第三篇章终极集合论宇宙（V=UltimateL） (9-7)

　　（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

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