ZFC | Brilliant Math & Science Wiki

ZFC, or Zermelo-Fraenkel set theory, is an axiomatic system used to formally define set theory (and thus mathematics in general). Specifically, ZFC is a ... Brilliant Home Courses Today Signup Login Thisholidayseason,sparkalifelongloveoflearning. GiftBrilliantPremium Excelinmathandscience. LoginwithFacebook LoginwithGoogle Loginwithemail JoinusingFacebook JoinusingGoogle Joinusingemail Forgotpassword? Newuser? Signup Existinguser? Login SignupwithFacebook or Signupmanually Alreadyhaveanaccount? Loginhere. RelevantFor... Probability > SetNotation AlexanderKatz, AFormerBrilliantMember, MichaelLivshits, and 2others JiminKhim EliRoss contributed ZFC,orZermelo-Fraenkelsettheory,isanaxiomaticsystemusedtoformallydefinesettheory(andthusmathematicsingeneral). Specifically,ZFCisacollectionofapproximately9axioms(dependingonconventionandpreciseformulation)that,takentogether,definethecoreofmathematicsthroughtheusageofsettheory.Moreformally,ZFCisapredicatelogicequippedwithabinaryrelation∈\in∈,whichreferstosetmembershipandisreadas"in".Tobeclear,itissaidthata∈ba\inba∈bwhenaaaisanelementofbbb. TheBanach-Tarskiparadox,inwhichaballcanberearrangedintotwoballsofthesamesizeastheoriginal,isacounterintuitiveconsequenceoftheaxiomofchoice. Historically,ZFCwasformulatedasameansofdefiningsettheoryinsuchawaythatparadoxessuchasRussell'sparadoxareavoided,thoughthereremainsomeunsatisfactoryaspectsofthetheory:inparticular,itcanbeshownthattheaxiomofchoiceisindependentoftheremainingaxioms,andthusZFCwiththeaxiomofchoiceomittedisstillaconsistenttheory.ThisisdenotedZFand,whileconsistent,ithasfallenoutofuseinfavorofthemorenaturalZFC. Contents Notation FormalDefinition(Axioms) AdvantagesandDisadvantages SeeAlso Ingeneral,statementsinsettheoryareexpressedusingfirst-orderlogic,whichusesanumberofquantifiers(orlogicalsymbols): ∈\in∈means"isin",asintheintroduction. ∀\forall∀means"forall";e.g.∀n∈R:n2≥0\foralln\in\mathbb{R}:n^2\geq0∀n∈R:n2≥0(\big((translated:forallrealnnn,n2≥0)n^2\geq0\big)n2≥0)isawayofexpressingthetrivialinequality. ∃\exists∃means"thereexists";e.g.∀x∈R (∃y∈R:y3=x)\forallx\in\mathbb{R}\(\existsy\in\mathbb{R}:y^3=x)∀x∈R (∃y∈R:y3=x)(\big((translated:forallrealxxx,thereexistsarealyyysuchthaty3=x)y^3=x\big)y3=x)isawayofstatingthateveryrealnumberhasarealcuberoot. ≡\equiv≡means"isequivalentto".Forexample,x3>0≡x>0x^3>0\equivx>0x3>0≡x>0isawayofexpressingthefactthatx3x^3x3ispositiveifandonlyifxxxispositive.   ⟹  \implies⟹means"implies".Forexample,x>0  ⟹  x2>0x>0\impliesx^2>0x>0⟹x2>0isawayofexpressingthefactthatthesquareofapositivenumberispositive.Notethatx>0≢x2>0x>0\not\equivx^2>0x>0​≡x2>0,sincex2>0  ⟹  x>0x^2>0\impliesx>0x2>0⟹x>0isafalsestatement(((e.g.forx=−1).x=-1).x=−1). ∧\land∧means"logicaland";e.g.n2>0∧n3<0n^2>0\landn^3<0n2>0∧n3<0isawayofexpressingthefactthatn2n^2n2ispositiveandn3n^3n3isnegative;i.e.nnnisnegative. ∨\lor∨means"logicalor";e.g.n>0∨n3≤0n>0\lorn^3\leq0n>0∨n3≤0isawayofsayingthateithernnnispositiveorn3n^3n3isnonpositive. ThisallowsfortheaxiomsinZFCtobestatedsuccinctlyusingsymbols,asinthefollowingsection. TheaxiomsofZFCcanbestatedinseveralequivalentways,andhaveslightlydifferentnamesandlogicalformulationsdependingonthesource.Ofcourse,eachindividualsourcewillhavearigorouscorrecttreatmentoftheaxioms,oneofwhichfollows: Axiomofextensionality: ∀u(u∈X≡u∈Y)  ⟹  X=Y\forallu(u\inX\equivu\inY)\impliesX=Y∀u(u∈X≡u∈Y)⟹X=Y Inotherwords,ifu∈X  ⟺  u∈Yu\inX\iffu\inYu∈X⟺u∈Yforalluuu,thenX=YX=YX=Y.Inplainlanguage,thisstatementisequivalentto"Iftwosetshavethesameelements,theyarethesameset." Axiomofpairing: ∀a∀b∃z∀x(x∈z≡(x=a∨x=b)),\foralla\forallb\existsz\forallx\big(x\inz\equiv(x=a\lorx=b)\big),∀a∀b∃z∀x(x∈z≡(x=a∨x=b)), where∨\lor∨denotesthelogicalorquantifier. Inotherwords,forallaaaandbbb,thereexistsazzzsuchthatforallxxx,x∈Zx\inZx∈Zisequivalenttothestatement"x=ax=ax=aorx=bx=bx=b".Inplainlanguage,thisstatementisequivalentto"Giventwoelements,thereexistsasetcontainingexactlythosetwoelements." Axiomofcomprehension:theelementsofAAAsatisfyingϕ\phiϕformanewsetBBB Axiomofcomprehension: ∀X∀p∃Y∀u(u∈Y≡(u∈X∧ϕ(u,p))),\forallX\forallp\existsY\forallu\left(u\inY\equiv\big(u\inX\land\phi(u,p)\big)\right),∀X∀p∃Y∀u(u∈Y≡(u∈X∧ϕ(u,p))), where∧\land∧isthelogicalandquantifier,andϕ\phiϕisanarbitraryproperty. Inplainlanguage,thisstatementisequivalentto"Givenanypropertyϕ\phiϕandsetXXX,thereexistsasetcontainingallelementsofXXXthatsatisfyϕ.