Set Theory - Stanford Encyclopedia of Philosophy

One can prove in ZFC—and the use of the AC is necessary—that there are non-determined sets. Thus, the Axiom of Determinacy (AD), which asserts ... StanfordEncyclopediaofPhilosophy Menu Browse TableofContents What'sNew RandomEntry Chronological Archives About EditorialInformation AbouttheSEP EditorialBoard HowtoCitetheSEP SpecialCharacters AdvancedTools Contact SupportSEP SupporttheSEP PDFsforSEPFriends MakeaDonation SEPIAforLibraries EntryNavigation EntryContents Bibliography AcademicTools FriendsPDFPreview AuthorandCitationInfo BacktoTop SetTheoryFirstpublishedWedOct8,2014;substantiverevisionTueFeb12,2019 Settheoryisthemathematicaltheoryofwell-determined collections,calledsets,ofobjectsthatarecalled members,orelements,oftheset.Puresettheory dealsexclusivelywithsets,sotheonlysetsunderconsiderationare thosewhosemembersarealsosets.Thetheoryofthe hereditarily-finitesets,namelythosefinitesetswhose elementsarealsofinitesets,theelementsofwhicharealsofinite, andsoon,isformallyequivalenttoarithmetic.So,theessenceof settheoryisthestudyofinfinitesets,andthereforeitcanbe definedasthemathematicaltheoryoftheactual—asopposedto potential—infinite. Thenotionofsetissosimplethatitisusuallyintroduced informally,andregardedasself-evident.Insettheory,however,as isusualinmathematics,setsaregivenaxiomatically,sotheir existenceandbasicpropertiesarepostulatedbytheappropriate formalaxioms.Theaxiomsofsettheoryimplytheexistenceofa set-theoreticuniversesorichthatallmathematicalobjectscanbe construedassets.Also,theformallanguageofpuresettheoryallows onetoformalizeallmathematicalnotionsandarguments.Thus,settheory hasbecomethestandardfoundationformathematics,asevery mathematicalobjectcanbeviewedasaset,andeverytheoremof mathematicscanbelogicallydeducedinthePredicateCalculusfrom theaxiomsofsettheory. Bothaspectsofsettheory,namely,asthemathematicalscienceof theinfinite,andasthefoundationofmathematics,areof philosophicalimportance. 1.Theorigins 2.Theaxiomsofsettheory 2.1TheaxiomsofZFC 3.Thetheoryoftransfiniteordinalsandcardinals 3.1Cardinals 4.Theuniverse\(V\)ofallsets 5.Settheoryasthefoundationofmathematics 5.1Metamathematics 5.2Theincompletenessphenomenon 6.Thesettheoryofthecontinuum 6.1DescriptiveSetTheory 6.2Determinacy 6.3TheContinuumHypothesis 7.Gödel’sconstructibleuniverse 8.Forcing 8.1Otherapplicationsofforcing 9.Thesearchfornewaxioms 10.Largecardinals 10.1Innermodelsoflargecardinals 10.2Consequencesoflargecardinals 11.Forcingaxioms Bibliography AcademicTools OtherInternetResources RelatedEntries 1.Theorigins Settheory,asaseparatemathematicaldiscipline,beginsinthe workofGeorgCantor.Onemightsaythatsettheorywasborninlate 1873,whenhemadetheamazingdiscoverythatthelinearcontinuum, thatis,therealline,isnotcountable,meaningthatitspoints cannotbecountedusingthenaturalnumbers.So,eventhoughtheset ofnaturalnumbersandthesetofrealnumbersarebothinfinite, therearemorerealnumbersthantherearenaturalnumbers,which openedthedoortotheinvestigationofthedifferentsizesof infinity.Seetheentryonthe