Ultimate L (第2/2页)

fact，aredherring.

    Thekeynotionwewillbestudyingisthefollowing:

    Definition.N\subseteqVisaweakextendermodelfor`δissupercompact’iffforallλ>δthereisanormalfineUonP_δ(λ)suchthat:

    P_δ(λ)\capN\inU，and

    U\capN\inN.

    ThisdefinitioncouplesthesupercompactnessofδinNdirectlywithitssupercompactnessinV.Inthemanuscript，thatNisaweakextendermodelfor`δissupercompact’isdenotedbyo^N_{mlong}(δ)=\infty.Notethatthisisaweaknotionindeed，inthatwearenotrequiringthatN=L[\vecE]forsome(long)sequence\vecEofextenders.TheideaistostudybasicpropertiesofNthatfollowfromthisnotion，inthehopesofbetterunderstandinghowsuchanL[\vecE]modelcanactuallybeconstructed.

    Forexample，finenessofUalreadyimpliesthatNsatisfiesaversionofcovering:IfA\subseteqλand|A|<δ，thenthereisaB\inP_{δ}(λ)\capNwithA\subseteqB.Butinfactasignificantlystrongerversionofcoveringholds.Toproveit，wefirstneedtorecallaniceresultduetoSolovay，whousedittoshowthat{\sfSCH}holdsaboveasupercompact.

    Solovay’sLemma.Letλ>δberegular.ThenthereisasetXwiththepropertythatthefunctionf:a\mapsto\sup(a)isinjectiveonXand，foranynormalfinemeasureUonP_δ(λ)，X\inU.

    ItfollowsfromSolovay’slemmathatanysuchUisequivalenttoameasureonordinals.

    Proof.Let\vecS=\leftbeapartitionofS^λ_\omegaintostationarysets.

    (WecouldjustaswelluseS^λ_{\le\gamma}foranyfixed\gamma<δ.Recallthat

    S^λ_{\le\gamma}=\{\alpha<λ\mid{mcf}(\alpha)\le\gamma\}

    andsimilarlyforS^λ_\gamma=S^λ_{=\gamma}andS^λ_{<\gamma}.)

    Itisawell-knownresultofSolovaythatsuchpartitionsexist.

    Hughactuallygaveaquicksketchofacrazyproofofthisfact:Otherwise，attemptingtoproducesuchapartitionoughttofail，andwecanthereforeobtainaneasilydefinableλ-completeultrafilter{\mathcalV}onλ.Thedefinabilityinfactensuresthat{\mathcalV}\inV^λ/{\mathcalV}，contradiction.Wewillencounterasimilardefinablesplittingargumentinthethirdlecture.

    LetXconsistofthosea\inP_δ(λ)suchthat，letting\beta=\sup(a)，wehave{mcf}(\beta)>\omega，and

    a=\{\alpha<\beta\midS_\alpha\cap\betaisstationaryin\beta\}.

    Thenfis1-1onXsince，bydefinition，anya\inXcanbereconstructedfrom\vecSand\sup(a).AllthatneedsarguingisthatX\inUforanynormalfinemeasureUonP_δ(λ).(ThisshowsthattodefineU-measure1sets，weonlyneedapartition\vecSofS^λ_\omegaintostationarysets.)

    Letj:VoMbetheultrapowerembeddinggeneratedbyU，so

    U=\{A\inP_δ(λ)\midj‘λ\inj(A)\}.

    Weneedtoverifythatj‘λ\inj(X).First，notethatj‘λ\inM.Lettingau=\sup(j‘λ)，wethenhavethatM\models{mcf}(au)=λ.Since

    M\modelsj(λ)\geauisregular，

    itfollowsthatau=j(\left).InM，theT_\betapartitionS^{j(λ)}_\omegaintostationarysets.Let

    A=\{\beta