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

859 Pages·2010·4.218 MB·English

by  W. Hugh Woodin

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
LibraryofCongressCataloging-in-PublicationData
Woodin,W.H.(W.Hugh)
The axiom of determinacy, forcing axioms, and the nonstationary
ideal/byW.HughWoodin.(cid:2)2ndrev.andupdateded.
p.cm.(cid:2)(DeGruyterseriesinlogicanditsapplications;1)
Includesbibliographicalreferencesandindex.
ISBN978-3-11-019702-0(alk.paper)
1.Forcing(Modeltheory) I.Title.
QA9.7.W66 2010
511.3(cid:2)dc22
2010011786
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de.
(cid:2)2010WalterdeGruyterGmbH&Co.KG,Berlin/NewYork
Typesetting:Da-TeXGerdBlumenstein,Leipzig,www.da-tex.de
Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen
(cid:3)Printedonacid-freepaper
PrintedinGermany
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Contents
1 Introduction 1
1.1 Thenonstationaryidealon! . . . . . . . . . . . . . . . . . . . . . 2
1
1.2 ThepartialorderP . . . . . . . . . . . . . . . . . . . . . . . . . . 6
max
1.3 P variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
max
1.4 ExtensionsofinnermodelsbeyondL.R/ . . . . . . . . . . . . . . . 13
1.5 Concludingremarks–theviewfromBerlinin1999 . . . . . . . . . . 15
1.6 TheviewfromHeidelbergin2010 . . . . . . . . . . . . . . . . . . . 18
2 Preliminaries 21
2.1 Weaklyhomogeneoustreesandscales . . . . . . . . . . . . . . . . . 21
2.2 Genericabsoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Thestationarytower . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 ForcingAxioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 ReflectionPrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Genericideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Thenonstationaryideal 51
3.1 Thenonstationaryidealandı1 . . . . . . . . . . . . . . . . . . . . . 51
(cid:2)2
3.2 ThenonstationaryidealandCH . . . . . . . . . . . . . . . . . . . . 108
4 TheP -extension 116
max
4.1 Iterablestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2 ThepartialorderP . . . . . . . . . . . . . . . . . . . . . . . . . . 136
max
5 Applications 184
5.1 Thesentence(cid:2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
AC
5.2 Martin’sMaximumand(cid:2) . . . . . . . . . . . . . . . . . . . . . . 187
AC
5.3 Thesentence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
AC
5.4 ThestationarytowerandP . . . . . . . . . . . . . . . . . . . . . 199
max
5.5 P(cid:2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
max
5.6 P0 . . . .(cid:2).(cid:3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
max
(cid:2)
5.7 TheAxiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
(cid:2)
5.8 HomogeneitypropertiesofP.! /=I . . . . . . . . . . . . . . . . . 274
1 NS
6 P variations 287
max
6.1 2P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
max
6.2 Variationsforobtaining! -denseideals . . . . . . . . . . . . . . . . 306
1
6.2.1 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
max
6.2.2 Q(cid:2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
max
vi Contents
6.2.3 2Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
max
6.2.4 WeakKurepatreesandQ . . . . . . . . . . . . . . . . . . 377
max
6.2.5 KTQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
max
6.2.6 Nullsetsandthenonstationaryideal . . . . . . . . . . . . . . 403
6.3 Nonregularultrafilterson! . . . . . . . . . . . . . . . . . . . . . . 421
1
7 Conditionalvariations 426
7.1 Suslintrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
7.2 TheBorelConjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 441
8 |principlesfor!1 493
8.1 CondensationPrinciples . . . . . . . . . . . . . . . . . . . . . . . . 496
8.2 P|NS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
max
8.3 Theprinciples,|C and|CC . . . . . . . . . . . . . . . . . . . . . . 577
NS NS
9 ExtensionsofL.(cid:2);R/ 609
C
9.1 AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
