the reals.

The prewellordering is universally Baire.

The second of these formulations involves the notion of a universally Baire set

of reals which originates in .Feng, Magidor, and Woodin 1992/. Universally Bairesets are discussed briefly in Section 10.3. We note here that if there exists a proper

class of Woodin cardinals then a set A R is universally Baire if and only if it is1-weakly homogeneously Suslin which in turn is if and only if it is 1-homogeneously

Suslin. Another relevant point is that if there exist infinitely many Woodin cardinals

with measurable above and ifA R is universally Baire, then

L.A;R/ AD

and soAbelongs to an inner model of AD. The converse can fail.

More generally one can ask for any bound provided of course that the bound is a

specific! which can be defined without reference to 2@0 .

For example every 12 prewellordering has length less than !2 and if there is a

measurable cardinal then every

1

3

prewellordering has length less than !3. A much

deeper theorem of .Jackson 1988/ is that if every projective set is determined then

every projective prewellordering has length less than !! . This combined with the

theorem of Martin and Steel on projective determinacy yields that if there are infinitely

many Woodin cardinals then every projective prewellordering has length less than !! .

The point here of course is that these bounds are valid independent of the sizeof2@0 .

The current methods do not readily generalize to even produce a forcing extension

ofL.R/ (without adding reals) in which ZFC holds and !3 < L.R/. Thus at this point

it is entirely possible that!3 is the bound and that this is provable in ZFC. If a large

cardinal admits an inner model theory satisfying fairly general conditions then most

likely the only (nontrivial) bounds provable from the existence of the large cardinal are

those provable in ZFC; i. e. large cardinal combinatorics are irrelevant unless the large

cardinal is beyond a reasonable inner model theory.

For example suppose that there is a partial order P 2L.R/such that for all transi-

tive modelsMof ADC containingR, ifG P isM-generic then

.R/MG D.R/M,

.

13/MG D.!3/

MG,

L.R/G ZFC,


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1.1 The nonstationary ideal on!1 5

where

13is the supremum of the lengths of

13prewellorderings ofR. The axiom AD

C

is a technical variant of AD which is actually implied by AD in many instances. As-

suming DC it is implied, for example, by ADR. It is also implied by AD ifV DL.R/.

By the results of .Woodin 2010b/ if inner model theory can be extended to the

level of one supercompact cardinal then the existence of essentially all large cardinalsis consistent with

13D!3.

It follows from the results of.Steel and VanWesep 1982/and .Woodin 1983/that

such a partial order P exists in the case of

12, more precisely, assuming

L.R/ AD;

there is a partial order P 2 L.R/such that for all transitive modelsM of ADC con-

taining R, ifG P isM-generic then

.R/MG D.R/M,

.12/MG D.!2/MG,

L.R/G ZFC.

Thus if a large cardinal admits a suitable inner model theory then the existence of the

large cardinal is consistent with

12 D !2. We shall prove a much stronger result in

Chapter 3, showing that if is a Woodin cardinal and if there is a measurable cardinal

above then there is a semiproper partial order Pof cardinality such that

VP

12 D!2:

This result which is a corollary of Theorem 1.1, stated below, and Theorem 2.64, dueto Shelah, shows that this particular instance of theEffective Continuum Hypothesis is

as intractable as theContinuum Hypothesis.

Foreman and Magidor initiated a program of proving that

12 < !2 from various

combinatorial hypotheses with the goal of evolving these into large cardinal hypothe-

ses, .Foreman and Magidor 1995/. By the (initial) remarks above their program if

successful would have identified a critical step in the large cardinal hierarchy.

Foreman and Magidor proved among other things that if there exists a (normal)

!3-saturated ideal on!2 concentrating on a specific stationary set then 12 < !2. In

Chapter 9 we improve this result slightly showing that this restriction is unnecessary;

if there is a measurable cardinal and if there is an !3-saturated (uniform) ideal on!2then

12< !2.

An early conjecture of Martin is that

1n D @n for alln follows from reasonable

hypotheses.

1nis the supremum of the lengths of

1nprewellorderings.

The following theorem proves the Martin conjecture in the case ofn D 2.

Theorem 1.1. Assume that the nonstationary ideal on !1 is !2-saturated and that

there is a measurable cardinal. Then

12 D!2 and further every club in !1 contains a

club constructible from a real. ut


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6 1 Introduction

As a corollary we obtain,

Theorem 1.2. AssumeMartins Maximum. Then

12 D !2 and every club in !1 con-

tains a club constructible from a real. ut

Another immediate corollary is a refinement of the upper bound for the consistencystrength of

ZFC C For every realx; x# exists. C !2 is the second uniform indiscernible.

Assuming in addition that larger cardinals exist then one obtains more information.

For example,

Theorem 1.3. Assume the nonstationary ideal on !1 is !2-saturated and that there

exist! many Woodin cardinals with a measurable cardinal above them all.

(1) Suppose thatA R,A 2 L.R/, and that there is a sequence hB W < !1i of

borel sets such thatAD [B j < !1:

ThenA is 12.

(2) Suppose thatXis a bounded subset ofL.R/ of cardinality !1. Then there exists

a setY 2L.R/of cardinality!1 in L.R/such thatX Y. ut

We note that assuming for every x 2 R,x# exists, the statement (1) of Theorem 1.3

implies that

12D!2; if

12 < !2 then every

13 set is an!1 union of borel sets.

1.2 The partial order Pmax

Theorem 1.3 suggests that if the nonstationary ideal is saturated (and if modest large

cardinals exist) then one might reasonably expect that the inner model L.P.!1//may

becloseto the inner modelL.R/. However if the nonstationary ideal is saturated one

can, by passing to a ccc generic extension, arrange that

P.R/ L.P.!1//

and preserve the saturation of the nonstationary ideal. Nevertheless this intuition was

the primary motivation for the definition ofPmax.The canonical model for :CH is obtained by the construction of this specific partial

order, Pmax. The basic properties ofPmaxare given in the following theorem.

