However,wecanreplacetheconditiononofbeingasetorconsistentmultiplicity - PDF document

However,wecanreplacetheconditiononofbeingasetorconsistentmultiplicitybycertainotherconditionsCwhichhaveaprecisemeaning,yieldingtheprincipleIftheinitialsegmentof CsatisestheconditionC,thenithasaleaststrictupperboundS()2 C.WewillcallsuchconditionsCexistenceconditions.Unlikethecaseof ,noobviouscontradictionarisesingeneralfromtheassumptionthatwecantaketheleastupperboundS( C).Wecanmerelyconcludethat CdoesnotsatisfytheconditionC(since,otherwise,S( C)2 CandsoS( C)S( C));sothat,inproceedingtoconstructS( C)andlargernumbers,wemustreplaceCbysomeotherexistenceconditionwhichtheysatisfy.Inthisway,weareledtoahierarchyofmoreandmoreinclusiveexistenceconditions,eachofwhichcanreplacethecondition\isaset"inCantor'sdenitionandsoyieldsaninitialsegmentofthetransnitenumbers,butnoneofwhichyields`allthenumbers'.Informulatingexistenceconditions,wecanmakeuseofthefactthateachtransnitenumbercanbearmedwiththestructurehR();2;rani,whereR()isobtainedbystartingwiththenullsetanditeratingtheoperationD7!P(D)oftakingpowersetstimes.(R(S())=SfR()j2g.)2=2\(R()R()).Therankofasetistheleaststrictupperboundoftheranksofitselements,sothatR()isthesetofallsetsofrank.ranistherankfunction,assigningtoeachsetitsrank,andranisitsrestrictiontoR().ThusranisafunctionwithdomainR()andrangeofvaluesthesetofordinals.WeavoidtakingthetransnitenumberstobeexternaltosettheorybyidentifyingthemwiththevonNeumannordinals,i.e.withthetransitivesetsoftransitivesets.(Thesetxistransitiveiy2x�!yx.)thenisthesubsetofR()consistingofallordinalsinR()andsimplymeans2.IntermsofvonNeumannordinals,S()isjustandanexistenceconditionispreciselyaconditionunderwhichweadmittheexistenceofasaset.Thus,wewillbediscussingaparticularconceptionofsettheorywhichisexplicitlyformulatedin[Godel,*1933o]and[Godel,1947]butisimplicitin[Zermelo,1930].Inpart,itistheconceptionaccordingtowhichsetsaretheobjectsinanymemberofthehierarchyofdomainsobtainedfromthenulldomainbyiteratingthepowersetoperation.Thus,thisideapresupposesaswell-understoodthemoreprimitive,logicalnotionofset,theonewhichisformalizedinsecond-orderpredicatelogic,namelythatofformingfroma2 2Morse-KelleySetTheoryInthissection,wewilldiscusstheexistenceconditionswhichleadtothe(impredicative)second-ordertheoryofsetsrstintroducedin[Morse,1965].GivenanexistenceconditionC, C()willdenotetheleastinitialseg-mentofordinalscontaining+1whichdoesnotpossessC.Inwhatfollows,willbeaninitialsegmentofordinals.IfTisanextensionofT0,thenbyanordinalofTweshallmeanansuchthatR()isamodelofT.Inthefollowing,weshallconsiderextensionsTofT0obtainedbyaddingaxiomswhichareformalexpressionsofexistenceconditions.Thus,theleastordinalofT�is C(),whereCistheexistenceconditioninquestion.Theexistenceconditionsthatweshallconsiderinthispaperareallfor-mulatedintermsofaformula(X),withonlyXfree.Fornow,theformulaisoneinthelanguageofbasicsettheoryandXisasecond-ordervariable.Thecorrespondingconditionisthat(A)istrueinR()forsomeAR()and,forno2is(A\R())trueinR().Theformalexpressionthatthisconditionisanexistenceconditionistheaxiom8X[(X)�!9(X\R())](1)where(X)istheresultofrestrictingtherst-andsecond-orderboundvariablesin(X)toR()andR(+1),respectively.Axiomsofthisformhavebeencalledre