第三篇章终极集合论宇宙（V=UltimateL）

终极集合论宇宙（V=UltimateL）

TheMostowskiCollapse and the lnner Model

program

w.Hughwoodin

usiversity of CaEtyaia BerLeley

October 11.2013

TheMostowskiCoollapse

Theorcm

Suppose Mis atransitivesetandΧ≺M.Tbenthereisa unique transitve set Nand isomorphism

π：N≅Χ.

ThegentrallＩatiorsof the Mostowski Collapseareubiquitousin set Theory.

The Universeofsets

Thepowersct

suppose Χ is a set 。The powerset of Χ is the set

Р（X）={Y丨Y is a subset of Χ}.

CumuJativeHierarchyofScts　TheunherseVofsetsisgeneratedbydeflningVαbyinductionontheordinalα：

1.V₀=∅.

2.Vα+1=Ρ（Vα）.

3.ifαisalimitordinalthenVα=Uᵦ＜αVᵦ.

⇨EverysetbeloegstoVαforsomeordinalα.

Logicaldefinabilityfromparameters

BchedDooset

SupposeΧisatransitiveset.AsubsetΥ⊆Χislogicallydeflnablein（Χ，∈）fromparmetersifforsomeformula φ[x₀……x₀]andforsomeparametersa₁……a₂∈Χ，

Υ={a∈Χ丨（Χ，∈）╞ φ[a，a₁……a₀]}

Thedefinablepowerset

ForeachsetΧ，Рᴅel（Χ）denotesthesetofallΥ⊆ΧsuchthatΧislogicallydtfinableinthestructure（Χ，∈）fromparametersinΧ.

⇨（AxiomofChoice）Рᴅel（Χ）=Р（X）ifandonlyifΧisflnite。

⇨Рᴅel（Vᴊ+1）∩Р（R）isexactlytheprojectivesets.

Theeffectivecumulativehierarchy：L

Gōdd＇sconstructibleuniverse.L

DefineLαbyinductiononαasfollows.

1.L₀=∅.

2.（Successorcase）Lα+1=Рᴅel（Lα）.

3.（Limitcase）Lα=∪{Lᵦ丨β＜α）.

ListheclassofallsetsΧsuchthatΧ∈Lαforsomeordinalα.

Theorem（Gōdel）

SupposeΧ≺Lα.Thenthereisauniqueordinalβand

isomorphism

π：Lᵦ≅Χ.

Theorem（Scott）

AssumeV=LSupposeMisatransitivesetandthat

Χ≺M

isanelementarysubstructuresuchthatΧ≅Vαforsomeα.ThenVα=Χ.

AxiomsewhichasserttheexistenceofΧ≺MwhereMistransitive.

Χ≅Vα

andΧ≠Vαyieldthemodernhierarchyoflargecardinalaxioms.

⇨TheseaxiocnsimplyV≠L.

Stiongaxcmsofinfinity：largecardinalaxioms

BzrJmpJateforlargecardinalaxioms

Acardinalκisalargecardinalifthereexistsanelementaryembedding.

j：V→M

suchthatMisatrarsitiveclassandκistheleastordinalsuchthatj（α）≠α.

⇨RequiningMbeclosetoVyitldsahierarchyoflargecardinalaxioms：

⇨simplestcaseiswhereκisameasurablecardinal.

⇨M=VcontradictstheAxiomofChoice.

ThelnnerModelprogramseeksenlargementsofLinlargecardinalscanexist.

⇨Theproblembecomesmorediffrcultasoneascendsthehierarchy.

Thehierarchyoflargecardinalaxioms-shortversion

⇨Thereisaproperclassofmeasurablecardinals.

⇨Thereisaproperclassofstrongcardinals.

⇨Thereisaproperclassofwoodincardinals.

⇨Thereisaproperclassofsuprstrongcardinals.

…………………

⇨Thereisaproperclassofsupercompactcardinals.

⇨Thereaproperclassofextendiblecardinals.

⇨Thereaproperclassofhugecardinals.

⇨Thereaproperclassofw-hugecardinals.

EnlargementsofL

Deflnition

SupposeEisaset（orclass）.Then

1.L₀[E]=∅.

2.（Successorcase）Lα+1[E]=Рᴅel（Z）Where

Z=Lα[E]∪{E∩Lα[E]}.

3.（Limitcase）Lα[E]=∪{Lᵦ[E]丨β＜α}.

