(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.UltimateLAdvertisementsREPORTTHISADILetδbesupercompact.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)byA\\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，andU\\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=\\left<S_\\alpha\\mid\\alpha<λight>beapartitionofS^λ_\\omegaintostationarysets.(WecouldjustaswelluseS^λ_{\\le\\gamma}foranyfixed\\gamma<δ.RecallthatS^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，anda=\\{\\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，soU=\\{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.SinceM\\modelsj(λ)\\geauisregular，itfollowsthatau<j(λ).Let\\left<T_\\beta\\mid\\beta<j(λ)ight>=j(\\left<S_\\alpha\\mid\\alpha<λight>).InM，theT_\\betapartitionS^{j(λ)}_\\omegaintostationarysets.LetA=\\{\\beta , )