Ultimate L (第1/2页)

    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，in