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=\left<s_\alpha\mid\alpha<λight>beapartitionofs^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(λ).let\left<t_\beta\mid\beta<j(λ)ight>=j(\left<s_\alpha\mid\alpha<λight>).inm，thet_\betapartitions^{j(λ)}_\omegaintostationarysets.let

    a=\{\beta