thefullseparationschemeInrecentyetunpublishedworkbySAwodeyCButzASimpsonandTStreicherABSSithasbeenshownthatsettheorywithboundedseparationieseparationrestrictedtoboundedformulasbutwithrepl ID: 206220
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Thelogicoftoposes(withNNO)isinherentlyintuitionisticandinHAHtheaxiomofchoiceimpliesclassicallogic.Therefore,wehavetogiveup(1)and(2)abovewhenconsideringGrothendieckandrealizabilitytoposesasmodelsofsomekindofsettheory.Butwhatabout(3),theaxiomofreplacement?Firstofallnoticethattherearemodelsofsettheorywithoutreplacementbutsatisfyingclassicallogicandchoice,namelyV!+!.2Ontheotherhandalotoftoposes,inparticularGrothendiecktoposesandrealizabilitytoposes,domodeltheaxiomofreplacementwhereasinmostcasestheyrefuteclassicallogicandtheaxiomofchoice.Moreprecisely,theabovementionedtoposesmodelIZF,i.e.ZFwithintuitionisticlogicandaxiomofregularityreformulatedas2-induction.3ThoughalargeclassoftoposesvalidatesIZFonestillmaycomplainthattheformulationofIZFsu ersfrom\epsilonitis",i.e.thatit\implements"in-formalmathematicsviathe2-relationratherthanaxiomatizingmathematicalpracticeintermsofitsbasicnotions.Soonemayaskwhatisthemathematicalrelevanceoftheset-theoreticreplacementaxiom?Maybeaset-theoristwouldanswer\forconstructingordinalsgreaterthan!+!"which,however,mayseemabitdisappointingbecausemostmathematicscanbeformulatedwithoutref-erencetotransniteordinals.4Actually,whataxiomofreplacementismainlyneededforinmathematicalpracticeistodenefamiliesofsetsindexedbysomesetIcarryingsomeinductivestructureas,typically,thesetNofnaturalnum-bers.Forexample,mostmathematicianswouldnothesitatetoconstructthesequence(Pn(N))n2Nby(primitive)recursionoverN.AlreadyinZC,however,thisisimpossiblebecausefhn;Pn(N)ijn2Ng62V!+!.Usually,inZFCthesequence(Pn(N))n2Nisconstructedbyapplyingtheaxiomofreplacementtoanappropriatelydenedclassfunctionfromthesetofnaturalnumberstotheclassofallsets.However,inasensethatdoesnotproperlyre ectthemathe-matician'sintuitionwhothinksof(Pn(N))n2NasafunctionffromNtosetsdenedrecursivelyasf(0)=Nandf(n+1)=P(f(n)).Thereis,however,a\little"problem,namelythatthecollectionofallsetsdoesnotformasetbutaproperclass.Notice,however,thataposterioritheimageoffdoesformaset thefullseparationscheme.Inrecent,yetunpublishedworkbyS.Awodey,C.Butz,A.SimpsonandT.Streicher[ABSS]ithasbeenshownthatsettheorywithboundedseparation,i.e.separationrestrictedtoboundedformulas,butwithreplacement(andevenstrongcollection)isequiconsistenttoHAHaslongastheunderlyinglogicisintuitionistic.OtherwisetheclassicalPrincipleofExcludedMiddleallowsonetoderivefullseparationfromreplacement.Inthecontextofthispaperwhenwesayreplacementwemeanthepowerofreplacementtogetherwithfullseparation(althoughthelatterdoesnotmakesensefromatype-theoreticpointofview!).2ButnoticethatV!