Proceedings of the International Congress of Mathemati - PDF document

2IshallbrieyreviewtheconstructionofL,thebasictemplateforlargecardinalaxioms,anddescribetheprogramwhichseeksenlargementsofLcompatiblewithlargecardinalaxioms.FinallyIwillintroduce -logic,explainhowonthebasisofthe ConjectureamultiverseconceptionofVisuntenable,andreviewtherecentdevelopmentsontheprospectsforanultimateversionofL.IwillendbystatinganaxiomwhichIconjectureistheaxiomthatVisthisultimateLeventhoughthedenitionofthisultimateLisnotyetknown.Thisaccountfollowsathreadovernearly100yearsbutneitheritnorthelistofreferencesisintendedtobecomprehensive,see[10]and[18]forfarmoreelegantandthoroughaccounts.2.TheprojectivesetsandtwoquestionsofLuzinTheprojectivesetsarethosesetsofrealnumbersARwhichcanbegeneratedfromtheopensubsetsofRinnitelymanystepsoftakingcomplementsandimagesbycontinuousfunctions,f:R!R:SimilarlyonedenestheprojectivesetsARnoronecansimplyuseaborelbijection,:R!RnanddeneARntobeprojectiveifthepreimageofAbyisprojective.Fromperspectiveofsettheoreticcomplexity,projectivesetsarequitesimpleandonemightexpectthattheirbasicpropertiescanbeestablisheddirectlyonthebasisoftheaxiomsZFC.TheprojectivesetsweredenedbyLuzinwhoposedtwobasicquestions,[13]and[14].Adenitionisrequired.SupposethatARR.AfunctionfuniformizesAifforallx2R,ifthereexistsy2Rsuchthat(x;y)2Athen(x;f(x))2A.TheAxiomofChoiceimpliesthatforeverysetARRthereexistsafunctionwhichuniformizesA.ButifAisprojectivetheAxiomofChoiceseemstooerlittleinsightintowhetherthereisafunctionfwhichuniformizesthesetAandwhichisalsoprojective(inthesensethatthegraphoffisaprojectivesubsetofRR).ThetwoquestionsofLuzinarethefollowingbutIhaveexpandedthescopeofthesecondquestion–thisisthemeasurequestion–toincludethepropertyofBaire.1.SupposeARRisprojective.CanAbeuniformizedbyaprojectivefunction?2.SupposeARisprojective.IsALebesguemeasurableanddoesAhavethepropertyofBaire?Luzinconjecturedthat“wewillneverknowtheanswertothemeasurequestionfortheprojec-tivesets”.Luzin'sreasonforsuchaboldconjectureistheobviousfactthatLebesguemeasurabilityisnotpreservedundercontinuousimagessinceanysetARisthecontinuousimageofaLebesguenullset.TheexactmathematicalconstructionsofG¨odel[6],[7]andCohen[1],[2]whichwereusedtoshowthattheContinuumHypothesiscanneitherbeprovednorrefutedonthebasisoftheZFCaxioms,showthattheuniformizationquestionfortheprojectivesetscanalsoneitherbeprovedorrefutedfromtheaxiomsZFC.ThemeasurequestionismoresubtlebuttheconstructionofG¨odelandarenementofCohen'sconstructionduetoSolovay[25]showthesameistrueforthemeasurequestion.AcuriouswrinkleisthatforSolovay'sconstructionamodestlargecardinalhypothesisisnecessary.ThestructureoftheprojectivesetsisoffundamentalmathematicalinterestsinceitissimplythestructureofthestandardmodelofSecondOrderNumberTheory:hP(N);N;+;