⇨L[E]istheclassofallsetsΧsuchthatΧ∈Lα[E]forsomeordinalα.

⇨lfE∩L=0thenL[E]=L

⇨ForeverysetΧthereisasetEsuchthatΧ∈L[E].

⇨ThisisequivalenttotheAxiomofChoice.

Thebuildingblocksforinnermodels：Extenders

supposethat

j：V→M

isarelementaryembeddingwithcniticalpointκ，κ＜η.andthat

Р（η）⊂M.

The（strong）extcnderEoflengthηdcrivedfromj

TheextenderEoflengthηdefinedfromjisthefunction：

E：Р（η）→Р（η）

whereE（A）=j（A）∩η.

TwoordinalsassociatedtotheextenderE：

⇨CRT（E）=min{α丨E（α）≠α}=κ.

⇨LTH（E）=ηwheredocn（E）=Р（η）.

Largecardinalaxiomsintermsofextenders

δ isastrongcardinalif

⇨foreach γ＞δ thereexistsanextenderEsuchthat

CRT（E）=δandLTH（E）≥ γ.

δisasupercompactcardinalif

⇨foreach γ＞δ thereexistsanextenderEsuchthat

E（CRT（E））=δandLTH（E）≥γ.

δisanextendiblecardinalif

⇨foreachγ＞δ thereexistsanextenderEsuchthat

CRT（E）=δ，E（δ）＞γ.andLTH（E）＞E（γ）.

weakextendermodelsandextendermodels

Foralargecardinalaxiom Φ：

Deflnition

AtrarsitiveclassNisaweakextendermodelforΦifΦiswitnessedtoholdinNbyextendersEofNsuchthat

E=F丨N

forsomeextenderF.

⇨lfΦholdsinVthenVisaweakextendermodelforΦ.

Deflnition

AtransitiveclassNisanextendermodelfor Φ ifforsomesequenceEofextenders：

1.N=L[E].

2.Nisaweakextendermodelfor Φ andthisiswitnessedbytheextendersonthesequtnceE.

ThelnnerModelprogram

ForaLargecardinalaxiom Φ andextendermodels.thesimplestgoalofthelnnerModelprogramistoanswerthequestion：

Question

Assumethat Φ holds.MustthereexistanextendermodelsuchthatN≠V？

Theorem（Martin-Steel）

Supposethereisaproperclassofwoodincardinals.ThenthereisanextendermodelNforaproperclassofwoodincardinalssuchthatN≠V.

Theorem（Martin-Steel）

SupposethereisaproperclassofsuptrstrongcardinalsandthelterationHypothesisholds.ThenthereisisanextendermodelNforaproperclassofsuperstrongcardinalssuchthatN≠V.

Beyondsuperstrong：theUniversalityTheorem

Thcorem（UniversaΓtyTheorcm）

SupposethatNisaweakextendermodelforδissupercompact.

supposethatFisanextendersuchthat：

⇨CRT（F）≥δandNisclosedunderF.

ThenF丨N∈N.

⇨ForanyextendtrF.LisclosedunderFbutF丨L∉L

⇨AnyweakextendermodelforδissupercompactinhenitsallLargecardinalsfromVwhichoccuraboveδ.

Conclution

TheextensionofthelnnerModelprogramtothelevelofonesupercompactcardinalmustyieldtheultimateinnermodel

⇨itmustyieldanultimateversionofL.

Gödel’stransitiveclassHOD

⇨ForeachsetΧ，TC（Χ）isthesmallesttransitivesetMwithΧ∈M.

Deflnition

Foreachordinalα.HODα+1isthesetofallsetsΧ⊆Vαsuchthat：

1.ΧisdefinableinVαfromordinalparameters.

2.lfＹ∈TC（Χ）thenＹisdtfinableinVαfromordinalparameters.

⇨ThedefinitionofHODα+1isamixtureofthedefinitionofLα+1andVα+1.

OefinlenM（Gödel）

HODistheclassofallsetsΧsuchthatΧ∈HODα+1forsomeα.

whatabutextendermodelsforsupercompactcardinals？

Deflnition

supposethatE=（Eα：α∈Ord）isasequence.

ThenEisweakly∑₂-definableifthereisaformua φ（x）suchthatforallβ∈ord.

⇨for all β＜η₁＜η₂＜η₃ .if

（E）ᵛᵉˢ丨β=（E）ᵛᵉˢ丨β

then（E）ᵛᵉ¹丨β=（E）ᵛᵉ²丨β=（E）ᵛᵉ³丨β.

where（E）ᵛ⁷={a∈Vα丨Vγ╞φ [a]}.