+!validatesZC,i.e.ZFCwithoutreplacement,which,however,isstillstrongerthanHAHasalreadyZprovestheconsistencyofHAH.3Axiomofregularityand2-inductionareequivalentonlyclassicallyasinIZFtheprincipleofexcludedmiddlefollowsfromtheaxiomofregularityjustasinHAHtheprincipleofexcludedmiddlefollowsfromtheleastnumberprinciple.4Therearenotableexceptionstypicallyintheareaofdescriptivesettheoryase.g.BoreldeterminacywhichisprovableinZF(asshownbyD.A.Martin)butnotinZ(asshownbyH.Friedman).EvenIZFdoesnotdecideBoreldeterminacyasitholdsinSetbutnotinHyland'se ectivetoposE .2 2UniversesinToposesWenowdeneanotionofuniverseinan(elementary)toposthatisstrongerthantheset-theoreticaxiomofreplacementbutadaptedtothe\spirit"oftypetheoryandthusfreedfrom\epsilonitis".Wedonotclaimanyoriginalityforthesubsequentnotionasitisinspiredbythecategoricalsemantics(seee.g.[Str1])foranimpredicativeversionofMartin-Lof'suniversesascanbefoundintheExtendedCalculusofConstructions(see[Luo]).CategoricalsemanticsofuniverseswasanticipatedbyJeanBenabou'sin- uentialpaper[Ben](from1971!)introducing(amongotherimportantthings)anotionoftoposinternaltoatopos.7Denition2.1AuniverseinatoposEisgivenbyaclassSofmorphismsinEsatisfyingthefollowingconditions(1)SisstableunderpullbacksalongmorphismsinE,i.e.foreverypullbackB -AJb? f-Ia? inEitholdsthatb2Swhenevera2S.(2)ScontainsallmonosofE.(3)Sisclosedundercomposition,i.e.iff:A!Iandg:B!AareinSthenfg=fg2S.(4)Sisclosedunderdependentproducts,i.e.iff:A!Iandg:B!AareinSthenfg2S(wherefisrightadjointtof:E=I!E=A).(5)InSthereisageneric8morphism,i.e.amorphismEl:E!UinSsuchthatforeverya:A!IinSwehaveA -EIa? f-UEl? forsomemorphismf:I!UinE,i.e.a=fEl.AuniverseSiscalledimpredicativei !1isinS.IfEhasanaturalnumbersobjectNthenwesaythatScontainsNi N!1isinS. 7Alas,laterworkoncategoricalsemanticsoftypetheoriesusuallydoesnotreferto[Ben]butratherimplicitlytosome\folklore"datingbacktotheearly70ieswhen[Ben]waswritten.8Noticethatwedonotrequireuniquenessoff,i.e.Elisnota\classifying"butonlya\generic"familyforS.Someauthorsusealsotheword\weaklyclassifying"insteadof\generic".4 Condition(3)saysthat(ineachbre)forafamilyof\small"setsindexedovera\small"setitssum(disjointunion)issmall,too.From(1)and(3)itfollowsthatfora:A!Iandb:B!IthebrewiseproductaIb:AIB!IisinS,too.If,moreover,condition(2)isassumedthenfora:A!IinSandarbitrarysubobjectsm:A0AthecompositeamisinS,too.Thus,underassumptionof(1),(2)and(3)foreveryobjectIinEthefullsubcategoryS=IofE=I(onmapsofSwithcodomainI)isnitelycompleteandinheritsitsnitelimitsfromE=I.11Condition(4)saysthatthefullsubcategoryasgivenbySisclosedunderdependentproductsand,therefore,underexponentiation.Underassumptionof(1),(2)and(3)condition(4)isequivalenttotherequirementthateveryS=IisclosedunderexponentiationinE=I,i.e.thatS=Iisafullsub-cartesian-closed-categoryofE=I.Underassumptionofconditions(1){(4)condition(5)isequivalenttotherequirementthatthesubbrationofPEasgivenbyS,!E2isequivalent12toasmallbration.Thisstillholdsevenifcondition(2)isweakenedtotherequire-mentthatforallI2EthesubcategoryS=IofE=Iisclosedunderequalisers.Ingeneral,agenericfamilyneednotbeclassifyingasitcanwellhappenthatfordistinctf1;f2:I!Uthefamiliesf1Elandf2ElareisomorphicasfamiliesoverI.ThefamilyElisclassifyingi