earlydevelopmentofsettheory foradiscussionoftheoriginof set-theoreticideasandtheirusebydifferentmathematiciansand philosophersbeforeandaroundCantor’stime. AccordingtoCantor,twosets\(A\)and\(B\)havethesamesize,or cardinality,iftheyarebijectable,i.e.,theelementsof \(A\)canbeputinaone-to-onecorrespondencewiththeelementsof \(B\).Thus,theset\(\mathbb{N}\)ofnaturalnumbersandtheset \(\mathbb{R}\)ofrealnumbershavedifferentcardinalities.In1878 CantorformulatedthefamousContinuumHypothesis(CH),which assertsthateveryinfinitesetofrealnumbersiseithercountable, i.e.,ithasthesamecardinalityas\(\mathbb{N}\),orhasthesame cardinalityas\(\mathbb{R}\).Inotherwords,thereareonlytwo possiblesizesofinfinitesetsofrealnumbers.TheCHisthemost famousproblemofsettheory.Cantorhimselfdevotedmucheffortto it,andsodidmanyotherleadingmathematiciansofthefirsthalfof thetwentiethcentury,suchasHilbert,wholistedtheCHasthefirst probleminhiscelebratedlistof23unsolvedmathematicalproblems presentedin1900attheSecondInternationalCongressof Mathematicians,inParis.TheattemptstoprovetheCHledtomajor discoveriesinsettheory,suchasthetheoryofconstructiblesets, andtheforcingtechnique,whichshowedthattheCHcanneitherbe provednordisprovedfromtheusualaxiomsofsettheory.Tothisday, theCHremainsopen. Earlyon,someinconsistencies,orparadoxes,arosefromanaive useofthenotionofset;inparticular,fromthedeceivinglynatural assumptionthateverypropertydeterminesaset,namelythesetof objectsthathavetheproperty.OneexampleisRussell’s Paradox,alsoknowntoZermelo: considerthepropertyofsetsofnotbeingmembersof themselves.Ifthepropertydeterminesaset,callit\(A\),then\(A\)is amemberofitselfifandonlyif\(A\)isnotamemberof itself. Thus,somecollections,likethecollectionofallsets,the collectionofallordinalsnumbers,orthecollectionofallcardinal numbers,arenotsets.Suchcollectionsarecalledproper classes. Inordertoavoidtheparadoxesandputitonafirmfooting,set theoryhadtobeaxiomatized.Thefirstaxiomatizationwasdueto Zermelo(1908)anditcameasaresultoftheneedtospelloutthe basicset-theoreticprinciplesunderlyinghisproofofCantor’s Well-OrderingPrinciple.Zermelo’saxiomatizationavoidsRussell’s ParadoxbymeansoftheSeparationaxiom,whichisformulatedas quantifyingoverpropertiesofsets,andthusitisasecond-order statement.FurtherworkbySkolemandFraenkelledtothe formalizationoftheSeparationaxiomintermsofformulasof first-order,insteadoftheinformalnotionofproperty,aswellasto theintroductionoftheaxiomofReplacement,whichisalsoformulated asanaxiomschemaforfirst-orderformulas(seenextsection).The axiomofReplacementisneededforaproperdevelopmentofthetheory oftransfiniteordinalsandcardinals,usingtransfiniterecursion (seeSection3).Itisalsoneededto provetheexistenceofsuchsimplesetsasthesetofhereditarily finitesets,i.e.,thosefinitesetswhoseelementsarefinite,the elementsofwhicharealsofinite,andsoon;ortoprovebasic set-theoreticfactssuchasthateverysetiscontainedina transitiveset,i.e.,asetthatcontainsallelementsofitselements (seeMathias2001fortheweaknessesofZermelosettheory).Afurther addition,byvonNeumann,oftheaxiomofFoundation,ledtothe standardaxiomsystemofsettheory,knownastheZermelo-Fraenkel axiomsplustheAxiomofChoice,orZFC. Otheraxiomatizationsofsettheory,suchasthoseofvon Neumann-Bernays-Gödel(NBG),orMorse-Kelley(MK),allowalsofor aformaltreatmentofproperclasses. 