9.2 TheP -extensionofL.(cid:3);R/ . . . . . . . . . . . . . . . . . . . . . 617
max
9.2.1 Thebasicanalysis . . . . . . . . . . . . . . . . . . . . . . . 618
9.2.2 Martin’sMaximumCC.c/ . . . . . . . . . . . . . . . . . . . 622
9.3 TheQ -extensionofL.(cid:3);R/. . . . . . . . . . . . . . . . . . . . . 633
max
9.4 Chang’sConjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
9.5 WeakandStrongReflectionPrinciples . . . . . . . . . . . . . . . . . 651
9.6 StrongChang’sConjecture . . . . . . . . . . . . . . . . . . . . . . . 667
9.7 Idealson! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683
2
10 Furtherresults 694
10.1 Forcingnotionsandlargecardinals . . . . . . . . . . . . . . . . . . . 694
10.2 CodingintoL.P.! // . . . . . . . . . . . . . . . . . . . . . . . . . 701
1
Q
10.2.1 Codingbysets,S . . . . . . . . . . . . . . . . . . . . . . . . 703
10.2.2 Q.X/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
max
10.2.3 P.;/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
max
10.2.4 P.;;B/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
max
10.3 BoundedformsofMartin’sMaximum . . . . . . . . . . . . . . . . . 784
10.4 (cid:4)-logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
10.5 (cid:4)-logicandtheContinuumHypothesis . . . . . . . . . . . . . . . . 813
10.6 TheAxiom.(cid:2)/C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827
10.7 TheEffectiveSingularCardinalsHypothesis . . . . . . . . . . . . . . 835
11 Questions 840
Bibliography 845
Index 849
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
page18withcommentsregardingthisedition.
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
Go¨del’sconstructibleuniverseLanditsrelativizationtothereals,L.R/,arecanonical
models though of course the assertion that L.R/is a canonical model is made in the
contextoflargecardinals. Ourclaimisvague,neverthelessthemodelweidentifycan
becharacterizedbyitsabsolutenessproperties. Thismodelcanalsobecharacterized
by certain homogeneity properties. From the point of view of forcing axioms it is
theultimatemodel atleastasfaras thesubsetsof ! areconcerned. Itisarguably a
1
completionofP.! /,thepowersetof! .
1 1
This model is a forcing extension of L.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
provideevidencethatthisisausefulmethodforachievingconsistencyresults,obtain-
inganumberofresultswhichseemoutofreachofthecurrenttechnologyofiterated
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 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
genericextensionsofspecificinnermodels(L.R/inmostcases),thesemodelscanbe
characterizedwithoutreferencetothis. Forexample,aswehaveindicatedabove,our
canonical model is a generic extension of L.R/. The corresponding partial order we
denote(cid:2)by(cid:3) Pmax. In Chapter 5(cid:2)w(cid:3)e give a characterization for this model isolating an
axiom (cid:2) . Theformulationof (cid:2) doesnotinvolveP ,nordoesitobviouslyreferto
(cid:2) (cid:2) max
L.R/. InsteaditspecifiespropertiesofdefinablesubsetsofP.! /.
1
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
(quitebyaccident)theeffectivefailureofCH. Thisandrelatedresultsarethesubject
ofChapter3. WediscusseffectiveversionsofCHbelow.
Gdel was the first to propose that large cardinal axioms could be used to settle
questionsthatwereotherwiseunsolvable. Thishasbeenremarkablysuccessfulpartic-
ularlyintheareaofdescriptivesettheorywheremostoftheclassicalquestionshave
nowbeenanswered. HoweveraftertheresultsofCohenitbecameapparentthatlarge
cardinalscouldnotbeusedtosettletheContinuumHypothesis. Thiswasfirstargued
byLevyandSolovay.1967/.
NeverthelesslargecardinalsdoprovidesomeinsighttotheContinuumHypothesis.
One example of this is the absoluteness theorem of Woodin .1985/. Roughly this
theoremstatesthatinthepresenceofsuitablelargecardinalsCH“settles”allquestions
withthelogicalcomplexityofCH.
MorepreciselyifthereexistsaproperclassofmeasurableWoodincardinalsthen
†2sentencesareabsolutebetweenallsetgenericextensionsofV whichsatisfyCH.