intheinternallogicofEitholdsthat8a;b2U�El(a)=El(b))a=Ub.NoticethatthisrequirementfailsalreadywhenEisSetandSisthefamily(a)a2UforsomeGrothendieckuniverseUinSetbecauseUwillcontainanawfullotofdistinctbutequipollentsets.This,ofcourse,canbeovercomebyrestrictingUtothecardinalnumbersinUwhich,however,isonlypossibleinpresenceoftheaxiomofchoiceandinanycasedoesnotseemverynatural.Oneoftheusefulconsequencesofcondition(5)isthatthemapsofSareclosedunder+whichcanbeseenasfollows.Supposea:A!Iandb:B!JaremapsinS.Bycondition(5)thereexistmapsf:I!Uandg:J!Uwitha=fElandb=gEl.Duetotheextensivitypropertiesoftoposeswethenhavea+b=fEl+gEl=[f;g]El2Sasdesired.NoticethatS=Mono(E)isaclassofmapssatisfyingconditions(1){(5).Thus,auniverseneednotcontaintheterminalprojection E!1E.However,ifitdoes,i.e.ifSis\impredicative",theneveryS=Icontainstheobject:I !IandthusS=IisasubtoposofE=IinthesensethatS=I,!E=Iisalogicalfunctor.Moreover,forE=SettheclassFoffamiliesofnitecardinalsgivesrisetoanimpredicativeuniversewhich,however,doesnotcontainN.OneoftheusefulconsequencesofimpredicativityisthatSisclosedunder 11Actually,aweakerconditionthan(2)sucesforthispurpose,namelythateveryregularmonomorphismisinS.Underassumptionof(1)and(3)thisweakeningof(2)isequivalenttotherequirementthatScontainsallisosandfg2Simpliesg2S.Thus,itfollowsinparticularthatmorphismsbetweensmallmapsaresmallthemselves.12Thisbrationneednotitselfbesmallasthereneednotbeaclassifyingfamily,onlyagenericone.ButitisequivalenttothesmallbrationarisingfromtheinternalcategoryinEwhosesetofobjectsisgivenbyUandwhosefamilyofmorphismsisgivenbytheexponentialElElinE=U.6 WesuggestthatareasonablenotionofmodelforimpredicativeconstructivemathematicsisprovidedbyatoposEwithanaturalnumbersobjectNtogetherwithasequence(Sn)n2NofimpredicativeuniversescontainingNsuchthateverySniscontainedinSn+1(inthesenseofDef.2.2above).AnappropriateinternallanguageforsuchastructureisgivenbyZ.Luo'sExtendedCalculusofConstructions(ECC),see[Luo],togetherwiththeAxiomofUniqueChoice(AUC),extensionalityforfunctions,thepropositionalextensionalityprinciple8p;q2Prop�(p,q))p=qandtheprincipleofproof-irrelevancestatingthatpropositionaltypes,i.e.typesinProp,containatmostoneelement.WecallthisformalsystemECCTasanacronymforExtendedCalculusofConstructionswithinaTopos.Determiningtheproof-theoreticstrengthofECCTisanopenproblemwhich,however,seemstobefairlydicultforthefollowingreasons.OntheonehandinSetanimpredicativeuniversecontainingNhasatleaststronglyinaccessiblecardinality(becauseitisinnite,regularandclosedunderP(�),i.e.2(�)).Thus,inSetaninnitecumulativesequenceofsuchuniversesrequirestheexistenceofinnitelymanystronglyinaccessiblecardinals.ForthisreasononemightexpectthatECCTisasstrongasIZFwithanexternal13cumulativesequenceofGrothendieckuniverses.PostulatingtheAxiomofChoice(AC)thishasbeenachievedbyB.Wernerin[Wer].However,intoposesACdoesnotholdunlesstheirlogicisboolean.Butin[JM]A.JoyalandI.Moerdijkhaveconstructedso-calledinitialZF-algebras,i.e.internalmodelsofIZF,fromuniversesSwhichontheonehandarealittleweakerthanournotion(seediscussioninsection4)butontheotherhandarestrongerinthesensethattheyvalidatetheso-calledtype-theoreticCollectionAxiom(CA)(8X)(8A:U)(8e:X!A)Epic(e))(9C:U)(9f:C!X)Epic(ef)whichisneededforverifyingthattheinitialZF-algebravalidatestheset