ectionprinciples,becausetheyexpressthefactthatR()'spossessionofacertainpropertyisre ectedbyR()'spossessionofitforsome2.Sincepredicatelogicassumesthatallmodelsarenon-null,T0alreadyexpressesthattheconditiongivenby(X)=8y(y6=y)isanexistencecondition.Ifwetake(X)=9(X=fg)then(1)isequivalenttotheaxiomofSuccessor8[[fgisaset]:Withthisaxiom,wecanderivetheaxiomsofUnorderedPairsandPowerset,sinceifxandyareinR(),thentheclassesfx;ygandfzjzxgareincludedinR(+1)andsoaresets.Sincewehavearbitraryunorderedpairs,wecannowintroducetheusualcoding(x;y)=ffxg;fx;ygg6 Rank2,sincerancannowbedenedby2-inductionandthesetwoaxiomscanbededucedfromtheremainingaxioms[Zermelo,1930].Asusual,weidentifycardinalnumberswithinitialordinals,i.e.withordinalnumberswhicharenotinone-to-onecorrespondencewithsmallerordinals.InT2wecanprovethateverysetxhas(i.e.isinone-to-onecorrespondencewith)acardinalnumberjxj.Let2=jP()j.Acardinaliscalledastronglimitcardinalifalwaysimpliesthat2.So,everyordinalofT2mustbeastronglimitcardinal.Anordinal iscalledsingulariitisoftheformSF(),wheredom(F)= ;otherwise, iscalledregular.Notethat0and!arebothregularstronglimitcardinals.Clearlyaregularordinalmustbeacardinaland,inviewoftheaxiomofReplacement,theordinalsofT2mustberegular.Conversely,ifisaregularstronglimitcardinal&#x-281;0,thenitisanordinalofT2.First,notethatwecandenetheordinalfunction(i.e.ordinal-valuedfunctiononordinals)inR()by(0)=0(+1)=2()andforlimitordinals ( )=[ ()For,()=jR()jandsox2R()=SR()impliesjxj2R().ToseethatR()satisestheaxiomofRegularity,letFbeafunctionwithdom(F)2R()andrang(F)R().Letgbeaone-to-onecorre-spondencefromjdom(F)jtodom(F).ThenG=ranFgisafunc-tionfromjdom(F)jandso,sinceisregular,Srang(G)=.Butrang(F)R()andsorang(F)2R().So,aswasrstprovedin[Zermelo,1930],theordinalsofT2arepreciselytheregularstronglimitcar-dinals&#x-460;0.isanexampleofanormalfunction,i.e.anordinalfunctionfwhichisorder-preserving,i.e.impliesf()f(),andcontinuous,i.e.,when isalimitordinal,thenf( )=S f().Whenfisanorder-preservingordinalfunction,thenf()forall.For,f()impliesf(f())f();andsotherecanbenosuchthatf()becausetherecanbenoleastsuch.Whenfisanormalfunction,thenithasxed-pointsf()=anygiven.Eitherisalreadyaxedpointorelse,usingthe8 3Zermelo'sconceptionofsettheoryItisanembarassmentinsettheory,asitisoftenunderstood,thatanab-solutedistinctionmustbedrawnbetweentotalitiessuchasthetotalityof`allordinals'or`allcardinals'or`allsets'|thetotalitieswhichCantorcalled`inconsistentmanifolds'andwecallproperclasses|ontheonehand,andthosetotalitieswhichformsets.Forwhenwetaketheformertotalitiestobewell-denedobjects,thenwemustmakethisabsolutedistinction:thetwokindsofobjectsmustbetreatedquitedierently.Butwhy,ifthetotalityofallsetshasawell-denedextension,isitnotasetinanmoreextensivetotality?Theonlygroundsforthedistinctionisthenegativeonethat,ifwetreatproperclasseslikesets,weareledtoinconsistency|intheexamplescited,tothefamiliar`paradoxesofsettheory'.Thus,onthisunderstandingofsettheory,theseparadoxestrulyareparadoxical:thereisnoaccountingforthem.Wecan,asCantordid,onlyintroducethedistinctionbetweenthosetotalitieswhichare`consistent'andthosewhicharenot.