Theorem 1.4. AssumeADL.R/ and that there exists a Woodin cardinal with a measur-

able cardinal above it. Then there is a partial orderPmaxin L.R/such that;

(1) Pmaxis !-closed and homogeneous(inL.R/),

(2) L.R/Pmax ZFC.


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1.2 The partial orderPmax 7

Further if is a 2 sentence in the language for the structure

hH.!2/; 2; INS i

and if

hH.!2/; 2; INS i

then

hH.!2/; 2;INS iL.R/Pmax : ut

The partial order Pmax is definable and thus, since granting large cardinals

Th.L.R//is canonical, it follows that Th.L.R/Pmax /is canonical.

Many of the open combinatorial questions at !1 are expressible as2 statements

in the structure

hH.!2/; 2; INS i

and so assuming the existence of large cardinals these questions are either false, orthey are true inL.R/Pmax .

In some sense the spirit ofMartins Axiom and its generalizations is to maximize

the collection of2 sentences true in the structure

hH.!2/; 2i

Indeed MA!1 is easily reformulated as a 2 sentence for hH.!2/; 2i.

By the remarks above, assuming fairly weak large cardinal hypotheses, any such

sentence which is true in some set generic extension ofV is true in a canonical generic

extension ofL.R/.

The situation is analogous to the situation of12 sentences andL. By Shoenfieldsabsoluteness theorem if a12 sentence holds inVthen it holds inL.

The difference here is that the model analogous to L is not an inner model but

rather it is a canonical generic extension of an inner model. This is not completely

unprecedented. Mansfields theorem on 12 wellorderings can be reformulated as fol-

lows.

Theorem 1.5(Mansfield). Suppose that is a 13 sentence which is true in V and

there is a nonconstructible real. Then is true in LP wherePis Sacks forcing .defined

inL/. ut

Of course the 13 sentence also holds inL so this is not completely analogous toour situation. :CH is a (consistent)2 sentence for hH.!2/; 2i which is false in any

of the standard inner models.

Nevertheless the analogy with Sacks forcing is accurate. The forcing notionPmaxis a generalization of Sacks forcing to !1.

The following theorem, slightly awkward in formulation, shows that any at-

tempt to realize in H.!2/ all suitably consistent 2 sentences, requires at least 12-

Determinacy.


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8 1 Introduction

Theorem 1.6. Suppose that there exists a model, hM; Ei, such that

hM; Ei ZFC

and such that for each2 sentence if there exists a partial orderPsuch that

hH.!2/; 2iVP ;

then

hH.!2/; 2ihM;Ei :

Assume there is an inaccessible cardinal. Then

V 12-Determinacy: ut

One can strengthen Theorem 1.4 by expanding the structure

hH.!2/; 2; INS i

by adding predicates for each set of reals in L.R/. This theorem requires additional

large cardinal hypotheses which in fact imply ADL.R/ unlike the large cardinal hy-

pothesis of Theorem 1.4.

Theorem 1.7. Assume there are ! many Woodin cardinals with a measurable above.Suppose is a2 sentence in the language for the structure

hH.!2/; 2; INS ; XI X 2L.R/; X Ri

and that

hH.!2/; 2; INS ; XI X 2L.R/; X Ri

Then

hH.!2/; 2; INS ; XI X2L.R/; X RiL.R/Pmax : ut

We note that since Pmaxis !-closed, the structure

hH.!2/; 2; INS ; XI X 2L.R/; X RiL.R/Pmax

is naturally interpreted as a structure for the language of

hH.!2/; 2;INS ; XI X2L.R/; X Ri:

The key point is that this strengthened absoluteness theorem has in some sense a

converse.

Theorem 1.8. Assume ADL.R/. Suppose that for each 2 sentence in the language

for the structure

hH.!2/; 2; INS ; XI X 2L.R/; X Ri

if

hH.!2/; 2; INS ; XI X 2L.R/; X RiL.R/Pmax

then

hH.!2/; 2;INS ; XI X2L.R/; X Ri :

Then

L.P.!1//D L.R/G

for someG Pmaxwhich isL.R/-generic. ut


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1.2 The partial orderPmax 9

If one assumes in addition that R# exists then Theorem 1.8 can be reformulated as

follows. For eachn 2 ! letUnbe a set which is1 definable in the structure

hL.R/; hi Wi < ni; 2i

where hi W i < ni is an increasing sequence of Silver indiscernibles ofL.R/, and

such thatUnis universal.

Theorem 1.9. AssumeADL.R/ and thatR# exists. Suppose that for each2sentence

in the language for the structure

hH.!2/; 2; INS ; UnI n < !i

if

hH.!2/; 2;INS ; UnI n < !iL.R/Pmax

then

hH.!2/; 2; INS ; UnI n < !i :

Then

L.P.!1//D L.R/G

for someG Pmaxwhich isL.R/-generic. ut

Thus in the statement of Theorem 1.9 one only refers to a structure of countable

signature.

These theorems suggest that the axiom:

./ AD holds inL.R/andL.P.!1//is a Pmax-generic extension ofL.R/;

is perhaps, arguably, the correct maximal generalization ofMartins Axiomat least as

far as the structure ofP.!1/is concerned. However an important point is that we do

not know if this axiom can always be forced to hold assuming the existence of suitablelarge cardinals.

Conjecture. Assume there are!2 many Woodin cardinals. Then the axiom./holds

in a generic extension ofV. ut

Because of the intrinsics of the partial order Pmax, this axiom is frequently easier to

use than the usual forcing axioms. We give some applications for which it is not clear

thatMartins Maximumsuffices. Another key point is:

There is no need in the analysis ofL.R/Pmax for any machinery of iterated forc-

ing. This includes the proofs of the absoluteness theorems.

Further

The analysis ofL.R/Pmax requires only ADL.R/.

For the definition ofPmaxthat we shall work with the analysis will require some iterated

forcing but only for ccc forcing and only to produce a poset which forces MA!1 .