-theoreticreplacementaxiomandtheyhaveveriedtheexistenceofsuchuniversesforallGrothendieckandrealizabilitytoposes.Thus,alas,thesecomparativelywell-knownmodelscannotservethepurposeofdisprovingtheclaimthatECCTprovesconsistencyofIZF.Ontheotherhandonehasgottheimpressionthatsomethinglike(CA)isneededforverifyingthattheinitialZF-algebradoesactuallyvalidatethereplacementaxiom.3ExistenceofUniversesinToposesAfterhavingintroducedthenotionofuniverseinatoposwenowdiscussthequestionoftheirexistence.WeareprimarilyinterestedintheexistenceofimpredicativeuniversescontainingNandfromnowonrefertothemsimplyasuniverses.Accordingly,weassumealltoposestohavean.n.o.denotedasN. 13ApparentlyECCTdoesnotprovetheconsistencyofIZFtogetherwiththeaxiomthateverysetisanelementofaGrothendieckuniverse.ForthispurposeonewouldhavetoextendECCTwithafurtheruniverseU!containingallUnforn!.8 pointistheexistenceofamorphismEl:E!UgenericforU.WedeneUasthepresheafoverCwithU(I)=U(C=I)opandfor :J!IinCweputU(f)=f=U(C=f)op.WedenethegenericfamilyEl:E!UastheobjectofbC=Ucorresponding(viabC=U'\Elts(U))tothepresheafE:Elts(U)op!Uwhichisdenedasfollows:witheveryobject(I;A)inElts(U)weassociatethesetE(I;A)=A(idI)inUandwitheverymorphismf:(J;fA)!(I;A)inElts(U)weassociatethemapE(f)=A(f):E(I;A)!E(J;fA).ThisconstructionforbCextendstosheaftoposesE=Sh(C;J)forGrothen-diecktopologiesJonCinthefollowingway.LetUbetheuniverseinbCasconstructedabove.WedeneauniverseUEinEastheintersectionofUandEbC.TheclassUEconsistsofallmapsofEthatareisomorphictosomearrowa(f)wheref2Uanda:bC!Sh(C;J)isthesheacationfunctorleftadjointtotheinclusionE=Sh(C;J),!bC.(OnereadilychecksthattheimageofUunderaiscontainedinU(usingtheassumptionthatCisinternaltoU)andthereforeUEcoincideswiththeimageofUundera.)FromthisobservationitfollowsthatUEisclosedundercompositionandasthesheacationfunctorapreservesnitelimitsitisimmediatethatUEisstableunderpullbacksalongarbitraryarrowsinE.Moreover,themapa(El)isgenericforUEbecauseElisgenericforU.AsapreservesmonosandUcontainsallmonosofbCitfollowsthatUEcontainsallmonosofE.ThatUEsatisescondition(4)ofDenition2.1canbeseenasfollows.Undertheconditions(1),(2)and(3)(alreadyestablishedforUE)condition(4)isequivalenttotherequirementthatforallA2EthesliceUE=AhasexponentialsinheritedfromE=A.Supposeb1:B1!Aandb2:B2!AareobjectsinUE=A.Thentheirexponentialbb12takeninbC=AstayswithinUasUisauniverseinbCanditstayswithinE=AasE=Aisasubtopos15ofbC=Aandsubtoposesareclosedunderexponentiation.Thusbb12isinUE=AconcludingtheargumentthatUE=AisclosedunderexponentiationtakeninE=A.NextweshowthattheuniverseUEisimpredicative.Firstnoticethata(�bC)isagenericmonoforE.16Thus,thereexistsamaps: E!a( bC)with�E=sa(�bC).Asthereisalsoamapp:a( bC)! Ewitha(�bC)=p�Eitfollowsthat(ps)�E=�E.Thus,wehaveps=id Eand,therefore,themaps: E!a( bC)isasplitmono.AsUisimpredicativeitcontainstheterminalprojectionof bCand,accordingly,UEcontainstheterminalprojectionofa( bC).Thus,asUEisclosedundersubobjectsinEitfollowsthatUEcontainsalsotheterminalprojectionof E,i.e.thatUEisimpredicative.AsUEcontainstheterminalprojectionofNE=a(N),then.n.o.ofE,theuniverseUEcontainsNE. 