(Ofcourse,relativetoanydomainR()thereareproperclasses,i.e.subclassesofthedomainwhicharenotcoextensivewithelementsofit;buttheseareproperclassesonlyinthisrelativesense:eachsubclassofR()willbecoextensivewithasetinR(+1)forexample.)Attheendof[1930],Zermelosketchedaquitedierentapproachtoun-derstandingsettheoryandtheso-calledparadoxes,onewhichprecludesthisembarassment.HebeginswiththeHilbertianthesisthatwemayspeakabouttheexistenceofthisorthatobjectinmathematicsonlywhenwehavespeciedaconsistentandcategoricaltheoryinwhichwecanspeakofsuchobjects.Moregenerally,wemayassertmathematicalpropositions(includingexistencepropositions)onlywithinsuchatheory.Thebackgroundofthisviewisthattherearenomathematicalphenomena(suchasKantianpureintuitions)onthebasisofwhichmathematicalpropositionsaremeaningful:howeveritisthatwecometoacceptabodyofpropositionsinsomemathe-maticalsphere,whetherderivedfromempiricalexperienceorbyanalogyorwhatever,therecanbenodenitivecriterionforexistenceortruthuntilwehavespeciedanaxiomaticbasisforthesepropositions.Therequirementofconsistencyaimsatmakingsurethatthedistinctionsbetweenexistenceandnon-existenceand,moregenerally,betweentruthandfalsetydonotcollapse.Therequirementofcategoricity,forHilbertandZermelo,probablyaimedatmakingsurethatquestionsoftruthhaddeterminateanswers.Sincethenon-trivialcategoricalaxiomatictheoriesareallsecond-orderandwenowknow10 thisprecise,wemaytaketherangeofthesecond-ordervariablestobetheisomorphicreplicaR(+1)ofR(+1)whichconsistsinreplacingeachelementxofthelatterbythepair(;x).Fory2R()and(;x)aclass,y2(;x)meanssimplythaty2x.Inthesameway,wecanequallywellconsiderquantiersoforderhigherthan2aswell.4FormulasofFiniteOrderInfact,inwhatfollows,wearegoingtowanttoconsidernotonlysecond-orderformulas,butformulasofarbitraryniteorder,wherethequantiersofordern+1(n�0)rangeoverclassesofobjectsofordern.Sowewillbeginwithsomedetailsthatwillapplygenerallyandnotjusttotheformulasofsecond-ordersettheory.Theorderofaformulaisthemaximumoftheordersoftheboundvari-ablesinit.Notethattheformulasoforder0arethosewhichcontainnoquantiers.TherelativizationoftheformulatoR()isextendedfromformulasofsecond-ordersettheorytothoseofarbitraryniteorderintheobviousway:allthequantiersofordern+1arerestrictedtoR(+n).Whentalkingaboutsetsandclasses,itismorenaturaltospeakoftheirtypesratherthantheirorders.Thetypeofanobjectofordern+1isn.LetAbeoftype1.FromthepointofviewofR(),AisA=A\R().For,whenAandBaretype1,thenAandBareequalrelativetoR()justincase8z2R()[z2A !z2B]iA=B.Itfollowsthat,fromthepointofviewofR(),BisB=fAjA2BgforBofarbitrarytype�1.So,forAoftype�0,(A)expressesthetruthof(A)inR().When= (A;:::;B)isaformulaofniteordercontainingnoparametersotherthanA;:::;B,eachofwhichisoftype�0,willdenote (A;:::;B).Inordertocodehigherorderandpossiblyheterogeneousrelationsasobjectsofnitetype,weintroducethefollowingoperations:WedeneA"oftypen+1forAofanytypen,andA#oftypen+1forAofanytypen+2,suchthatA"#=A.A"=fxjx=Ag:12 Denition1Letn0.Aformulaoforderniscalledan0formulaandan0formula.Anm+1isoneoftheform8Y (Y)where isanmformulaandYisavariableoftypen.Anm+1formulaisthenegationofanmformula.Ifisalimitordinal,thenR()isclosedunderalltheoperationsX",X#,XnandX=n.Wenotealso,forlateruseLemma1Foralimitordinal,eachofthefollowingholdwhenbothsidesoftheequationaredened:(A")=A"(A#)=A#An=(A)nA=n=(A)=n(A+B)=A+B:5Re ectiononaSecond-orderParameterWhenisrestrictedtonmandthetypeofXto1,thenwedenotetheaxiomschema(1)byRF(n;m):Denition2Anordinal issaidtobe-indescribableif(1)holdsinR( ).Ifisaclassofformulas,then iscalled-indescribableifitis-indescribableforeach2containingonlythefreevariableXoforder2.14 Denition3AbinarytreeisaclassToffunctionsfsuchthat,forsomeordinal,f:�!2andsuchthat,iff2Tandfhasdomain,thenfrestrictedtoanyordinallessthanisinT.AbinarytreeTispath-boundediforeveryfunctionF: �!2,thereisansuchthatFrestrictedtoisnotinT.TisboundedithereisansuchthatforallF: �!2,FrestrictedtoisnotinT.Thebinarytreepropertyisthateverypath-boundedbinarytreeisbounded.TheinstanceTisbounded�!9[Tisbounded]ofRF(1;1)impliesthebinarytreeproperty.So,sinceacardinalisweaklycompactjustincaseitisinaccessibleandR()hasthebinarytreeproperty,RF(1;2)impliestheexistenceofastationaryclassofweaklycompactcardi-nals.Wecanofcoursegoontoconstructastationaryclassofhyper-weaklycompactcardinals;andsoon.WenotedthatRF(1;0)notonlyimplies,butisequivalentinT0totheaxiomsofSuccessor,ReplacementandInnity.Likewise,RF(1;1)notonlyimplies,butisequivalentinT3tothebinarytreeproperty.Inotherwords,inT0,inaccessibilityisequivalentto10-indescribabilityandweakcompactnessisequivalentto11-indescribability([HanfandScott,1961]).Form;n�1,thereisasinglenmformula(X;y)suchthateverynmformulacontainingatmostthesecond-ordervariableXfreeisequivalentinT1to(X;e)forsomeniteordinale.Iteasilyfollowsthat,notjustform;n=1,butforalln;m�0,RF(n;m)isequivalenttothesinglenm+1formula n;m=8X8n2![(X;n)�!9(X;n)]inT1.So n;mexpressesnm-indescribability.[Kanamori,1994,Section6.9]SobyLemma2,RF(n;m+1)impliesthattheclassofnm-indescribablecardinalsisstationary.Inotherwords,theclassofnm-indescribablecardinalslessthanagivennm+1-indescribablecardinalisstationaryin.SotheprincipleRF(n;m)strictlyincreasesinstrengthasmincreases.16 Aswehavenoted,relativetoR(),thesecond-orderclassAisA=A\R()andsotherelativizationBofathird-orhigher-orderclassBisB=fAjA2Bg:Withthisdenition,(A)(=(A)),whereAisofarbitrarynitetype�0,expressesthat(A)istrueinR()and(1)hasmeaningforXofarbitrarynitetype.Butthereisaproblemwiththegeneralized(1),evenforXofthird-order:whenUistheclassofallbounded(or,alternatively,ofallunbounded)second-orderclasses,forexample,wehavethetruesentence(U)thateveryclassinUisbounded(orunbounded);whereasforevery,(U)isfalsesinceUisjustR(+1)and,inparticular,containsbothR()andthenullset.Therefore,wemustrestrict(1)tospecialclassesofformulas.Oneplausiblewaytothinkaboutthedierencebetweenre ecting(A)whenAissecond-orderandwhenitisofhigher-orderisthat,intheformercase,re ectionisassertingthat,if(A)holdsinthestructurehR();2;Ai,thenitholdsinthesubstructurehR();2;Aiforsome.