1
Theresultsofthisbookcanbeviewedcollectivelyasaversionofthisabsoluteness
theoremforthenegationoftheContinuumHypothesis(:CH).
1.1 The nonstationary ideal on !
1
Webeginwiththefollowingquestion.
Isthereafamily¹S j ˛ < ! ºofstationarysubsetsof! suchthatS \S is
˛ 2 1 ˛ ˇ
nonstationarywhenever˛ ¤ˇ?
Theanalysisofthisquestionhasplayed(perhapscoincidentally)animportantrole
insettheory particularly inthe studyof forcing axioms, large cardinals and determi-
nacy.
The nonstationary ideal on ! is ! -saturated if there is no such family. This
1 2
statement is independent of the axioms of set theory. We let I denote the set of
NS
subsets of ! which are not stationary. Clearly I is a countably additive uniform
1 NS
idealon! . Ifthenonstationaryidealon! is! -saturatedthenthebooleanalgebra
1 1 2
P.! /=I
1 NS
isacompletebooleanalgebrawhichsatisfiesthe! chaincondition.Kanamori.2008/
2
surveyssomeofthehistoryregardingsaturatedideals,theconceptwasintroducedby
Tarski.
The first consistency proof for the saturation of the nonstationary ideal was ob-
tainedbySteelandVanWesep.1982/.Theyusedtheconsistencyofaverystrongform
oftheAxiomofDeterminacy(AD),see.Kanamori2008/andMoschovakis.1980/for
thehistoryoftheseaxioms.
1.1 Thenonstationaryidealon!1 3
SteelandVanWesepprovedtheconsistencyof
ZFCC “Thenonstationaryidealon! is! -saturated”
1 2
assumingtheconsistencyof
ZFCADRC“‚isregular”:
ADRistheassertionthatallrealgamesoflength! aredeterminedand‚denotesthe
supremumoftheordinalswhicharethesurjectiveimageofthereals. Thehypothesis
waslaterreducedbyWoodin.1983/totheconsistencyofZFCAD. Thearguments
ofSteelandVanWesepweremotivatedbytheproblemofobtainingamodelofZFCin
which! istheseconduniformindiscernible.ForthisSteeldefinedanotionofforcing
2
whichforcesoverasuitablemodelofADthatZFCholds(i.e.thattheAxiomof Choice
holds)andforcesboththat! istheseconduniformindiscernibleand(byargumentsof
2
VanWesep)thatthenonstationaryidealon! is! -saturated. Themethodof.Woodin
1 2
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
quotientalgebraP.! /=I hasaparticularlysimpleform,
1 NS
P.! /=I ŠRO.Coll.!;<! //:
1 NS 2
WehaveprovedthatthisinturnimpliesADL.R/ andsothehypothesisused(thecon-
sistencyofAD)isthebestpossible.
Thenextprogressontheproblemofthesaturationofthenonstationaryidealwas
obtainedinaseriesofresultsbyForeman,Magidor,andShelah.1988/. Theyproved
thatageneralizationofMartin’sAxiomwhichtheytermedMartin’sMaximumactually
implies that the nonstationary ideal is saturated. They also proved that if there is a
supercompact cardinal then Martin’s Maximum is true in a forcing extension of V.
LaterShelahprovedthatifthereexistsaWoodincardinaltheninaforcingextension
ofV thenonstationaryidealissaturated. Thislatterresultismostlikelyoptimalinthe
sensethatitseemsveryplausiblethat
ZFCC“Thenonstationaryidealon! is! -saturated”
1 2
isequiconsistentwith
ZFCC“ThereexistsaWoodincardinal”
see.Steel1996/.
Therewaslittleapparentprogressonobtainingamodelinwhich! isthesecond
2
uniform indiscernible beyond the original results of .Steel and VanWesep 1982/ and
.Woodin1983/. Recallthatassumingthatforeveryrealx,x# exists,theseconduni-
formindiscernibleisequalto(cid:2)ı12,thesupremumofthelengthsof(cid:3)(cid:5)12prewellorderings.
Thustheproblemofthesizeoftheseconduniformindiscernibleisaninstanceofthe