In Chapter 5 we give three other presentations ofPmax based on the stationary

tower forcing. The analysis of these (essentially equivalent) versions ofPmax require

no local forcing arguments whatsoever. This includes the proof of the absoluteness

theorems.


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10 1 Introduction

Also in Chapter 5 we shall discuss methods for exploiting ./, giving a useful

reformulation of the axiom. This reformulation doesnot involvethe definition ofPmax.

We shall also prove that, assuming./,

L.P.!1// AC:

This we accomplish by finding a 2 sentence which if true in the structure,hH.!2/; 2i;

implies (in ZF C DC) that there is a surjection

W!2! R

which is definable in the structure

hH.!2/; 2i

from parameters. This sentence is a consequence ofMartins Maximum and an anal-

ogous, but easier, argument shows that assuming ADL.R/, it is true inL.R/Pmax . Thus

the axiom./implies2@0 D @2. Actually we shall discuss two such sentences,AC

andAC. These are defined in Section 5.1 and Section 5.3 respectively.

1.3 Pmaxvariations

Starting in Chapter 6, we shall define several variations of the partial order Pmax. In-

terestingly each variation can be defined as a suborder of a reformulation ofPmax. The

reformulation is Pmaxand it is the subject of Section 5.5. A slightly more general refor-

mulation is P0max and in Section 5.6 we prove a theorem which shows that essentially

any possible variation, subject to the constraint that

2@0 D2@1

in the resulting model, is a suborder ofP0max.

The variations yield canonical models which can be viewed as constrained versions

of the Pmaxmodel. Generally the constrained versions will realize any 2 sentence in

the language for the structure

hH.!2/; INS ; 2i

which is (suitably) consistent with the constraint; i. e. unless one takes steps to prevent

something from happening it will happen. This is in contrast to the usual forcing

constructions where nothing happens unless one works to make it happen.

One application will be to establish the consistency with ZFC that the nonstationaryideal on !1 is !1-dense. This also shows the consistency of the existence of an !1-

dense ideal on !1with :CH. Further for these results only the consistency of ZF CAD

is required. This is best possible for we have proved that if there is an!1-dense ideal

on!1 then

L.R/ AD:

More precisely we shall define a variation ofPmax, which we denote Qmax, and

assuming ADL.R/ we shall prove that

L.R/Qmax ZFC C The nonstationary ideal on!1 is !1-dense:


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1.3 Pmax variations 11

Again ADL.R/ suffices for the analysis ofL.R/Qmax and there are absoluteness theo-

rems which characterize the Qmax-extension.

Collectively these results suggest that the consistency of ADL.R/ is an upper bound

for the consistency strength of many propositions at !1, over the base theory,

ZFC C For allx 2 R,x#

exists C 12 D!2:

However there are two classes of counterexamples to this.

Suppose that R# exists and thatL.R#/ AD. For each sentence such that

L.R/ ;

the following:

There exists a sequence, hB; W < < !1i, of borel sets such that

R# D[

\>

B;

;

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 ofany set generic extension

ofL.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 thereis a partial order P 2L.R/such that

L.R/P ZFC C R# exists:

The second class of counterexamples is a little more subtle, as the following exam-

ple illustrates. If the nonstationary ideal on!1 is!1-dense and ifChangs Conjecture

holds then there exists a countable transitive set,M, such that

M ZFC C There exist! C 1many Woodin cardinals;

(and soM ADL.R/ and much more). The application ofChangs Conjectureis onlynecessary to produce

X2 H.!2/

such thatX\ !2 has ordertype!1. The subtle and interesting aspect of this example

is thatL.R/Qmax Changs 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 Changs Conjecture,

is equivalent to a strong form of the consistency of AD. This is the subject of Sec-

tion 9.4.


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12 1 Introduction

The statement that the nonstationary ideal on!1 is !1-dense is a2 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) consistent2 sentences which are mutually inconsistent. This is

in contrast to the case of2 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 Chap-

ter 8, (see Theorem 10.159), where we discuss problems related to the problem of therelationship betweenMartins Maximumand 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:

SupposeC !1 is closed and unbounded. Then there exists x 2 R such that

< !1 jLx is admissible 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 Pmaxvariations. The two variations considered in Chapter 7 were selected

simply to illustrate the possibilities. The examples in Chapter 8 were chosen to high-

light quite different approaches to the analysis of a Pmaxvariation, 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 ofPmax to re-

produce 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 thenH .!2/is contained in the limit of a directed system of count-

able 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


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1.4 Extensions of inner models beyondL.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 inMis a Suslin tree in V.

If AD holds inL.R/and ifG SmaxisL.R/-generic then inL.R/Gthe following

strengthening of the sentence holds:

For every A !1 there exists B !1 such that A 2 LB and such that if

T 2LBis a Suslin tree inLB, thenTis a Suslin tree.

InL.R/G 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 consis-

tent.

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 forcingor

Mathias forcing, etc. In Section 7.2, we define a variation ofPmax which forces the

Borel Conjecture. The definition of this forcing notion does not involve Laver forc-

ing, Mathias forcing or any variation of these forcing notions. In the model obtained,

a version ofMartins Maximum holds. Curiously, to prove that theBorel Conjecture

holds in the resulting model we do use a form of Laver forcing. An interesting tech-

nical 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 theBorel

Conjectureholds and in which additional interesting combinatorial facts also hold.

1.4 Extensions of inner models beyondL.R/

In Chapter 9 we again focus primarily on the Pmax-extension but now consider exten-

sions of inner models strictly larger than L.R/. These yield models of./ with rich

structure forH.!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 formL.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 Martins Maximum holds for

partial orders of cardinalityc , this isMartins Maximum.c/, and in which!2 exhibits

some interesting combinatorial features.


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14 1 Introduction

Actually in the models obtained,Martins MaximumCC.c/ holds. This is the asser-

tion thatMartins MaximumCC holds for partial orders of cardinalityc whereMartins

MaximumCC is a slight strengthening ofMartins Maximum. These forcing axioms,

first formulated in.Foreman, Magidor, and Shelah 1988/, are defined in Section 2.5.