2.Theaxiomsofsettheory ZFCisanaxiomsystemformulatedinfirst-orderlogicwith equalityandwithonlyonebinaryrelationsymbol\(\in\)for membership.Thus,wewrite\(A\inB\)toexpressthat\(A\)isamemberof theset\(B\).Seethe SupplementonBasicSetTheory forfurtherdetails.Seealsothe SupplementonZermelo-FraenkelSetTheory foraformalizedversionoftheaxiomsandfurthercomments.We statebelowtheaxiomsofZFCinformally. 2.1TheaxiomsofZFC Extensionality:Iftwosets\(A\)and\(B\)have thesameelements,thentheyareequal. NullSet:Thereexistsaset,denotedby \({\varnothing}\)andcalledtheemptyset,whichhasno elements. Pair:Givenanysets\(A\)and\(B\),there existsaset,denotedby\(\{A,B\}\),whichcontains\(A\)and\(B\)asits onlyelements.Inparticular,thereexiststheset\(\{A\}\)whichhas \(A\)asitsonlyelement. PowerSet:Foreveryset\(A\)thereexistsa set,denotedby\(\mathcal{P}(A)\)andcalledthepowersetof \(A\),whoseelementsareallthesubsetsof\(A\). Union:Foreveryset\(A\),thereexistsaset, denotedby\(\bigcupA\)andcalledtheunionof\(A\),whose elementsarealltheelementsoftheelementsof\(A\). Infinity:Thereexistsaninfiniteset.In particular,thereexistsaset\(Z\)thatcontains\({\varnothing}\)and suchthatif\(A\inZ\),then\(\bigcup\{A,\{A\}\}\inZ\). Separation:Foreveryset\(A\)andeverygiven property,thereisasetcontainingexactlytheelementsof\(A\)that havethatproperty.Apropertyisgivenbyaformula \(\varphi\)ofthefirst-orderlanguageofsettheory. Thus,Separationisnotasingleaxiombutanaxiomschema,that is,aninfinitelistofaxioms,oneforeachformula \(\varphi\). Replacement:Foreverygivendefinable functionwithdomainaset\(A\),thereisasetwhoseelementsare allthevaluesofthefunction. Replacementisalsoanaxiomschema,asdefinablefunctionsare givenbyformulas. Foundation:Everynon-emptyset\(A\)contains an\(\in\)-minimalelement,thatis,anelementsuchthatnoelementof \(A\)belongstoit. ThesearetheaxiomsofZermelo-Fraenkelsettheory,orZF.The axiomsofNullSetandPairfollowfromtheotherZFaxioms,sothey maybeomitted.Also,ReplacementimpliesSeparation. Finally,thereistheAxiomofChoice(AC): Choice:Foreveryset\(A\)of pairwise-disjointnon-emptysets,thereexistsasetthatcontains exactlyoneelementfromeachsetin\(A\). TheACwas,foralongtime,acontroversialaxiom.Ontheone hand,itisveryusefulandofwideuseinmathematics.Ontheother hand,ithasratherunintuitiveconsequences,suchasthe Banach-TarskiParadox,whichsaysthattheunitballcanbe partitionedintofinitely-manypieces,whichcanthenberearrangedto formtwounitballs.Theobjectionstotheaxiomarisefromthefact thatitassertstheexistenceofsetsthatcannotbeexplicitly defined.ButGödel’s1938proofofitsconsistency,relativeto theconsistencyofZF,dispelledanysuspicionsleftaboutit. TheAxiomofChoiceisequivalent,moduloZF,tothe Well-orderingPrinciple,whichassertsthateverysetcanbe well-ordered,i.e.,itcanbelinearlyorderedsothateverynon-empty