;2i: 44.Thehierarchyoflargecardinals,determinacy,andtheanswerstoLuzin'squestionsSupposeAR.Thereisanassociatedinnitegameinvolvingtwoplayers.Theplayersalternatechoosingi2f0;1g.Afterinnitelymanymovesaninnitebinarysequencehi:i2Niisdened.PlayerIwinsthisrunofthegameif1i=1i=2i2AotherwisePlayerIIwins.Eitherplayercouldchoosetofollowastrategywhichissimplyafunction:SEQ!f0;1gwhereSEQisthesetofallnitebinarysequencesh1;:::;ni.Thestrategyisawinningstrategyforthatplayerifbyfollowing,thatplayerwinsnomatterhowtheotherplayermoves.Triviallyif[0;1]AtheneverystrategyisawinningstrategyforPlayerIandifA\[0;1]=;theneverystrategyisawinningstrategyforPlayerII.ThesetAisdeterminedifthereisawinningstrategyforoneoftheplayersinthegameassociatedtoA.GaleandStewart[5]provedthatifAisaclosedsetthenAisdeterminedandtheyaskedwhetherthisisalsotruewhenAisborel.MycielskiandSteinhaus[20]tookamuchbolderstepandformu-lated50yearsagotheaxiomAD.Denition1(Mycielski,Steinhaus).AxiomofDeterminacy(AD):EverysetARisdetermined.utTheaxiomADisrefutedbytheAxiomofChoiceandsoitisfalse.Butrestrictedversionshaveproventobequiteimportantandprovidetheanswers(yes)toLuzin'squestions,[18],[19],and[20].Denition2.ProjectiveDeterminacy(PD):EveryprojectivesetARisdetermined.utTheorem3.Assumeeveryprojectivesetisdetermined.(1)(Mycielski,Steinhaus)EveryprojectivesethasthepropertyofBaire.(2)(Mycielski,Swierczkowski)EveryprojectivesetisLebesguemeasurable.(3)(Moschovakis)EveryprojectivesetARRcanbeuniformizedbyaprojectivefunction.utTheaxiomPDyieldsarichstructuretheoryfortheprojectivesetsandmodulonotionsofinnitynoquestionabouttheprojectivesetsisknowntobeunsolvableonthebasisofZFC+PD.ButisPDevenconsistentandifconsistentisPDtrue?Theanswerstobothquestionsisyesbutthisinvolvesanotherfamilyofaxioms,thesearelargecardinalaxiomswhichareaxiomsofstronginnity.ThebasicmodernformoftheseaxiomsisasfollowswhereaclassMistransitiveifeachelementofMisasubsetofM(justasfortransitivesets).Acardinalisalargecardinalifthereexistsanelementaryembedding,j:V!MsuchthatMisatransitiveclassandistheleastcardinalsuchthatj(),.Thisisthecriticalpointofj,denotedCRT(j).ByrequiringmoresetstobelongtoM,possiblyinawaythatdependsonactionofjonthecardinals,oneobtainsahierarchyofnotions.Theobviousmaximumhere,takingM=V,isnotpossible(itisrefutedbytheAxiomofChoicebyatheoremofKunen).Insomecasesthelargecardinalaxiomofinterestholdingatisspeciedbytheexistenceofmanyelementaryembeddingsandpossiblyelementaryembeddingswithsmallercriticalpointsthatthecardinal.ThecarefulreadermightobjecttothereferencetoclassesbutinallinstancesofinterestonecanrequireMandjbedenableclasses(fromparametersbutinasimplemanner)andjneedonlybeelementarywithrespecttorathersimpleformulas.ThesituationisanalogoustothatinNumberTheorywhereonefrequentlyreferstoinnitecollectionssuchasthesetofprimenumbers.This 6ItisaconsequenceoftheaxiomsZFCthatforeachordinal,VexistsandmoreoverthatforeachsetXthereexistsanordinalsuchthatX2V.ThesetVistherankinitialsegmentofVdeterminedbytheordinal.ThiscalibrationofVsuggeststhattounderstandVoneshouldsimplyproceedbyinductionon,analyzingV.TheintegersappearinV!,therealsappearinV!+1,andallsetsofrealsappearinV!+2.TheprojectivesetsintheirincarnationasrelationsofSecondOrderNumberTheoryappearineectinV!+1sinceV!+1islogicallybi-interpretablewithhP(N);N;+;