Recasting the Pmaxvariation for the Borel Conjecture in this context we obtain, inthe spirit ofMartins Maximum, a model in which the Borel Conjectureholds together

with the largestfragment ofMartins Maximum.c/ which is possibly consistent withtheBorel Conjecture.

Another reason for considering extensions of inner models larger than L.R/is that

one obtains more information about extensions ofL.R/. For example the proof that

L.R/Qmax Changs Conjecture;

requires considering the.Qmax/N-extension of inner models Nsuch that

.R \ N /# 2N

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

12 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 cannotbe realized in any such structure defined in an ex-

tension ofL.R/.

The results of Chapter 9 suggest a strengthening of the axiom./:

Axiom ./C: For each set X !2there exists a set A R and a filter G Pmaxsuch that

(1) L.A;R/ ADC,

(2) G is L.A;R/-generic andX 2L.A;R/G.

This is discussed briefly in Chapter 10 which explores the possible relationships be-

tween Martins 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

Martins MaximumCC.c/ C Strong Changs Conjecture

together withall the 2 consequences of./for the structure

hH.!2/;Y; 2 WY R; Y 2L.R/i

doesnotimply./. We shall also prove an analogous theorem which shows that cofi-

nally many sets from P.R/ \ L.R/must be added; for each setY02 P.R/ \ L.R/,

Martins MaximumCC.c/ C Strong Changs Conjecture


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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 ofMartins

MaximumCC.

1.5 Concluding remarks the view from Berlin in 1999

The following question resurfaces with added significance.

Assume ADL.R/. IsL.R/ !3?

The point is that if it is consistent to have ADL.R/ andL.R/ > !3 then presum-

ably this can be achieved in a forcing extension ofL.R/. This in turn would suggest

there are generalizations ofPmaxwhich produce generic extensions ofL.R/in which

c > !2. There are many open questions in combinatorial set theory for which a (posi-

tive) solution requires building forcing extensions in which c > !2.

The potential utility ofPmaxvariations 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 Jacksons analysis of measures andultrapowers inL.R/under the hypothesis of ADL.R/.

Theorem 1.10(Jackson). Assume the nonstationary ideal on !1 is!2-saturated andthat 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 inV, or;

(2) There exists a setA of regular cardinals, above !2, such that

a) jAj D @1,

b) jpcf.A/j D @2. ut

One of the main open problems of Shelahs 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 Pmaxvariations 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,


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16 1 Introduction

recently, a more general class of variations has been identified for which Theo-

rem 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 theexposition 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 actu-

ally 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 theo-

rems in terms of 2-compactness but restricting to the case of extensions ofL.R/,

.Shelah and Zapletal 1999/.

More recently Ketchersid, Larson, and Zapletal.2007/isolate a family of explicitNamba-like forcing notions which can, under suitable circumstances, change the value

of

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 gen-

eralization 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-logic etc.

We (very briefly) discuss -logic (updated to 2010) in Section 10.4 and Sec-

tion 10.5. In some sense the entire discussion ofPmax 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 Pmaxand 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 a2 sentence in the language for the structure hH.!2/; 2;INS i

and that

ZFC C hH.!2/; 2;INS i


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1.5 Concluding remarks the view from Berlin in 1999 17

is-consistent, then

hH.!2/; 2;INS iL.R/Pmax : ut

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 vari-ations, this holds for all the variations which we discuss in this book, is that each

provides a finite axiomatization, over ZFC, of the theory ofH .!2/(in -logic). ForPmax, 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/ :. ut

This particular feature underscores the fundamental difference between the method

ofPmax 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

byany Pmaxvariation. Theorem 10.185 indicates such an example, the essential feature

is that

12 < !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/; 2iVG 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 structurehH.!2/; 2iV G. There is at present no known example which works,

say from some large cardinal assumption, independent of the choice of the groundmodel.

-logic provides the natural setting for posing questions concerning the possibility

of such generalizations ofPmax, to for example !2, i. e. for the structureH .!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 Pmaxvariations to the context of CH; more precisely, if CH holds then the theory ofH .!2/cannotbe 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.


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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, 19971998, 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 W. Hugh Woodin

1.6 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 onthis 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 theo-

rem that Martins Maximum implies !.!2/, have all been rendered irrelevant by a

remarkable theorem of.Shelah 2008/ which shows that !.!2/ is a consequence of2!1 D !2. Shelahs result shows that assuming Martins Maximum.c/, or simply as-

suming that2!1 D!2, then the nonstationary ideal at !2 cannot be semi-saturated onthe 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 consis-

tency of this theory follows from simply the existence of a Woodin cardinal which isa limit of Woodin cardinals. Therefore in this edition the consistency results for semi-

saturated ideals at !2 are simply stated without proof. The proofs of these theoremsare 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 semi-

saturated is not correct that error was due to a careless misconception regarding


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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 ofH .!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 ofL. This enlargement ofL 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 exten-

der 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 (non-

trivial) 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 thestructure theory ofL.R/in the context of AD to a structure theory ofL.VC1/in the

context of an elementary embedding,

j WL.VC1/! L.VC1/

with critical point below require that Vnot be a generic extension of any inner model

which is not countably closed withinV . 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 Vis a strategic extender

model based on natural structural principles. This of course would rule out that the

axiom./ holds though ifVis a strategic extender model (with a Woodin cardinal)

then the axiom./holds in a homogeneous forcing extension ofVand so the axiom

./has a special connection to V as an axiom which holds in a canonical companion

toVmediated by an intervening model of ADC which is the manifestation of-logic.

An appealing aspect to this scenario is that the relevant axiom forVcan 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:


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20 1 Introduction

(1) There is a supercompact cardinal.

(2) There exist a universally Baire setA R and < L.A;R/ such that

V .HOD/L.A;R/ \ V

for all2-sentences (equivalently, for all2-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


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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 withthe standard Euclidean space,.1; 1/. If at some point the discussion is particularly

sensitive to the manifestation ofR 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 ifa 2 H .!1/then the seta can be coded by coding the structure

hb [ !;a; 2i

whereb is the transitive closure ofa.