15inthegeometricsensebecausea=A:bC=A!E=AisanitelimitpreservingleftadjointtotheinclusionE=A,!bC=A16Ingeneral,sheacationdoesnotpreservesubobjectclassiers.Actually,itdoesifandonlyifthecorrespondingLawvere-Tierneytopologyjpreservesimplicationinthesensethatj!=!(jj).10 with2SA;02S,suchthatforeveryr:RABwithr;02Sthereexistsaunique:B!PS(A)withR -2SAABr?? A-APS(A)?? Obviously,thesubobjectPS(A)consistsofthosesubsetsofAwhicharesmallinthesenseofS.In[Sim1]ithasbeenshownthatexistenceofsuchsmallpowerobjectsentailsthedescentpropertyforS.19Currentlyitisnot(yet)clear(tous)whethertheuniversesintroducedinsection3aredenable.ThoughthisseemstobeverylikelythecaseforGrothendiecktoposesthequestionforrealizabilitytoposesseemstobemuchharder20.However,insteadofauniverseSonemightinsteadconsideritsstackcom-pletionS`stillsatisfyingconditions(1){(4)ofDenition2.1butinsteadofcondition(5)only(5.1)(descent)foreverya:A!Iifea2S`forsomeepie:JIthenalreadya2S`(5.2)(weaklygenericfamily)thereexistsamapEl:E!UinS`suchthatforeverya:A!IinS`thereisanepie:JIandf:J!Uwithea=fEl.WeleaveitforfutureinvestigationstondoutwhetherthestackcompletionsS`oftheuniversesSconstructedinsection3dovalidatethetype-theoreticComprehensionAxiom(CA)discussedatendofsection2. 19Ifb2SandB e0--AJb? e--Ia? thena= (2SA;0)wheretheclassifyingmap :I!PS(A)isgivenbyx= (i)()9j2J:i=e(j)^x=e0!(b�1[j])wheree0!isthedirectimagemapfore0andb�1istheinverseimagemapforb,i.e. (i)=fx2Aj9y2B:e(b(y))=i^e0(y)=xg:20OnewouldhavetoshowforexamplethatfortheclassSofsmallmapsintherealizabilitytoposE asconsideredin[JM]thereexistsagenericfamilyandnotonlyaweaklygenericone.Butthisisdicultasonedoesn'tevenknowthesizeof��1(1),i.e.howmany(uptoisomorphism)objectsX2E existsuchthatXhaspreciselyoneglobalelement.12 References[ABSS]S.Awodey,C.Butz,A.Simpson,T.StreicherRelatingsettheories,toposesandcategoriesofclasses.paperinpreparation(2004).[Ben]J.BenabouProblemesdanslestopos.LectureNotesofaCoursefrom1971takenbyJ.-R.RoisinandpublishedasTech.Rep.,Univ.Louvain-la-Neuve(1973).[Jac]B.JacobsCategoricalLogicandTypeTheory.NorthHolland(1999).[HS]M.Hofmann,T.StreicherLiftingGrothendieckUniverses.unpublishednote(199?)availableelectronicallyathttp:==www:mathematik:tu�darmstadt:de=~streicher=NOTES=lift:dvi:gz.[JM]A.Joyal,I.MoerdijkAlgebraicSetTheory.LondonMathematicalSocietyLectureNotesSeries,220.CambridgeUniversityPress(1995).[Luo]Z.LuoComputationandReasoning.ATypeTheoryforComputerScience.OxfordUniversityPress(1994).[MM]S.MacLane,I.MoerdijkSheavesinGeometryandLogic.SpringerVerlag(1994).[McC]Ch.McCartyRealizabilityandRecursiveMathematics.PhDThesis,Oxford(1984).[Sim1]A.SimpsonElementaryaxiomsforcategoriesofclasses.Proc.ofLICS'99,pp.77-85,IEEEPress(1999).[Sim2]A.SimpsonComputationalAdequacyforRecursiveTypesinModelsofIntuitionisticSetTheory.acceptedforAnnalsofPureandAppliedLogic(2003).[Str1]T.StreicherSemanticsofTypeTheory.Birkhauser(1991).[Str2]T.StreicherFibredCategoriesalaJeanBenabou.lecturenotes(2003)availableelectronicallyathttp:==www:mathematik:tu�darmstadt:de=~streicher=FIBR=FibLec:ps:gz.[Wer]B.WernerSetsinTypes,TypesinSets.Proc.ofTACS'97,SLNCS1281.14