;2i.Giventheamountofmathemati-caleortanddevelopmentwhichwasrequiredtounderstandV!+1justtothepointwhereonecouldidentifythecorrectaxiomsforV!+1,andnotingthatthisisaninnitesimalfragmentoftheUniverseofSets,theprospectsforunderstandingVtothissamedegree,orevenjustV!+2whichwouldrevealwhethertheContinuumHypothesisistrue,isadauntingtask.Itakeastrong,perhapsunreasonableposition,onthis.ThestatementthatProjectiveDetermi-nacyisconsistentisanewmathematicaltruth.Itpredictsfactsaboutourworld,forexamplethatinthenext1000years,sobyICM3010,therewillbenocontradictiondiscoveredfromProjec-tiveDeterminacybyanymeans.Ofcourseonecouldrespondwiththeobservationthatwitheachnewtheoremofmathematicscomessuchaprediction.ForexamplefromWiles'proofofFermat'sLastTheorem,onehasthesuperciallysimilarpredictionthatnocounterexampletoFLTwillbediscovered.Butthisprediction,whilecertainlyanewprediction,isreduciblebynitemeans(i.e.theproof)toapreviousprediction–namelythattheaxioms(whatevertheyare)necessaryforWiles'proofwillnotbediscoveredtobecontradictory.ThisisnotthecaseforthepredictionIhavemadeabove.Thatpredictionisagenuinelynewpredictionwhichisnotreduciblebynitemeanstoanypreviouslyheldprediction(sayfrombefore1960).Thisisthenatureoftheinvestigationoflargecardinalaxiomswhichsetsitapartfromothermathematicalenterprises.Butnowthereisadilemma.Theclaimthatalargecardinalaxiomisconsistent,suchastheclaimthattheexistenceofWoodincardinalsisconsistent,wouldseemultimatelytohavetobefoundedonaconceptionoftruthfortheUniverseofSetswhichincludestheexistenceoftheselargecardinals.ButifouraxiomsforthisUniverseofSetsfailtoresolveeventhemostbasicquestionsabouttheUniverseofSets,suchasthatoftheContinuumHypothesis,thenultimatelywhatsenseistheretotheclaimthatlargecardinalsexist?Thisisperhapstolerableonatemporarybasisduringaperiodofaxiomaticdiscoverybutitcertainlycannotbethepermanentstateofaairs.Thealternativeposition–thatconsistencyclaimscanneverbemeaningfullymade–issimplyarejectionoftheinnitealtogether.Andwhatifmypredictioniscorrectandaninstanceofanevolvingseriesofeverstrongersimilarlycorrectpredictions?Howwillthisskepticexplainthat?InanycaseanincrementalapproachmightbeprudentandsoIshallrestrictattentiontosen-tencesabouttheuniverseofsetsofaparticularform.Asentenceisarank-universalsentenceifforsomesentence ,assertsthatV forallordinals.Similarlyasentenceisarank-existentialsentenceifforsomesentence ,assertsthatthereexistsanordinalsuchthatV .Foranysentence ,theassertionthatV!+2 isbothrank-universalandrank-existentialandsotheContinuumHypothesisisexpressibleasbotharankuniversalsentenceandarankexistentialsentence.Thereisnothingparticularlyspecialabouttheordinal!hereorforthatmatterabout2either.Forexampleif0istheleastWoodincardinalthenforanysentence ,theassertionthatV0+! isbothrank-universalandrank-existential,etc. 8Theorem10(Martin,Steel,Woodin).AssumethereareinnitelymanyWoodincardinalswithameasurablecardinalabove.ThenL(R)AD.utTheproofofthetheoreminvolvescombining[15]withmethodsfrompreviousresultsandsomethinglikethemeasurablecardinalisnecessarybutonlyjustbarely.Thefollowingtheoremclariesthesituationbyprovidinganexactmatchtotheaxiom“L(R)AD”withinthehierarchyoflargecardinalsaxiomsandfromtheperspectiveoftheformalconsistencyoftheories.Theorem11(Woodin).Thefollowingtheoriesareequiconsistent.