Ultimate L

    Luminy–HughWoodin:UltimateL(I)

    TheXIInternationalWorkshoponSetTheorytookplaceOctober4-8，2010.ItwashostedbytheCIRM，inLuminy，France.IamverygladIwasinvited，sinceitwasagreatexperience:TheWorkshophasatraditionofexcellence，andthistimewasnoexception，withseveralverynicetalks.Ihadthechancetogiveatalk(availablehere)andtointeractwiththeotherparticipants.Thereweretwomini-courses，onebyBenMillerandonebyHughWoodin.Benhasmadetheslidesofhisseriesavailableathiswebsite.

    WhatfollowsaremynotesonHugh’stalks.Needlesstosay，anymistakesaremine.Hugh’stalkstookplaceonOctober6，7，and8.Thoughthetitleofhismini-coursewas“Longextenders，iterationhypotheses，andultimateL”，Ithinkthat“UltimateL”reflectsmostcloselythecontent.ThetalkswerebasedonatinyportionofamanuscriptHughhasbeenwritingduringthelastfewyears，originallytitled“Suitableextendersequences”andmorerecently，“Suitableextendermodels”which，unfortunately，isnotcurrentlypubliclyavailable.

    ThegeneralthemeisthatappropriateextendermodelsforsupercompactnessshouldprovablybeanultimateversionoftheconstructibleuniverseL.Theresultsdiscussedduringthetalksaimatsupportingthisidea.

    UltimateL

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    I

    Letδbesupercompact.ThebasicproblemthatconcernsusiswhetherthereisanL-likeinnermodelN\subseteqVwithδsupercompactinN.

    Ofcourse，theshapeoftheanswerdependsonwhatwemeanby“L-like”.Thereareseveralpossiblewaysofmakingthisnontrivial.Here，weonlyadopttheverygeneralrequirementthatthesupercompactnessofδinNshould“directlytraceback”toitssupercompactnessinV.

    Recall:

    WeuseP_δ(X)todenotetheset\{a\subseteqX\mid|a|<δ\}.

    Anultrafilter(ormeasure)UonP_δ(λ)isfineiffforall\alpha<λwehave\{a\inP_δ(λ)\mid\alpha\ina\}\inU.

    TheultrafilterUisnormaliffitisδ-completeandforallF:P_δ(λ)oλ，ifFisregressiveU-ae(i.e.，if\{a\midF(a)\ina\}\inU)thenFisconstantU-ae，i.e.，thereisan\alpha<λsuchthat\{a\midF(a)=\alpha\}\inU.

    δissupercompactiffforallλthereisanormalfinemeasureUonP_δ(λ).

    Itisastandardresultthatδissupercompactiffforallλthereisanelementaryembeddingj:VoMwith{mcp}(j)=δ，j(δ)>λ，andj‘λ\inM(or，equivalently，{}^λM\subseteqM).

    Infact，givensuchanembeddingj，wecandefineanormalfineUonP_δ(λ)by

    A\inUiffj‘λ\inj(A).

    Conversely，givenanormalfineultrafilterUonP_δ(λ)，theultrapowerembeddinggeneratedbyUisanexampleofsuchanembeddingj.Moreover，ifU_jistheultrafilteronP_δ(λ)derivedfromjasexplainedabove，thenU_j=U.

    AnothercharacterizationofsupercompactnesswasfoundbyMagidor，anditwillplayakeyroleintheselectinthisreformulation，ratherthanthecriticalpoint，δappearsastheimageofthecriticalpointsoftheembeddingsunderconsideration.Thisversionseemsideallydesignedtobeusedasaguideintheconstructionofextendermodelsforsupercompactness，althoughrecentresultssuggestthatthisis，infact，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