A realx codesa ifx decodes setsA ! andE ! ! such that

hb [ !;a; 2i h! ;A ;Ei;

where againb is the transitive closure ofa.

Suppose thatM 2H.cC/and letNbe the transitive closure ofM. Fix a reason-

able decoding of a setX R to produce an element ofP.R/ P.R R/ P.R R/:

A setX R codesM ifXdecodes setsA R,E RR and RR such

that is an equivalence relation on R, A R, E R R, AandE are invariant

relative to , and such that

hN;M; 2i hR=; A=; E=i:

We shall be interested in setsMwhich are coded in this fashion by setsX R such

thatXbelongs to a transitive inner model in which the Axiom of Choicefails.

2.1 Weakly homogeneous trees and scales

For any setX,X

`.s/ denotes the length ofs , which formally is simply the domain ofs . A tree T on

a setXis a set of finite sequences fromXwhich is closed under initial segments. So

T X

We abuse this convention slightly and say that a tree Tis a tree on! whereis an ordinal ifTis a set of pairs .s;t /such that


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22 2 Preliminaries

(1) s 2!

(2) `.s/D `.t/,

(3) for alli < `.s/,.sji; t ji /2 T.

Suppose thatTis a tree on! . Fors 2!

Ts D t 2

and for eachx 2!! ,

Tx D [Txjk jk 2! :

Thus for eachx 2!! ,Tx is a tree on . We let

T D .x; f /j x 2!!; f 2!; and for allk 2!; .xjk; fjk/ 2 T

denote the set of infinite branches ofTand we let

pTD x 2!! j.x; f /2 T for somef 2!:

ThuspT !! , it is theprojectionofT, and clearly

pT D x 2!! jTx is not wellfounded:

A set of reals, A, is Suslin ifA D pT for some tree T. Of course assuming

theAxiom of Choice every set is Suslin. One can obtain a more interesting notion by

restricting the choice of the tree. This can done two ways, by definability or by placing

combinatorial constraints on the tree. The first route is the descriptive set theoretic one.

Apointclassis a set P.!!/. Suppose that is a pointclass and that for any

continuous function

F W!! !!!

ifA 2 then F1A 2 ; i. e. suppose is closed under continuous preimages.

Then has an unambiguous interpretation as a subset ofP.X /whereXis any space

homeomorphic with!! . The point of course is that this does not depend on the home-

omorphism. We shall use this freely. Similarly if in addition, is closed under finite

intersections and contains the closed sets, then has an unambiguous interpretation as

a subset ofP.X/whereXis any space homeomorphic with a closed subset of!! .

If is a pointclass closed under preimages by borel functions then has an unam-biguous interpretation as a subset ofP.X /whereXis any space homeomorphic with

a borel subset of!

!

. If the borel set is uncountable, i. e. ifXis uncountable, then thepointclass, , is uniquely determined by this interpretation. More generally ifX is a

topological space for which there is an isomorphism

W hX; .Z.X//i ! h!!;B.!!/i

where.Z.X// is the -algebra generated by the zero sets ofX, and B.!!/ is the

-algebra of borel subsets of!! , then again has an unambiguous interpretation as a

subset ofP.X/which again uniquely determines . This includes any space we shall

ever need to interpret in. We shall almost exclusively be dealing with pointclasses

closed under preimages by borel functions.


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2.1 Weakly homogeneous trees and scales 23

Suppose that is a pointclass. Then : denotes the pointclass obtained from

complementing the sets in ,

: D !! n Aj A 2 :

Clearly if is closed under continuous preimages then so is the dual pointclass, :.Moschovakis introduced the fundamental notion in descriptive set theory of a scale,

(see.Moschovakis 1980/). We recall the definition.

Definition 2.1. Suppose that is a pointclass closed under continuous preimages.

(1) Suppose thatA 2 . The setAhas a -scaleif there is a sequence

hi Wi 2!i

of prewellorderings onA such that the following conditions hold.

a) The set

hi ; x ; yi ji 2! ; x i y

belongs to .

b) There existsY 2 : such that

Y ! !! !!

and such that for alli < !,

i DYi \ .R A/:

whereYi D .x;y/j .i; x;y/ 2 Y is the section given byi .

c) Suppose that hxi Wi < !i is a sequence of reals in A which converges to x.

Suppose that for each i there existsi such thatxj i xi andxi i xjfor allj i . Thenx 2A and for alli < !,

x i xi :

(2) The pointclass has thescale propertyif every set in has a -scale. ut

The notion of a scale is closely related to Suslin representations.

Remark 2.2. (1) If the pointclass is a -algebra closed under continuous pre-

images and if contains the open sets then a set A 2 has a -scale if and

only if there is a sequence hi Wi < !i of prewellorderings onA such that each

belongs to and the condition (c) of the definition holds.

(2) If is a-algebra closed under both continuous preimages and continuous im-

ages then a set A 2 has a -scale if and only ifA D pT for some tree Twhich is coded by a set in . ut


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24 2 Preliminaries

Recall that a setA 2 P.R/ \ L.R/is 21-definable inL.R/if and only if it is1definable inL.R/with parameter R.

Assuming the Axiom of Choice fails in L.R/, then it is easily verified that there

must exist a setA2 P.R/ \ L.R/, such that R n Ais21-definable inL.R/and such

thatAis notSuslin inL.R/.The following theorem of Martin and Steel.1983/shows that assuming.AD/L.R/,

the pointclass.21/L.R/ has the scale property. By the remarks above this is best pos-

sible. In fact it follows by Wadge reducibility that, assuming .AD/L.R/, every set

A2 P.R/ \ L.R/

which is Suslin inL.R/, is necessarily.21/L.R/.

This theorem will play an important role in the analysis of the Pmax extension of

L.R/.