(1)ZFC+“L(R)AD”.(2)ZFC+“ThereareinnitelymanyWoodincardinals”.utTheaxiom,L(R)AD,givesacompleteanalysisofL(R)extendingtheanalysisthattheax-iom,allprojectivesetsaredetermined,providesfortheprojectivesets.ForexampleMoschovakis'stheoremonuniformizationgeneralizestoshowthatformanyordinals,assumingallsetsinL(R)\P(R)aredetermined,uniformizationholdsinL(R).Thisincludesallcountableandquiteabitmore.SubsequentworkofSteelhasexactlycharacterizedtheseordinals.OfcourseassumingV=L,uniformizationholdsinL(R)sinceinthiscaseL(R)=L.ButifuniformizationholdsinL(R)thentheAxiomofChoicemustholdinL(R)andsoinL(R),uni-formizationimpliesthatL(R)6AD.ThusthereismathematicaltensionbetweenuniformizationandtheregularitypropertiessuchasLebesguemeasurabilityandhavingthepropertyofBaire.Theorem12(Woodin).SupposethatuniformizationholdsinL(R)andthat=!1
forsomelimitordinal.Thenthefollowingareequivalent.(1)EverysetA2L(R)\P(R)isLebesguemeasurableandhasthepropertyofBaire.(2)EverysetA2L(R)\P(R)isdetermined.utTheproofofthetheoremusesratherelaboratemachinerytoconstructgivenA2L(R)\P(R)andassuming(1),acountabletransitivesetMsuchthatA\M2MandsuchthatinMthereareWoodincardinalssucienttoestablishthatA\Misdetermined.ThereisanadditionalrequirementthatM\AbecorrectaboutwhetherastrategyisawinningstrategyinthegameassociatedtoAandsothedeterminacyofA\MwithinMyieldsthedeterminacyofA.ByaremarkabletheoremofSteeltherestrictiononisnecessary,inparticularTheorem12doesnotholdwith=!1andthisfactarguesstronglythatthereisnoelementaryproofofthetheoremeveninspeciccasessuchas=!1
!1wherethetheoremdoeshold.Theprevioustheoremisnowoneofmanyanalogoustheoremswhichhavebeenproved,in-cludingrecentdramaticresultsofSargsyan[21].Thesetheoremscollectivelyconrmthattheun-derstandingofdeterminacyplaysacentralroleinmodernSetTheory.TheubiquityofProjectiveDeterminacyininnitarycombinatoricsisoftencitedasanindependentconrmationofitstruth.8.TheuniversallyBairesetsForanysetEthereisanassociatedenlargementofL,denotedL[E],whichisdenedasfollows.Foreachordinal,L[E]isrstdenedbyinductionon:1.L0[E]=;2.(Successorcase)L+1[E]=PDef(L[E][fE\L[E]g),3.(Limitcase)L[E]=[fL[E]jg. 10ThereisanordinalmeasureofcomplexityfortheuniversallyBairesets–thiscanbedenedanumberofwaysandIdeneasomewhatcoarsenotionusingadenitionwhichisjustforthisaccount.SupposeAandBaresubsetsofR.DeneAtobeborelreducibletoB,writtenAborelB,ifthereisaborelfunction:R!RsuchthateitherA=�1[B]orA=Rn�1[B].DeneAborelBifAborelBbutBborelA.FinallydeneAandBtobeborelbi-reducibleifbothAborelBandBborelA.TheboreldegreeofAistheequivalenceclassofallsetswhichareborelbi-reduciblewithA.TheboreldegreeofasetARisanalogoustotheTuringdegreeofasetAN.ThefollowinglemmaisanimmediatecorollaryoftheratherremarkableWadge'sLemmafromthetheoryofdeterminacytogetherwiththedeterminacyoftheuniversallyBairesets.Thesubse-quenttheoremissimilarlyacorollaryofafundamentaltheoremofMartinontheWadgeorder.Lemma18.AssumethereisaproperclassofWoodincardinals.SupposethatAandBareuniver-sallyBairesubsetsofR.