⇨Thesequtnce（HOD∩Vα：α∈Ord）isweakly∑₂-dtfnable.

Aseriousobstruction

⇨Assumethereisaproperclassofsupercompactcardinals

Byclassforcingonecanarrangethatthefollowinghold

1.V=HODandthereisaproperclassofsupercompactcardinals.

2.SupposeEisanextendersequencesuchthat

（a）L[E]isanextendermodelforδisasupercompact

（b）Eisweakly∑₂-deflnable.

ThenV⊆L[E].

Ramiflcations

RulesoutdeVelopingthelnnerModelprogramtothelevelofconstructingextendermodelsfor δ issupercompact.

⇨lnfactonecannotgobeyondtheMartin-Steelextendermodelsinanyessentialway.

Рartial-extendersandpartial-extendermodels

A partial-extender E of length η isobtainel from an elementary embedding.

j：N→M

whereN∩Р（η）=M∩Р（η）：

1.E has domain N∩P（η）：

2.E（A）=j（A）∩η.

Deflnition

AtransitiveclassNisapartial-extendermodelsequenceEofpartial-extenders：

1.N=L[E].

2.Nisaweakextendermodelfor Φ andthisiswitnessedbythe ∽₁：alextendersonthesequenceE.

Goodpertial-extendermodels

⇨Eveyweakextendermodelcanbere-organiＩedasapartial-extendermodel.therefore：

⇨ReguireagererakＩationoftheMostowskiCollapse.

Defmition

SupposeL[E]isapartial-extendermodel.ThenL[E]ispartial-extendermodelifforall

η＜α.if

X≺（Lα[E].E∩Lα[E]）

istheelementarysubstructuregivenbytheelementswhicharedeflnablewithparametensfromηthen.

Χ≅（Lᵦ[E].E∩Lᵦ[E]）

for some β.

⇨lfL[E]isagoodpartial-extendermodelthenthecontinuumHypothesisholdsinL[E].

Mitchell-Steelmodels

⇨Thebasicframewcrkforgoodpartial-extendersmodelsforlargecardinalsuptothelevelofsuperstrongcardinalsoriginatesintheconstuctionsofMitchellandSteel.

⇨ThereisanimportantvaiationduetoJensenwhichisequivalentbutyiekjsmodelswithstrongercondensationproperties.

Theeorem（MitchellSteeletal）

Assumethereisaproperclassofwoodincardinals.Thenthereisapartial-extendermodelL[E]foraproperclassofwoodincardinalssachthat

（1）Eisweakly∑₂-definable.

（2）L[E]isagoodpartial-extendermodel.

Theorem（Mitchell-Steeletal）

AssumetheltenationHypothesisandthatthereisaproperclassofsuperstrongcardinals.Thenthereisapartial-extendermodelL[E]foraproperclassofsuperstrongcardinalssuchthat

（1）Eisweakly∑₂-definable。

（2）L[E]isagoodpartial-extendermodel.

Conjecture

AssumethelterationHypothesisandthatthereisanextendibiecardinal.Thenthereisapartial-extendermodelL[E]forasupercompactcardinalsuchthat

（1）Eisweakly∑₂-definable.

（2）L[E]isagoodpartial-extendermodel.

Afirststep

Theorem

AssumethereisasupercompactcardinalandthatthelterationHypothesisholds.Thenthereisapartial-extendermodelL[E]suchthat

（1）Eisweakly∑₂-deflnable。

（2）L[E]isagoodpartial-extendermodel.

（3）L[E]isaweakextendermodelfortheexistenceofκsuchthatκisκᵒⁿ-supercompactforalln＜ω.

⇨Thetheoremshowsthattheobstructionscanbesuccessfullydealtwith.

⇨Theconstructionsseemtoindicatehowtohandlethegeneralcase.

TheGeneric-Multiverse

Definition

SupposethatMisacountabletransitivesetandthat

M╞ZFC.

Thegeneric-multiversegeneratedbyMisthesmallestsetVᴍofcountabletransitivesetssuchthatforallpairs（N₀，N₁）ofcountabletransitivesetsif

1.N₁isagenericextensionofN₀

2.eitherN₀∈VᴍorN₁∈VᴍthenbothN₀∈VᴍandN₁∈Vᴍ.