subsethasaminimalelement. Althoughnotformallynecessary,besidesthesymbol\(\in\)one normallyusesforconvenienceotherauxiliarydefinedsymbols.For example,\(A\subseteqB\)expressesthat\(A\)isasubsetof \(B\),i.e.,everymemberof\(A\)isamemberof\(B\).Othersymbolsare usedtodenotesetsobtainedbyperformingbasicoperations,suchas \(A\cupB\),whichdenotestheunionof\(A\)and\(B\),i.e.,the setwhoseelementsarethoseof\(A\)and\(B\);or\(A\capB\),which denotestheintersectionof\(A\)and\(B\),i.e.,thesetwhose elementsarethosecommonto\(A\)and\(B\).Theorderedpair \((A,B)\)isdefinedastheset\(\{\{A\},\{A,B\}\}\).Thus,two orderedpairs\((A,B)\)and\((C,D)\)areequalifandonlyif\(A=C\)and \(B=D\).AndtheCartesianproduct\(A\timesB\)isdefinedas thesetofallorderedpairs\((C,D)\)suchthat\(C\inA\)and\(D\in B\).Givenanyformula\(\varphi(x,y_1,\ldots,y_n)\),andsets \(A,B_1,\ldots,B_n\),onecanformthesetofallthoseelementsof\(A\) thatsatisfytheformula\(\varphi(x,B_1,\ldots,B_n)\).Thissetis denotedby\(\{a\inA:\varphi(a,B_1,\ldots,B_n)\}\).InZFonecan easilyprovethatallthesesetsexist.Seethe SupplementonBasicSetTheory forfurtherdiscussion. 3.Thetheoryoftransfiniteordinalsandcardinals InZFConecandeveloptheCantoriantheoryoftransfinite(i.e., infinite)ordinalandcardinalnumbers.Followingthedefinitiongiven byVonNeumannintheearly1920s,theordinalnumbers,or ordinals,forshort,areobtainedbystartingwiththeempty setandperformingtwooperations:takingtheimmediatesuccessor,and passingtothelimit.Thus,thefirstordinalnumberis \({\varnothing}\).Givenanordinal\(\alpha\),itsimmediate successor,denotedby\(\alpha+1\),istheset\(\alpha\cup\{ \alpha\}\).Andgivenanon-emptyset\(X\)ofordinalssuchthatfor every\(\alpha\inX\)thesuccessor\(\alpha+1\)isalsoin\(X\),one obtainsthelimitordinal\(\bigcupX\).Oneshowsthatevery ordinalis(strictly)well-orderedby\(\in\),i.e.,itislinearly orderedby\(\in\)andthereisnoinfinite\(\in\)-descending sequence.Also,everywell-orderedsetisisomorphictoaunique ordinal,calleditsorder-type. Notethateveryordinalisthesetofitspredecessors.However, theclass\(ON\)ofallordinalsisnotaset.Otherwise,\(ON\)wouldbe anordinalgreaterthanalltheordinals,whichisimpossible.The firstinfiniteordinal,whichisthesetofallfiniteordinals,is denotedbytheGreekletteromega(\(\omega\)).InZFC,oneidentifies thefiniteordinalswiththenaturalnumbers.Thus,\({\varnothing}=0\), \(\{{\varnothing}\}=1\),\(\{{\varnothing},\{{\varnothing}\}\}=2\), etc.,hence\(\omega\)isjusttheset\(\mathbb{N}\)ofnatural numbers. Onecanextendtheoperationsofadditionandmultiplicationof naturalnumberstoalltheordinals.Forexample,theordinal\(\alpha +\beta\)istheorder-typeofthewell-orderingobtainedby concatenatingawell-orderedsetoforder-type\(\alpha\)anda well-orderedsetoforder-type\(\beta\).Thesequenceofordinals, well-orderedby\(\in\),startsasfollows 0,1,2,…,\(n\),…,\(\omega\),\(\omega+1\), \(\omega+2\),…,\(\omega+\omega\),…,\(\omega\cdotn\), …,\(\omega\cdot\omega\),…,\(\omega^n\),…, \(\omega^\omega\),… Theordinalssatisfytheprincipleoftransfinite induction:supposethat\(C\)isaclassofordinalssuchthat