(1)EitherAborelBorBborelA,(2)SupposeAborelB.Thenthereisaborelfunction:R!RsuchthatA=�1[B].utTheorem19.AssumethereisaproperclassofWoodincardinals.ThereisnosequencehAi:i!iofuniversallyBairesetssuchthatforalli!,Ai+1borelAi.utThus,assumingthereisaproperclassofWoodincardinals,theboreldegreesoftheuniversallyBairesetsarelinearlyorderedbyborelreducibilityandmoreoverthislinearorderisawellorder.IillustratetherelevanceofthistotheInnerModelProgram.SupposethatthereisproperclassofWoodincardinalsandconsidertheenlargementofL(R)givenbyL(R)[]asdiscussedabove.ThenthesetsinL(R)[]\P(R)arealluniversallyBaire.SupposethatA;B2L(R)[]\P(R)andthatforsomeordinal,A2L(R)[]butBL(R)[].ThenAborelB.Ingeneral,therankingoftheuniversallyBairesetsgivenbyborelreducibilitymustrenetheorderofgenerationofthesesetsinanypossibleenlargementofLadaptedtodeneanenlargementofL(R).ThepointhereisthatforanytransitivesetX,ifAborelBandB2PDef(X)thenA2PDef(X).Insummary,thesetsgeneratedbyanypossibleenlargementofL(subjectonlytoverygeneralconstraints)adaptedtodeneanenlargementofL(R)denesaninitialsegmentoftheuniversallyBairesetsrelativetotheorderofborelreducibility.Theextentofthatinitialsegmentisdeterminedbytheextentofthelargecardinalaxiomswhichholdintheinitialsegmentsofthatenlargement.9. -logicandthe ConjectureThefoundationalissuesoftruthinSetTheoryarisebecauseofCohen'smethodofforcingandIshallreferinthispapertoextensionsobtainedbythemethodofforcingasCohenextensions.CohenextensionsarethesourceoftheprofoundunsolvabilityofproblemssuchasthatoftheContinuumHypothesiswhichmakestheseproblemsseemsointractable.ThisisincontrasttoLuzin'squestionsabouttheprojectivesetswhichwehaveseenareresolvedbysimplyinvokingstrongnotionsofinnity.Perhapsthenthebestonecandoisamultiverseconceptionoftheuniverseofsets.ToillustratesupposethatMisacountable(transitive)modelofZFC(ofcourseonecannotprovesuchasetexistswithoutappealingtolargecardinalaxioms).LetV(M)bethesmallestcollectionofcountabletransitivemodelssuchthatif(M0;M1)isanypairofcountabletransitivemodelswithM1aCohenextensionofM0,ifeitherM02V(M)orM12V(M)thenbothmodelsareinV(M).V(M)isthegenericmultiversegeneratedbyM.TakingMtobeVitself,thisdenesthegeneric-multiverse.Ofcourseoneisinterestedinthecorrespondingnotionoftruth.Soasentenceistrueinthegeneric-multiversegeneratedbyVifistrueineachuniverseofthegeneric-multiversegeneratedbyV.Thiscanbemadeperfectlyprecise(withoutquantifyingoverclasses)andIshallgivearelativelysimplereformulationatleastforrank-universalsentences. 12LetT0bethesetofsentences suchthat“V!