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
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).
Itisastandardresultthatδissupercompactiffforall
更多内容加载中...请稍候...
若您看到此段落，代表章节内容加载失败，请关闭浏览器的阅读模式、畅读模式、小说模式，以及关闭广告屏蔽功能，或复制网址到其他浏览器阅读！