whenever\(C\)containsallordinals\(\beta\)smallerthansomeordinal \(\alpha\),then\(\alpha\)isalsoin\(C\).Thentheclass\(C\)contains allordinals.UsingtransfiniteinductiononecanproveinZFC(and oneneedstheaxiomofReplacement)theimportantprincipleof transfiniterecursion,whichsaysthat,givenanydefinable class-function\(G:V\toV\),onecandefineaclass-function\(F:ON\toV\) suchthat\(F(\alpha)\)isthevalueofthefunction\(G\)appliedtothe function\(F\)restrictedto\(\alpha\).Oneusestransfiniterecursion, forexample,inordertodefineproperlythearithmeticaloperations ofaddition,product,andexponentiationontheordinals. Recallthataninfinitesetiscountableifitis bijectable,i.e.,itcanbeputintoaone-to-onecorrespondence,with \(\omega\).Alltheordinalsdisplayedaboveareeitherfiniteor countable.Butthesetofallfiniteandcountableordinalsisalsoan ordinal,called\(\omega_1\),andisnotcountable.Similarly,theset ofallordinalsthatarebijectablewithsomeordinallessthanor equalto\(\omega_1\)isalsoanordinal,called\(\omega_2\),andisnot bijectablewith\(\omega_1\),andsoon. 3.1Cardinals Acardinalisanordinalthatisnotbijectablewithany smallerordinal.Thus,everyfiniteordinalisacardinal,and \(\omega\),\(\omega_1\),\(\omega_2\),etc.arealsocardinals.The infinitecardinalsarerepresentedbytheletteraleph(\(\aleph\))of theHebrewalphabet,andtheirsequenceisindexedbytheordinals.It startslikethis \(\aleph_0\),\(\aleph_1\),\(\aleph_2\),…,\(\aleph_\omega\), \(\aleph_{\omega+1}\),…,\(\aleph_{\omega+\omega}\),…, \(\aleph_{\omega^2}\),…,\(\aleph_{\omega^\omega}\),…, \(\aleph_{\omega_1}\),…,\(\aleph_{\omega_2}\),… Thus,\(\omega=\aleph_0\),\(\omega_1=\aleph_1\),\(\omega_2=\aleph_2\), etc.Foreverycardinalthereisabiggerone,andthelimitofan increasingsequenceofcardinalsisalsoacardinal.Thus,theclass ofallcardinalsisnotaset,butaproperclass. Aninfinitecardinal\(\kappa\)iscalledregularifitis nottheunionoflessthan\(\kappa\)smallercardinals.Thus, \(\aleph_0\)isregular,andsoareallinfinitesuccessorcardinals, suchas\(\aleph_1\).Non-regularinfinitecardinalsarecalled singular.Thefirstsingularcardinalis\(\aleph_\omega\),as itistheunionofcountably-manysmallercardinals,namely \(\aleph_\omega=\bigcup_{n\kappa\),whichimpliesthatthecofinalityof thecardinal\(2^{\aleph_0}\),whateverthatcardinalis,mustbe uncountable.ButthisisessentiallyallthatZFCcanproveaboutthe valueoftheexponential\(2^{\aleph_0}\). Inthecaseofexponentiationofsingularcardinals,ZFChasalot moretosay.In1989,Shelahprovedtheremarkableresultthatif \(\aleph_\omega\)isastronglimit,thatis, \(2^{\aleph_n}\alpha\)andeverysequenceofelementsof\(M\)oflength \(\alpha\)belongsto\(M\). Woodincardinalsfallbetweenstrongandsupercompact.Every supercompactcardinalisWoodin,andif\(\delta\)isWoodin,then \(V_\delta\)isamodelofZFCinwhichthereisaproperclassof strongcardinals.Thus,whileaWoodincardinal\(\delta\)neednotbe itselfverystrong—thefirstoneisnotevenweakly compact—itimpliestheexistenceofmanylargecardinalsin \(V_\delta\). Beyondsupercompactcardinalswefindtheextendible cardinals,thehuge,thesuperhuge,etc. Kunen’stheoremaboutthenon-existenceofanon-trivialelementary