Theorem 2.3(MartinSteel). Suppose that

L.R/ AD:Then every setA Rwhich is 21-definable in L.R/ has a scale which is 21-definable

inL.R/. ut

Suppose thatXis a nonempty set. We let m.X/denote the set of countably com-

plete ultrafilters on the boolean algebra P.X/. Our convention is that is ameasure

on X if 2 m.X/. As usual for 2 m.X/ and A X, we write .A/ D 1 to

indicate thatA 2 .

Suppose thatX DY

is a unique k 2 ! such that .Yk/ D 1. Suppose that 1 and2 are measures on

Y

k2/D 1. Then2projects to

1 ifk1 < k2and, for allA Yk1 ,1.A/D 1 if and only if2.A/D 1 whereA D

s 2Yk2 js jk1 2A

:

We write1 < 2 to indicate that2projects to1.

For each 2 m.X/there is a canonical elementary embedding

j WV !MwhereMis the transitive inner model obtained from taking the transitive collapse of

VX=. Suppose that 1 2 m.Y

canonical elementary embedding

j1;2 WM1 !M2such thatj2 Dj1;2 j1 .

Suppose that hk W k 2 !i is a sequence of measures on Y

k 2 !, k.Yk/ D 1. The sequence hk W k 2 !i is a tower if for all k1 < k2,

k1 < k2 . The tower, hk W k 2 !i, is countably complete if for any sequence

hAk W k 2 !i such that for all k < !, k.Ak/ D 1, there exists f 2 Y! such that

fjk 2 Ak for allk 2 !. It is completely standard that ifhk W k 2 !iis a tower ofmeasures onY

of the sequence hMk Wk < !i under the system of maps,

jk1 ;k2 WMk1 !Mk2 .k1 < k2 < !/;

is wellfounded.


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We come to the key notions of homogeneous trees and weakly homogeneous trees.

These definitions are due independently to Kunen and Martin.

Definition 2.4. Suppose that is an ordinal and 0. Suppose thatT is a tree on

! .

(1) The treeT is-weakly homogeneousif there is a partial function

W!

such that

a) if.s;t / 2 dom./ then .s; t/.Ts/ D 1and.s;t/ is a -compete mea-sure,

b) for allx 2!! ,x 2pT if and only if there exists y 2!! such that

.xjk; yjk/ j k < ! dom./, h.x jk; yjk/ W k 2!i is a countably complete tower.

(2) The treeT is such that

a) ifs 2dom./then.s/.Ts/D 1 and.s/is a-compete measure,

b) for allx 2!! ,x 2pT if and only if

xjk jk 2! dom./,

h.x jk/ W k 2!i is a countably complete tower.

(2) The treeT is


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26 2 Preliminaries

Lemma 2.6. Suppose thatT is a tree on ! . ThenT is -weakly homogeneous

if and only if there exists a countable set m.

in is -complete and such that for all x 2 !!, x 2 pT if and only if there is a

countably complete towerhk W k 2 !i of measures in such that for all k 2 !,

k.Txjk/D 1. utHomogeneity is a rather restrictive condition on a tree, weak homogeneity, how-

ever, is not. For example if is a Woodin cardinal andTis a tree on ! for some

then there exists an ordinal < such that ifG Coll.!;/is V-generic then in

V G,T is


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Definition 2.10. (1) WH

is the set of all A R such that A is -weakly homo-

geneously Suslin. WH


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28 2 Preliminaries

Theorem 2.14(Steel). Suppose that0 < 1 are Woodin cardinals and

A2 WHC1

:

ThenA has a scale in WH


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Definition 2.18. Suppose that is a pointclass which is a boolean subalgebra ofP.R/and that is closed under continuous preimages and under continuous images.

(1) N is the set of all sets Xsuch that

hY;X; 2i hR=; P =; E=iwhere

a) Yis the transitive closure ofX,

b) is an equivalence relation on R,

c) P R andE R R,

d) ; P ; E are each in .

(2) M is the set of all X 2 N such that the following holds where Y is thetransitive closure ofX.

a) Suppose that

0 W R !N

and

1 W R !N

are functions inN . Then

.x;y/2 R R j0.x/ D 1.y/ and0.x/ 2 Y 2: ut

ClearlyM andN are each transitive. With our coding conventions N is simply

the set of all setsXwhich are coded by a set in .

Remark 2.19. Suppose that P.R/is a pointclass as in Definition 2.18.

(1) Suppose thatY 2N is transitive and that

hY; 2i hR=; E=i

where

a) is an equivalence relation on R,

b) E is a binary relation on R,

c) ; E are each in .

Let W R !Ybe the associated surjection. Then 2N .

(2) M is a transitive set which is closed under the Godel operations. Even with

determinacy assumptions on we do not know if this is true ofN . ut

Remark 2.20. (1) If D P.R/then

M DN DH .cC/:


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30 2 Preliminaries

(2) If D P.R/ \ L.R/then

M DN DL.R/

where is as computed in L.R/; i. e. where is the least ordinal such that in

L.R/there is no surjection

W R !

of the reals onto. ut

The following theorems summarize some of the relationships between M and N .

Theorem 2.21. Suppose that P.R/is a pointclass such that for each A 2 ,

LA.A;R/ \P.R/

where for eachA 2 , A is the least ordinal admissible relative to the pair.A;R/.Then

M DN D [LA.A;R/j A 2 : ut

Assuming AD one obtains some nontrivial information in the general case (weaker

closure on ), using various generalizations of theMoschovakis Coding Lemma.

Theorem 2.22. Suppose thatis a pointclass which is a boolean subalgebra ofP.R/and that is closed under continuous preimages and under continuous images. Sup-

pose that every set in is determined. Then:

(1) M \ OrdD N \ Ord.

(2) Suppose thatT 2N is a wellfounded subtree ofR

. Let WT !Ord

be the associated rank function. Then 2N .