+2 ”isageneric-multiversetruthandletTbethesetofallrank-universalsentenceswhicharegeneric-multiversetruths.ClearlyT0isreducibletoT.Thesecondtheoremshowsthatassumingthe Conjecture(andthatthereisaproperclassofWoodincardinals)thenthesetwosetshavethesamecomputationalcomplexitybyshowingthatTisreducibletoT0(andtheproofgivestheexplicitreduction).Theorem26.SupposethatthereisaproperclassofWoodincardinalsandassumethe Conjectureholds.ThenTisrecursivelyreducibletoT0.utWhyisthisaproblem?Assumingthe Conjecture(andthatthereisaproperclassofWoodincardinals),thenthesecondtheoremshowsthatthewholehierarchyofrank-universaltruth–inthegeneric-multiverseconceptionoftruth–collapsestosimplythetruthsofV!+2.Moreoveraugmentedbyasecondconjecture,the Conjectureyieldsastrongformofthersttheorem–namelythatthissetofsentencesisactuallydenableinV!+2.ThiscollapseiscompletelycountertothefundamentalprinciplesconcerninginnityonwhichSetTheoryisfounded.MoreoversinceV!+2isinessencejustthestandardstructureforThirdOrderNumberTheory,thiscollapseshowsthatthegeneric-multiverseconceptionoftruth(forrank-universalsentences)issimplyaversionofthirdorderformalism.Ifthe Conjectureistruethenthegeneric-multiverseconceptionoftruthisuntenable.Noviablealternativemultiverseconceptionoftruthisknownthatsurvivesthechallengeposedbythe ConjectureandthisseemstoargueforamultiverseofoneuniversewhichleadsusbacktosearchingforgeneralizationsoftheaxiomV=LandtheInnerModelProgram.Perhapsthisallissimplyevidencethatthe Conjectureisfalse.The ConjectureisinvariantacrossthegenericmultiversegeneratedbyVandsoareasonableconjectureisthatifthe Conjec-turecanfailthenitmustberefutedbysomelargecardinalaxiom.Butthe ConjectureholdsinalltheenlargementsofLproducedbytheInnerModelProgramandsototheextentthisprogramsucceedsinanalyzinglargecardinalaxioms,nolargecardinalaxiomcanrefutethe Conjecture.10.Extendermodels,supercompactcardinals,andHODItisScott'stheoremthatifV=LthentherearenomeasurablecardinalswhichnecessitatesthesearchforgeneralizationsofthedenitionofLinwhichlargecardinalaxiomscanhold.ThisisreinforcedbyG¨odel'stheoremthatshowsthatifV=Lthenonecannothavethetruetheoryoftheprojectivesets:projectivedeterminacymustfailandmoreovertherearepathologicalprojectivesets.ButhowshouldoneenlargeL?TheenlargementsareoftheformL[˜E]forsomeset(orclass)˜E.Theproblemistoidentifysets˜EforwhichL[˜E]isageneralizationofLfromtheperspectiveofdenability.Sincetheissueislargecardinalaxioms,thesesetsshouldsomehowbederivedfromlargecardinals.Therelevantkeynotionisthatofanextender,themodernformulationisduetoJensenandbasedonanearlierformulationduetoMitchell.ThereareprecursorsduetoPowell(inamodeltheoreticcontext)andtoJensen,see[10]formoredetails.TosimplifythisexpositionIdeviatefromthestandarddenitionofanextenderanduseadenitionwhichisinsomewaysmorerestricted,inotherwaysmoregeneral,butinallwayslesstechnicaltostate.Denition27.Afunction,E:P( )!P( )where isanordinal,isanextenderoflength ifthereexistsanelementaryembeddingj:V!Msuchthat1.CRT(j) andV +!M,2.forallA ,E(A)=j(A)\ .utIfEisanextenderitisconvenienttodeneCRT(E)=CRT(j)wherej:V!MwitnessesthatEisanextender.Thisiswell-denedandCRT(E)iseasilycomputedfromEitself. 