\phi.ϕ."Ininformalterms,asubsetofasetcanbeconstructedbyasuccinctrule;e.g.therule"xxxeven"appliedtothesetofintegersresultsinanewset. Axiomofunion: ∀X∃Y∀u(u∈Y≡∃z(z∈X∧u∈z)),\forallX\existsY\forallu\big(u\inY\equiv\existsz(z\inX\landu\inz)\big),∀X∃Y∀u(u∈Y≡∃z(z∈X∧u∈z)), where∧\land∧isthelogicalandqualifier. Inotherwords,forallXXXthereexistsaYYYsuchthatforalluuu,u∈Yu\inYu∈Yisequivalenttothestatement"Thereexistszzzsuchthatz∈Xz\inXz∈Xandu∈z.u\inz.u∈z."Inplainlanguage,thereexistsasetYYYconsistingoftheunionofallelementsofXXX. Axiomofpowerset: ∀X∃Y∀u(u∈Y≡u⊆X)\forallX\existsY\forallu\big(u\inY\equivu\subseteqX\big)∀X∃Y∀u(u∈Y≡u⊆X) Inotherwords,foranysetXXX,thereexistsasetYYYwhoseelementsaresubsetsofXXX.Inplainlanguage,thisaxiomstatesthatthepowersetofXXXexists. Axiomofinfinity: ∃S(∅∈S∧(∀x∈S(x∪{x}∈S)))\existsS\left(\emptyset\inS\land\big(\forallx\inS(x\cup\{x\}\inS)\big)\right)∃S(∅∈S∧(∀x∈S(x∪{x}∈S))) Insimplerterms,aninfinitesetexists. Axiomofreplacement: IfFFFisanyfunction,thenforanysetXXXthereexistsasetY=F(X)={F(x),x∈X}Y=F(X)=\{F(x),x\inX\}Y=F(X)={F(x),x∈X}.Thestatementinlogicalquantifiersismorecomplex. AfunctiontakesanysetAAAtoanewsetB=F(A)B=F(A)B=F(A) Axiomofregularity: ∀S(S≠∅  ⟹  (∃x∈S:S∩x=∅))\forallS\big(S\neq\emptyset\implies(\existsx\inS:S\capx=\emptyset)\big)∀S(S​=∅⟹(∃x∈S:S∩x=∅)) Inotherwords,forallnon-emptysetsSSS,thereexistsanelementofSSSthatisdisjointwithSSS(sharesnoelementswithSSS).Thishastwomajorconsequences: Nosetcanbeanelementofitself.ThisresolvesRussell'sparadox. Everysethasasmallestelementwithrespectto∈\in∈. These8axiomsdefineaconsistenttheory,ZF(though,ofcourse,itisverydifficulttoprovethatthissystemisconsistent).Whentheaxiomofchoiceisaddedtotheeightaxiomsabove,thetheorybecomesZFC(the"C"forchoice),anditisthissystemthatiscommonlyusedasthefoundationofmathematics. ZFCisonlyoneofmanyaxiomaticsystemsthatcanbeusedastheformulationofmathematics,andassuchhascertainadvantagesanddisadvantagesoverothersimilarsystems. ThemainadvantageofZFCistofacilitatethestudyofsettheoryitself,thoughfromcertainperspectivesthisisadisadvantageaswell.Inparticular,mostofmodernmathematicscanbeproveninweakeraxiomaticsystemslikethePeanoaxioms,andallofitcanbecarriedoutinZC:Zermelosettheorywiththeaxiomofchoice(Zermelotheoryisobtainedfromdroppingtheaxiomofreplacement).Thus,insomesense,ZFCis"toostrong"forthesakeofmakingsettheoryeasier. Ontheotherhand,anothercommoncriticismisthatZFCistooweakwhencomparedtootheraxiomaticsystems.Forinstance,thecontinuumhypothesis(alongwithahandfulofotherproblems)canbeshowntobeindependentofZFC,meaningthatitcanbeneitherprovednordisprovedwiththegivenaxioms.ThussomechoosetoadoptvariousextensionsofZFC,orcloselyrelatedsystemssuchasZF+AD--wheretheaxiomofchoiceisreplacedbytheaxiomofdeterminacy. Additionally,thereareanumberofobjectionstosettheoryingeneral(seethesettheorypagefordetails). PredicateLogic AxiomofChoice AxiomofDeterminacy Citeas: ZFC. Brilliant.org. Retrievedfrom https://brilliant.org/wiki/zfc/ JoinBrilliant Excelinmathandscience. Signup Signuptoreadallwikisandquizzesinmath,science,andengineeringtopics. LoginwithFacebook LoginwithGoogle Loginwithemail JoinusingFacebook JoinusingGoogle Joinusingemail Forgotpassword? Newuser? Signup Existinguser? Login × ProblemLoading... NoteLoading... SetLoading...