(3) Let DM \ Ord. Then

P./ \ N D P./ \ M

and for each pair.X; Y /of elements ofP./ \ M,

L. X ;Y / M : ut

Remark 2.23. (1) We do not know if one can prove that M D N , assumingeither every set in is determined or even assuming

L.;R/ AD:

(2) Note that if0 1 are each (boolean) pointclasses closed under continuous

images and preimages then

N0 N1 :

However the relationship between M0 andM1 is less clear, even with deter-

minacy assumptions.


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2.2 Generic absoluteness 31

(3) Generally we shall only be interested inM \P.Ord/unless in fact satisfies

the closure requirements of Theorem 2.21. Thus the distinction between M and

N will never really be an issue for us.

Given a pointclass with the closure properties of Definition 2.18 we define a new

pointclass21./.

Definition 2.24. Suppose that is a pointclass which is a boolean subalgebra ofP.R/and that is closed under continuous preimages and under continuous images.

21./is the set of all Y R such thatY is1definable in the structure

hM \ V!C2; R; 2i

from real parameters. ut

It is easily verified that the pointclass

21./ is closed under finite unions, inter-

sections, continuous preimages and continuous images. It is not closed under comple-ments and further it isR-parameterized; i. e. it has a universal set. IfM D N then

21./is the set of all Y R such thatY is1 definable in the structure

hM ; R; 2i

from real parameters. We generally will only consider21./ when satisfies the

closure conditions of Theorem 2.21; i. e. whenM DN .

Definition 2.25. (1) Suppose is an ordinal and that the pointclass WH


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32 2 Preliminaries

Lemma 2.27. SupposeT1 is a tree on! 1,T2 is a tree on! 2, and

pT1D pT2:

SupposeT1 andT2 are-weakly homogeneous. Then

.pT1/

VG

D.pT2/

VG

whereG P isV-generic for a partial orderP 2V . ut

SupposeA R and let

D B R jB is projective inA:

Suppose every set in is -weakly homogeneously Suslin. Suppose .x1; x2/ is a

formula in the language of the structure

hH.!1/;A; 2i

anda 2 R. Let

B D t 2 R j hH.!1/;A; 2i t;a:ThusB 2.

SupposeP 2 V is a partial order and thatG P isV-generic. LetAG andBGbe the interpretations ofA andB inV G. Then

BG D

t 2 R j hH.!1/VG; AG ; 2i t;a

:

This is an easy consequence of Lemma 2.26 and Lemma 2.27. Alternate formula-

tions are given in the next two lemmas.

Lemma 2.28. SupposeA R and letB R be the set of reals which code elements

of the first order diagram of the structurehH.!1/; 2; Ai:

SupposeSandTare trees on! such that

(1) SandT are-weakly homogeneous,

(2) AD pSandB DpT .

Suppose P 2V andG P isV-generic. LetAG DpSand letBG DpT, each

computed inV G. Then inV G, BG is the set of reals which code elements of the

first order diagram of the structure

hH.!1/VG; 2; AGi: ut

Lemma 2.29. SupposeA R and suppose that each setB Rwhich is projective

inA, is-weakly homogeneously Suslin.

SupposeZ V is a countable elementary substructure such that C! < ,

2Z and such thatA 2 Z .

LetMZ be the transitive collapse ofZ and letZ be the image of under the

collapsing map. Suppose P 2 .MZ/Z is a partial order and thatg P is MZ-

generic.


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2.2 Generic absoluteness 33

Then

(1) A \ MZg 2 MZg,

(2) hV!C1 \ MZg; A \ MZg; 2i hV!C1; A; 2i.

Suppose further that

A2 . WH /V :

Then

A \ MZg 2 .WH

Z/MZ g : ut

Suppose that A R and that every set B 2 P.R/\ L.A;R/ is -weakly ho-

mogeneously Suslin. Then.A;R/# is-weakly homogeneously Suslin. This is easily

verified by noting that.A;R/# is a countable union of sets inL.A;R/.

This observation yields the following generic absoluteness theorem.

Theorem 2.30. Suppose thatA R and that every set in P.R/\L.A;R/ is -weaklyhomogeneously Suslin. Suppose thatT is a -weakly homogeneous tree such that

AD pT

and thatP 2V is a partial order.

Suppose thatG P isV-generic. Then there is a generic elementary embedding

jG WL.A;R/! L.AG;RG/

such that

(1) jG.A/D AG DpT V G,

(2) RG D RVG,

(3) L.AG ;RG/D jG.f /.a/j a 2 RG; f W R! L.A;R/andf 2L.A;R/.

Further the properties(1)(3)uniquely specifyjG . ut

One corollary of Theorem 2.30 is the following generic absoluteness theorem

which we shall need.

Theorem 2.31. Suppose that is a limit of Woodin cardinals and that there a measur-

able cardinal above. Suppose that

G PisV-generic where Pis a partial order such thatP 2V .

Then

L.R/V L.R/VG:

Proof. By Theorem 2.13, each set

X2 P.R/ \ L.R/

is


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34 2 Preliminaries

The next theorem shows, in essence, that the key property of weakly homogeneous

trees given in Lemma 2.26 is equivalent in the presence of large cardinals to weak

homogeneity.

Theorem 2.32. Suppose is a Woodin cardinal. Suppose thatS andTare trees on! such that ifG Coll.!;/is V-generic then,

.pT /V G D RV G n .pS/V G:

ThenSandTare eachthere existsb 2a such thatf b

(2) A setc isclosedif there exists a function

f W.[c/

such that

c D b [c jf b

(3) A setb isclosed and unbounded in a ifb D c \ a for some closed setc such

that [c D [a.

(4) A setb is stationary ina ifb is stationary,b a and if[aD [b. ut

The following elementary facts concerning stationary sets are easy to verify.

(i) (projection) Supposea is stationary andx [a. Then

\ x j2a

is stationary.

(ii) (normality) Supposea is stationary and that [a ;. Suppose

f Wa ! [a

is a choice function; i. e. for all 2a n ;,f ./2 . Then for somet 2 [a,

jf ./D t

is stationary ina.