embedding\(j:V\toV\)actuallyshowsthattherecannotbeanelementary embedding\(j:V_{\lambda+2}\toV_{\lambda+2}\)differentfromthe identity,forany\(\lambda\). Thestrongestlargecardinalnotionsnotknowntobeinconsistent, moduloZFC,arethefollowing: Thereexistsanelementaryembedding\(j:V_{\lambda+1}\to V_{\lambda+1}\)differentfromtheidentity. Thereexistsanelementaryembedding\(j:L(V_{\lambda+1})\to L(V_{\lambda+1})\)differentfromtheidentity. Largecardinalsformalinearhierarchyofincreasingconsistency strength.Infacttheyarethesteppingstonesoftheinterpretability hierarchyofmathematicaltheories.Seetheentryon independenceandlargecardinals formoredetails.Givenanysentence\(\varphi\),exactly onethefollowingthreepossibilitiesholdsaboutthetheoryZFCplus \(\varphi\): ZFCplus\(\varphi\)isinconsistent. ZFCplus\(\varphi\)isequiconsistentwithZFC. ZFCplus\(\varphi\)isequiconsistentwithZFCplustheexistenceof somelargecardinal. Thus,largecardinalscanbeusedtoprovethatagivensentence \(\varphi\)doesnotimplyanothersentence\(\psi\),moduloZFC,by showingthatZFCplus\(\psi\)impliestheconsistencyofsomelarge cardinal,whereasZFCplus\(\varphi\)isconsistentassumingthe existenceofasmallerlargecardinal,orjustassumingthe consistencyofZFC.Inotherwords,\(\psi\)hashigherconsistency strengththan\(\varphi\),moduloZFC.Then,byGödel’ssecond incompletenesstheorem,ZFCplus\(\varphi\)cannotprove\(\psi\), assumingZFCplus\(\varphi\)isconsistent. Aswealreadypointedout,onecannotproveinZFCthatlarge cardinalsexist.Buteverythingindicatesthattheirexistencenot onlycannotbedisproved,butinfacttheassumptionoftheir existenceisaveryreasonableaxiomofsettheory.Foronething, thereisalotofevidencefortheirconsistency,especiallyforthose largecardinalsforwhichitispossibletoconstructaninner model. 10.1Innermodelsoflargecardinals AninnermodelofZFCisatransitiveproperclassthat containsalltheordinalsandsatisfiesallZFCaxioms.Thus,\(L\)is thesmallestinnermodel,while\(V\)isthelargest.Somelarge cardinals,suchasinaccessible,Mahlo,orweakly-compact,mayexist in\(L\).Thatis,if\(\kappa\)hasoneoftheselargecardinal properties,thenitalsohasthepropertyin\(L\).Butsomelarge cardinalscannotexistin\(L\).Indeed,Scott(1961)showedthatif thereexistsameasurablecardinal\(\kappa\),then\(V\neL\).Itis importanttonoticethat\(\kappa\)doesbelongto\(L\),since\(L\) containsallordinals,butitisnotmeasurablein\(L\)becausea \(\kappa\)-completenon-principalmeasureon\(\kappa\)cannotexist there. If\(\kappa\)isameasurablecardinal,thenonecanconstructan \(L\)-likemodelinwhich\(\kappa\)ismeasurablebytakinga \(\kappa\)-completenon-principalandnormalmeasure\(U\)on\(\kappa\), andproceedingasinthedefinitionof\(L\),butnowusing\(U\)asan additionalpredicate.Theresultingmodel,called\(L[U]\),isaninner modelofZFCinwhich\(\kappa\)ismeasurable,andinfact\(\kappa\)is theonlymeasurablecardinal.Themodeliscanonical,inthesense thatanyothernormalmeasurewitnessingthemeasurabilityof\(\kappa\) wouldyieldthesamemodel,andhasmanyofthepropertiesof\(L\).For instance,ithasaprojectivewellorderingofthereals,andit