（meta）Definition

TheGeneric-Multiverseisthegeneric-multiversegeneratedbyV.

Mitchell-SteelmodelsandtheGeneric-Multiverse

Lemma（V=L）

VistheminimumuniverseoftheGeneric-Multiverse.

Thcorem

SupposeL[E]isan（iterable）Mitchell-Steelmodeland

L[E]╞ＴbctelsawoodincardinΓ.

ThenthereisaMitchell-SteelmodelL[F]⊂L[E]suchthatL[E]isageneΙcextensionofL[F].

ThesametheoremappliestotheextensionofMitchell-Steelmodelsbeyondsuperstrong.

lsUltimate-LageneralizedMitchell-Steelmodel？

AssumetheHerationHypothesisholdsinVandthatthereisaproperclassofmeasurablewoodincardinals.

⇨ltisnotknownifthereexistsaMitchell-SteelmodelL[E]foraproperclassofmeasurablswoodincardinalswithinwhichEisdefinablecevenfromparameters）.

⇨SupposeL[E]isaMitchell-Steelmodelwithinwhichthereexistsawoodincardinal.TheinductivefirstorderrequirementsonLα[E]arevtrycomplicated：

⇨thingsoelygetworseforthegentraliＩedMitchel-Steelmodels.

Twoquestions

1.lsthereasimplecandidatefortheaxiomⅤ=Ultimate-L”？

2.lsUltimate-Levenagoodpartial-extendermodel？

UniversallyBairesets

Definition（Feng-Magidoe-woodin）

AsetA⊆RisuniversallyBaireifforalltopologicalspacesΩandforallcontinuousfunctions：Ω→R.thepreimageofAbyπhasthepropertyofBaireinthespaceΩ.

⇨UniversallyBairesetsareanabstractgeneraliＩationoftheborelsets.

Theorcm

SupposethatthereisaproperclassofwoodincardinalsandthatA⊆RisuniversallyBaire.Theneveryset

B∈L（A，R）∩Р（R）

isuniversallyBaire.

HODᴸ（ᴬᴿ）andlargecardinalaxioms

Definition

SupposethatA⊆RisuniversallyBaire.

ThenΘᴸ（ᴬᴿ）isthesupremumoftheordinalsαsuchthatthereisasurjection.π：R→α.suchthatπ∈L（A，R）.

⇨Θᴸ（ᴬᴿ）isameasureofthecomplexityofA.

Relnrme

SupposethatthereisaproperclassofwoodincardinalsandthatAisuniversallyBaire.

ThenΘᴸ（ᴬᴿ）isawoodincardinalinHODᴸ（ᴬᴿ）.

HODᴸ（ᴬᴿ）andthelnnerModelprogram

Theorcm（Steel）

Supposethatthereisaproperclassofwoodincardinalsandletδ=Θᴸ（ᴿ）.

ThenHODᴸ（ᴿ）∩VδisaMitchell-Steelmodel.

Theorcm

Supposethatthereisaproperclassofwoodincardinals.

ThenHODᴸ（ᴿ）isnotaMitchell-Steelmodel.

Thereisanotherclassofsolutionstotheinnermodelproblemforlargecardinals.

⇨strategicpartial-extendermodels

⇨previouslyuhknown.

TheaxiomforV=Ultimate-L

（meta）Conjecture：TheaxiomforV=Ultimate-L

⇨Thereisastrongcardinalandaproperclassofwoodincardinals.

⇨Foreach∑₃-sentence φ，if φ holdsin V thenthereisauniversallyBairesetA⊆Rsuchthat

HODᴸ（ᴬᴿ）∩VΘ╞φ

where Θ=Θᴸ（ᴬᴿ）.

⇨Theaxiomsettles（moduloaxiomsofinfinity）allsentencesaboutp（R）（andmuchmore）whichhavebeenshowntobeindependentbyCohen’smethod.

Theorcm（V=Ultimate-L）

TheComtinuumHypothesisholds。

MoreconsequencesofV=Ultimate-L

Theorem（V=Ultimate-L）

Foreachcardinalκ.ifV[G]isaset-genericextensionofVthenthereexistsanelementaryembedding

π：（H（κ¹））ᵛ→N

u：kN+1（π，N）∈VandsuchthatN∈HODᵛ[ᶜ].

corollary（V=Ultimate-L）

V=HOD.

corollary（V=Ultimate-L）

Vistheminimumuniverse of the Generic-Multiverse.