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2.3 The stationary tower 35

Definition 2.34 (Stationary Tower). Supposea and b are stationary sets. Then a bif[b [aand

\ .[b/ j 2a b:

(1) For each ordinal, P


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36 2 Preliminaries

Suppose.M; E/is wellfounded and letNbe the transitive collapse of.M; E/. In

this case (1) asserts that for each x 2 V , jx 2 N. Therefore for each < ,

j jV 2Nand so by (2),G \ V 2N.

If.M; E/is not wellfounded these conclusions still hold. (1) implies that for each

< , V belongs to the wellfounded part of.M; E/ and so by (2), G \V alsobelongs to the wellfounded part of.M; E/.

The next theorem indicates a key influence of large cardinals.

Theorem 2.36. Suppose is a Woodin cardinal and thatG Q


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2.4 Forcing Axioms 37

(2) P issemiproperif for all sufficiently large ; if

XH .C/

is a countable elementary substructure with P 2 X, then for eachp0 2 P\ X

there existsp12 Psuch thatp1p0 and such that for each term2VP \ X;

ifG P isV-generic withp1 2G then eitherIG./ !V1 orIG. /2 X. ut

Remark 2.39. (With notation as in Definition 2.38.)

(1) Definition 2.38(1) asserts simply that ifp1 2 G thenXcan be expanded to an

elementary substructure

X H .C/G

such thatG 2 X and such that X \ C D X\ C. For sufficiently largethis in turn is equivalent to requiring thatX \ H.C/D X.

(2) Definition 2.38(2) asserts that ifp1 2G thenXcan be expanded to an elemen-

tary substructure

X H .C/G

such thatG 2X and such thatX \ !1 DX\ !1. ut

There are several equivalent definitions of proper partial orders. One elegant ver-

sion is given in the next lemma.

Lemma 2.40(Shelah). Suppose thatP is a partial order. The following are equiva-

lent.

(1) Pis proper.

(2) For all stationary setsa such thata P!1.[a/,

VP ais stationary: ut

Definition 2.41. (1) (Baumgartner, Shelah)Proper Forcing Axiom.PFA/: Suppose

that P is a proper partial order and that D P.P/ is a collection of dense

subsets ofPwith

jDj !1:

Then there exists a filter F Psuch that

F \ D ;

for allD 2 D .


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38 2 Preliminaries

(2) (Shelah)Semiproper Forcing Axiom .SPFA/: Suppose thatP is a semiproper

partial order and that D P.P/is a collection of dense subsets ofPwith

jD j !1:


F \ D ;

for allD 2 D . ut

Definition 2.42(ForemanMagidorShelah). Suppose that P is a partial order. The

partial order P isstationary set preservingif

.INS

/V D.INS

/VP

\ V: ut

Definition 2.43(ForemanMagidorShelah). Martins Maximum: Suppose thatP is

a partial order which is stationary set preserving.Suppose that D P.P/is a collection of dense subsets ofPwith

jD j !1:


F \ D ;

for allD 2 D . ut

In factMartins Maximumis equivalent to SPFA.

Theorem 2.44(Shelah). The following are equivalent.

(1) Martins Maximum.

(2) SPFA. ut

There are several variations of these forcing axioms which we shall be interested

in. We restrict our attention to variations ofMartins Maximum.

Definition 2.45(ForemanMagidorShelah). (1) Martins MaximumC: Suppose

that Pis a partial order which is stationary set preserving.

Suppose that D P.P/is a collection of dense subsets ofPwith

jD j !1;

and that2VP is a term for a stationary subset of!1. Then there exists a filter

F Psuch that:

a) for allD 2 D , F \ D ;;

b) < !1 j for somep 2 F; p 2 is stationary in!1.


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2.4 Forcing Axioms 39

(2) Martins MaximumCC: Suppose that Pis a partial order which is stationary set

preserving.

Suppose that D P.P/is a collection of dense subsets ofPwith

jD j !1;

and that h W < !1i is a sequence of terms for stationary subsets of!1. Then

there exists a filter F Psuch that:

a) For allD 2 D , F \ D ;;

b) For each < !1,

< !1 j for somep 2 F; p 2

is stationary in!1. ut

The following lemma notes useful consequences of these axioms which are quite

relevant to the themes of this book. These consequences ofMartins Maximum and

of Martins MaximumCC are not equivalences; however they are equivalences for

boundedversions of these forcing axioms, see Lemma 10.93 and Lemma 10.94 of

Section 10.3.

Lemma 2.46. Suppose thatPis a partial order which is stationary set preserving.

(1) (Martins Maximum)Then

hH.!2/; 2i 1 hH.!2/; 2iVP :

(2) (Martins MaximumCC)Then

hH.!2/;INS ; 2i 1 hH.!2/;INS ; 2iVP : ut

Definition 2.47(ForemanMagidorShelah). (1) Martins MaximumC.c/: Mar-

tins MaximumC holds for partial orders Pwith jPj c .

(2) Martins MaximumCC.c/:Martins MaximumCC holds for partial orders Pwith

jPj c . ut

Remark 2.48. One can naturally define SPFA.c/. One subtle aspect of the equiva-lence ofMartins Maximumand SPFA is that Martins Maximum.c/ is notequivalent

to SPFA.c/; Martins Maximum.c/ implies that INS (the nonstationary ideal on !1)is !2-saturated whereas SPFA.c/ does not. One strong indication of the difference

follows from the results of Section 9.5:

AssumeMartins Maximum.c/. ThenProjective Determinacyholds.

The consistency of SPFA.c/ can be obtained from that of the existence of a strong

cardinaland so SPFA.c/ does not imply even12-Determinacy. ut


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40 2 Preliminaries

We end this section with the definition of a somewhat technical variation ofMar-

tins Maximum.c/. For many applications whereMartins Maximum.c/ is used, this

variation suffices. For example, it implies that INS

is!2-saturated. However we shall

see in Section 9.2.2 that this forcing axiom is (probably) significantly weaker than

Martins Maximum.c/. We re

woodin - the axiom of determinacy, forcing axioms and the non-stationary ideal

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