(Wearenolongerconsideringtherankfunctionranaspartofthestructure,sinceitisdenableinT3.)But,whenAishigher-order,sayofthird-order,thisisnolongerso.NowweareconsideringthestructurehR();R(+1);2;AiandhR();R(+1);2;Ai.Butthelatterisnotasubstructureoftheformer,i.e.the`inclusionmap'ofthelatterstructureintotheformerisnolongersinglevalued:forsubclassesXandYofR(),X6=YdoesnotimplyX6=Y.Likewise,forX2R(+1),X62AdoesnotimplyX62A.Forthisreason,theformulasthatwecanexpecttobepreservedinpassingfromtheformerstructuretothelattermustbesuitablyrestrictedandinparticularshouldnotcontaintherelation62betweensecond-andthird-orderobjectsnortherelation6=betweensecond-orderobjects.Inthegeneralcaseofre ecting(A)whenAisofordern2,oneshouldnotadmitin(X)therelation62betweenkth-orderandk+1th-orderobjectsortherelation6=betweenkth-orderobjects,for1kn.However,wewillnotpursuethequestionofthemostgeneralgeneralformulationofthere ectionprinciplewithhigher-orderparametershere.Rather,weshallconsider(1)foraveryspecialclassofformulas(X).Let beaformula.A-instanceof istheresultofsubstitutingforeachfreevariableofordern(thenameof)anobjectofordernoverR(),foreachn.Denition518 �istheclassofrst-orderformulaspositiveintheextendedsense.�nistheclassofformulas8Y19Z1


8Yn9Zn where 2�=�0andtheYiaresecond-order.(TheZimaybeofanyorder.)Sothe�nformulasareallpositiveintheextendedsenseandso,byLemma3,re ectableforallregularcardinals.WewanttostudythecardinalsforwhichR()satises(1)forallformulasin�n.Moregenerally:Denition7Disn-re ectivei8X[�!92D]forall2�ncontainingjustthefreevariableXofarbitraryorder�1.Weshallinvestigatetheclassofsubsetsofcardinalswhicharen-re ective.Inparticular,weshallproveTheorem2IfDisn-re ective,thenD�!(Stationary)n+1.7n-StationarityDenition8Forn�0,afunctionKdenedon[]nsuchthat,for�1�:::�n;K(1;:::n)R(n),iscalledann-sequence(on).When,morestrictly,K(1;:::n)isalwaysn,thenKiscalledathinn-sequenceon.LetKbeann-sequenceon.AsubsetHofiscalledhomogeneousforKithereisaBR()suchthatK(1;:::n)=B\R(n)forall(1;:::n)2[H]nwith1�:::�n.20 Wedenethenotionofann-boxforD.Ann-boxisofordern+2.{A0-boxforDisaCLUBclassCsuchthatC\D=;.{Ann+1-boxforDisaclassToftriples(K;X;S)forsomexed1-sequenceK,calledthewitnessforT,suchthat,foreverysecond-orderclassX,thereisanSwith(K;X;S)2TandSisann-boxfor[K;X]\D.LetTbeofordern+2.Wedenethe�nformulan(T)byinductiononn:{0(T) !Tisanunboundedsubsetof{n+1(T) !8X9K9S[(K;X;S)2T^n(S)]TheorderoftheboundvariablesX,AandSare,respectively,2;2andn+2.Notethatn+1(T)canbeputintheformofa�n+1formulabymeansofcontractingoperations.Lemma5a)Disnotn-stationaryiithasann-box.b)IfTisann-boxforsomeD,thenn(T)istrue(inR()).c)IfTisann-boxforD,thenn(T)isfalseforall2D.d)IfDisn-re ective,thenitisn-stationary.Theproofofa)andb)isimmediatebyinductiononn.d)followsimmediatelyfroma)-c).Forc),letTbeann-boxforD,2D,andassumethatn(T)istrue.Wederiveacontradiction,byinductiononn.n=0.0(T)assertsthatT=T\isunboundedin.SinceTisCLUB,2T,contradictingD\T=;.n=m+1.LetKbethewitnessforT.ThenthereareK0andS0suchthathK0;K();S0i2Tandm(S0).K0=K,K()=AforsomeA,andS0=S,wherehK;A;Si2T.SoSisanm-boxforD\[K;A],m(S)istrue,and2D\[K;A]|acontradiction.SowehaveprovedhalfofLemma6Disn-re ectiveiitisn-stationary.22 Lemma8Ifn�0andDisn-stationarythenDisn-ineable.