14SupposeNisatransitiveclass.ThenE(N:V)denotestheclassofallF\NsuchthatFisanextender,F\N2N,andsuchthatF\NisanextenderinN.Theorem31.SupposeL[˜E]isanextendermodelsuchthatissupercompactinL[˜E]andthisiswitnessedinL[˜E]byE(L[˜E]:V).SupposeFisanextenderofstronglyinaccessiblelengthsuchthatL[˜E]isclosedunderFandsuchthatCRT(F).ThenF\L[˜E]2L[˜E].utInotethatLisclosedunderallextendersandmoregenerallyifNisanytransitiveclasswhichcontainstheordinalsandifsucientlargecardinalsexistinVthennecessarilyNisclosedunderFforarichclassofextendersFofstronglyinaccessiblelength.ThereforetherequirementsthatL[˜E]beclosedunderFandFhavestronglyinaccessiblelengtharenotaveryrestrictiverequirements.TodatethebasicmethodologyofextendermodelsissuchthatifFisanextenderinL[˜E],thenFisgivenbyanintialsegmentofanextenderonthesequence˜E,[22].ThishasalwaysseemedanessentialfeatureofthedetailedanalysisofL[˜E]anditiscloselyrelatedtowhyeachnewconstructionofanextendermodelhascomewithanassociatedgeneralizationofScott'sTheorem(thattherearenomeasurablecardinalsinL).Theprevioustheoremeasilyyieldsacompletereversalofthisatthelevel1ofonesupercompactcardinal.Forexampleif�isanextendiblecardinalthenmustbeasupercompactcardinalinL[˜E]andthisgeneralizestoessentiallyalllargecardinalnotions.Thenexttheorem–whichisalsocloselyrelatedtotheprevioustheorem–givesyetanothermea-sureofthetranscendenceofextendermodelsatthelevelofonesupercompactcardinal.Theseex-tendermodels,atleastinabackgrounduniverseofsucientlargecardinalstrength,mustcorrectlycomputetheproofrelationfor -logic.Theorem32.Supposethereisaproperclassofextendiblecardinals.SupposeL[˜E]isanextendermodelsuchthatissupercompactinL[˜E]andthisiswitnessedinL[˜E]byE(L[˜E]:V).Thenforallrank-universalsentencesthefollowingareequivalent.(1)` .(2)L[˜E]“` ”.utIrequireanotherdenitionduetoG¨odel.ThisdenitionisoftheclassHODofallhereditar-ilyordinaldenablesetsandhereIgiveanequivalentreformulationofG¨odel'sdenitionwhichhighlightsitassomesortofmergeofthedenitionsofthecumulativehierarchyandthatofL.Denition33.HODistheclassofallsetsXsuchthatthereexist2OrdandAsuchthatAisdenableinVwithoutparametersandsuchthatX2L[A].utIfV=LthenHOD=LbutifforexampleL(R)ADthenHOD,L.TheclassHODisnotingeneralcanonical,forexamplebypassingtoaCohenextensionofVonecanarrangethatanydesignatedsetofVbeanelementofHODasdenedintheextension.ThereisaremarkabletheoremofVopenkawhichconnectsHODandCohen'smethodofforcing,see[10].ThistheoremillustrateswhyCohen'smethodissocentralinSetTheoryandforreasonsotherthansimplyestablishingindependenceresults.IfGOrdthenHODGissimplyHODdenedallowingGasaparameter(soG2HODG).Theorem34(Vopenka).ForeachsetGOrd,ifGHODthenHODGisaCohenextensionofHOD.ut 1Therearesomeveryrecentdevelopmentswhichshowthatthecriticalstageisactuallybelowthelevelofonesupercompactcardinalandwhichplausiblyidentifyexactlythecriticalstage. 