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.
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