satisfiestheGCH. Buildingsimilar\(L\)-likemodelsforstrongerlargecardinals,such asstrong,orWoodin,ismuchharder.Thosemodelsareoftheform \(L[E]\),where\(E\)isasequenceofextenders,eachextender beingasystemofmeasures,thatencodetherelevantelementary embeddings. Thelargest\(L\)-likeinnermodelsforlargecardinalsthathave beenobtainedsofarcancontainWoodinlimitsofWoodincardinals (Neeman2002).However,buildingan\(L\)-likemodelforasupercompact cardinalisstillachallenge.Thesupercompactbarrierseemstobe thecrucialone,forWoodinhasshownthatforakindof\(L\)-like innermodelforasupercompactcardinal,whichhecallsthe Ultimate-\(L\),allstrongerlargecardinalsthatmayexistin\(V\),suchas extendible,huge,I1,etc.wouldalsoexistinthemodel.The constructionofUltimate-\(L\)isstillincomplete,anditisnotclear yetthatitwillsucceed,foritrestsuponsometechnicalhypotheses thatneedtobeconfirmed. 10.2Consequencesoflargecardinals Theexistenceoflargecardinalshasdramaticconsequences,even forsimply-definablesmallsets,liketheprojectivesetsofreal numbers.Forexample,Solovay(1970)proved,assumingthatthere existsameasurablecardinal,thatall\(\mathbf{\Sigma}^1_2\)setsof realsareLebesguemeasurableandhavetheBaireproperty,which cannotbeprovedinZFCalone.AndShelahandWoodin(1990)showed thattheexistenceofaproperclassofWoodincardinalsimpliesthat thetheoryof\(L(\mathbb{R})\),evenwithrealnumbersasparameters, cannotbechangedbyforcing,whichimpliesthatallsetsofreal numbersthatbelongto\(L(\mathbb{R})\)areregular.Further,undera weakerlarge-cardinalhypothesis,namelytheexistenceofinfinitely manyWoodincardinals,MartinandSteel(1989)provedthatevery projectivesetofrealnumbersisdetermined,i.e.,theaxiomofPD holds,henceallprojectivesetsareregular.Moreover,Woodinshowed thattheexistenceofinfinitelymanyWoodincardinals,plusa measurablecardinalaboveallofthem,impliesthateverysetofreals in\(L(\mathbb{R})\)isdetermined,i.e.,theaxiom\(AD^{L(\mathbb{R})}\) holds,henceallsetsofrealnumbersthatbelongto\(L(\mathbb{R})\), andthereforeallprojectivesets,areregular.Healsoshowedthat Woodincardinalsprovidetheoptimallargecardinalassumptionsby provingthatthefollowingtwostatements: ThereareinfinitelymanyWoodincardinals. \(AD^{L({\BbbR})}\). areequiconsistent,i.e.,ZFCplus1isconsistentifandonlyif ZFCplus2isconsistent.Seetheentryon largecardinalsanddeterminacy formoredetailsandrelatedresults. Anotherareainwhichlargecardinalsplayanimportantroleisthe exponentiationofsingularcardinals.Theso-calledSingular CardinalHypothesis(SCH)completelydeterminesthebehaviorof theexponentiationforsingularcardinals,modulotheexponentiation forregularcardinals.TheSCHfollowsfromtheGCH,andsoitholds in\(L\).AconsequenceoftheSCHisthatif \(2^{\aleph_n}. [Thiswasthepreviousentryonsettheoryinthe StanfordEncyclopediaofPhilosophy—seethe versionhistory.] RelatedEntries settheory:continuumhypothesis| settheory:earlydevelopment| settheory:independenceandlargecardinals| settheory:largecardinalsanddeterminacy Copyright©2019by JoanBagaria OpenaccesstotheSEPismadepossiblebyaworld-widefundinginitiative. 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