16BycombiningthethreenotionsofuniversallyBairesets,relativeconstructibility,andHOD,IcanformulatewhatIconjecturewillbetheaxiomthatVisultimate-L.IdothisinthecontextthatthereisasupercompactcardinalandaproperclassofWoodincardinalsthoughthelatterisultimatelyirrelevant.Theformulationofthisaxiominvolvesonelastdenition.SupposethatARisuniversallyBaire.ThenL(A;R)isthesupremumoftheordinalssuchthatthereisasurjection,:R!,suchthat2L(A;R).TheconnectionbetweenthedeterminacyoftheprojectivesetsandWoodincardinalsgeneralizestoastructuralconnectionillustratedbythefollowingtheoremwhereHODL(A;R)denotesHODasdenedwithinL(A;R).Theorem35(Woodin).SupposethatthereisaproperclassofWoodincardinalsandthatAisuniversallyBaire.ThenL(A;R)isaWoodincardinalinHODL(A;R).utTheconnectionrunsmuchdeeperasindicatedbythefollowingtheoremofSteelandInowrefertoextendermodelsintheirtrueform(andmoreoverexpandedtoincludeelementarysubstructures)andnotthesimpleapproximationthatIhavedenedpreviously.TheMitchell-SteelextendermodelsarethesolutionsoftheInnerModelProgramatthelevelofWoodincardinalswhichIalludedtointhediscussionafterthetwotestquestions.Theorem36(Steel).SupposethatthereisaproperclassofWoodincardinals.Let=L(R).ThenHODL(R)\VisaMitchell-Steelextendermodel.utTheorem37(Woodin).SupposethatthereisaproperclassofWoodincardinals.ThenHODL(R)isnotaMitchell-Steelextendermodel.utButthenwhatisHODL(R)?Itbelongstoadierent,previouslyunknown,classofextendermodels,thesearethestrategicextendermodels.ForasignicantinitialsegmentoftheuniversallyBairesets,HODL(A;R)hasbeenveriedtobeastrategicextendermodelandthereisverystrongevidencethatthiswillbetrueforalluniversallyBairesets.Untilrecentlyitwasnotclearatallwhatlargecardinalaxiomscouldholdinthesemodels.ButonthebasisofthefoundationalquestionswhichIhavebeendiscussingcombinedwithassociatedmathematicaldevelopments,[29],thereiscompellingevidence(tome)thattheseinnermodelsHODL(A;R)\Vwhere=L(A;R)cannotbelimitinginanyway:theonlyissue(assumingthesearestrategicextendermodels)iswhetherstrate-gicextendermodelscanexistatthelevelofonesupercompactcardinalforthenjustasisthecaseforextendermodels,theyaretranscendentforlargecardinals.Thereisabsolutelycompellingevidencethatstrategicextendermodelsexistwhicharetranscendentfor -logicinthesenseofTheorem32andfromthisperspectiveitseemsperhapsobviousthattheremustexiststrategicextendermodelsatthelevelofonesupercompactcardinalaswell.TheunderlyingpointhereisthatthefamilyofinnermodelsHODL(A;R)\Vwhere=L(A;R)andAisuniversallyBairearecollectivelytranscendentfor -logic.Thereforeiftheseinnermodelsarestrategicextendermodelsthenstrategicextendermodelsaretranscendentfor -logicaswell.Extendingthetheoryofextendermodelstothelevelofonesupercompactcardinalseemsdi-cultenough,whyshouldtherebeanyoptimismthatthiscanbedoneforstrategicextendermodelsthetheoryofwhichhasgenerallybeenmoredicult.Thereisakeyandfundamentaldierence.ThestructureandtheoryofstrategicextendermodelswillbefullyrevealedbytheinnermodelsHODL(A;R)whereAisuniversallyBaire.Sothemathematicalproblemisnotoneofndingthecorrectdenitiontosatisfyapossiblyvaguegoal,butratheroftheanalysisofstructureswecanalreadydene.Moreoverwehavearichframeworkprovidedbydeterminacyinwhichtoundertakethatanalysis.IshouldemphasizethatpriortotheproofofTheorem37,itwasnotknownifstrategicextendermodelscouldexistinanyreasonableform. 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