(Whenn=1,theconversealsoholds.)Letn=m+1withm�0andletKbeann-sequence(thinorotherwise.)ChooseAsothatD\[K0;A]ism-stationaryandhencem-ineable.SoK00(A)hasastationaryhomogeneousclassHD\[K0;A].SoHishomogeneousforK.9BoundsonHigher-OrderRe ectionAverygenerousupperboundfortheleastn-re ectivecardinalissimplyobtained:Theorem4LetbemeasurableandletUbeanormalultralteron.Then,foreverythin1-sequenceK,thereisasetAsuchthat[K;A]2U.SoeveryD2Uisn-stationary,i.e.n-re ective.Thesecondassertioneasilyfollowsfromtherst:EverysetinUisstationary,i.e.0-stationary,sinceUisanormalultralter.AssumethateverysetinUisn-stationaryandletKbeathin1-sequence.ThereisanAsuchthat[K;A]2U.SoforD2U,[K;A]\DisinUandsoisn-stationary.HenceDisn+1-stationary.Toprovetherstassertion,letKbeathin1-sequenceanddeneB=fj2Kg.Aisdenedby2A,B2ULetCbeBifthelattersetisinUandletitbe�B,otherwise.Inanycase,C2U.Then2[K;A]iK=A\i8[2B,B2Ui8[2C]i2
C,where,sinceUisanormalultraltercontainingeachC,
C=fj8[2C]gisinU.PeterKoellnerhasshownthatthereisacardinallessthantheErdoscardinal(!)(()beingtheleastsuchthat�!())whichisn-re ectiveforeachn.Asimpleextensionofthenotionofn-re ectivenessisobtainedasfollows:,let�mnbetheclassofformulas8Y19Z1


8Yn9Zn 24 Cantor,G.[1883].Uberunendliche,linearePunktmannigfaltigkeiten,5,MathematischeAnnalen21:545{586.In[Cantor,1932].Cantor,G.[1932].GesammelteAbhandlungenmathematischenundphilosophischenIn-halts,Berlin:Springer.ed.E.Zermelo.Ewald,W.[1996].FromKanttoHilbert:ASourceBookintheFoundationsofMathemat-ics.TwoVolumes,Oxford:OxfordUniversityPress.Godel,K.[*1933o].Thepresentsituationinthefoundationsofmathematics,[Godel,1995].Godel,K.[1947].WhatisCantor'scontinuumproblem?,AmericanMathematicalMonthly54:515{525.Reprintedin[Godel,1990][Godel,1964]isarevisedandexpandedversion.Godel,K.[1964].WhatisCantor'scontinuumproblem?,[BenacerrafandPutnam,1983],pp.258{273.Revisedandexpandedversionof[Godel,1947].Reprintedin[Godel,1990].Inthesecondeditionof[BenacerrafandPutnam,1983],thepagesare470{485.Godel,K.[1990].CollectedWorks,Vol.II,Oxford:OxfordUniversityPress.Godel,K.[1995].CollectedWorks,Vol.III,Oxford:OxfordUniversityPress.Hanf,W.andScott,D.[1961].Classifyinginaccessiblecardinals(abstract),NoticesoftheAmericanmathematicalSociety8:445.Kanamori,A.[1994].TheHigherInnite,Berlin:Springer-Verlag.Morse,A.[1965].ATheoryofSets,NewYork:AcademicPress.Tait,W.[1998a].Foundationsofsettheory,inH.DalesandG.Oliveri(eds),TruthinMathematics,pp.273{290.Zermelo,E.[1908].Neuerbeweisefurdiemoglichkeiteinerwohlorddnung,MathematischeAnnalen65:107{128.Zermelo,E.[1930].UberGrenzzahlenundMengenbereiche.NeueUntersuchungenuberdieGrundlagenderMengenlehre,FundamentaMathematicae16:29{47.Translatedin[Ewald,1996,1219-1233].26