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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 Advertisements REPORTTHISAD 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
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