Zermelo-Fraenkel Axioms -- from Wolfram MathWorld

The system of axioms 1-8 is called Zermelo-Fraenkel set theory, denoted "ZF." The system of axioms 1-8 minus the axiom of replacement (i.e., axioms 1-6 plus 8) ... Algebra AppliedMathematics CalculusandAnalysis DiscreteMathematics FoundationsofMathematics Geometry HistoryandTerminology NumberTheory ProbabilityandStatistics RecreationalMathematics Topology AlphabeticalIndex InteractiveEntries RandomEntry NewinMathWorld MathWorldClassroom AboutMathWorld ContributetoMathWorld SendaMessagetotheTeam MathWorldBook WolframWebResources » 13,781entries Lastupdated:FriDec172021 Created,developed,andnurtured by Eric WeissteinatWolfram Research FoundationsofMathematics > Axioms > FoundationsofMathematics > SetTheory > General SetTheory > MathWorldContributors > Szudzik > Zermelo-FraenkelAxioms TheZermelo-FraenkelaxiomsarethebasisforZermelo-Fraenkelsettheory.Inthefollowing(Jech1997,p. 1),standsfor exists,meansforall,standsfor"is anelementof,"forthe emptyset,forimplies, forAND,forOR,and for"isequivalent to." 1.AxiomofExtensionality:Ifandhavethesameelements, then. (1) 2.AxiomoftheUnorderedPair:Foranyandthereexistsa setthatcontainsexactlyand.(alsocalledAxiom ofPairing) (2) 3.AxiomofSubsets:Ifisaproperty (withparameter),thenforany andthereexistsa setthatcontainsall thosethathavetheproperty.(alsocalled AxiomofSeparationorAxiomofComprehension) (3) 4.AxiomoftheSumSet:Foranythereexistsa set,theunionofallelements of.(alsocalledAxiomofUnion) (4) 5.AxiomofthePowerSet:Foranythereexistsa set,thesetofallsubsetsof. (5) 6.AxiomofInfinity:Thereexistsaninfinite set. (6) 7.AxiomofReplacement:Ifisafunction, thenforanythereexistsaset. (7) 8.AxiomofFoundation:Everynonemptysethasan-minimalelement.(alsocalledAxiom ofRegularity) (8) 9.AxiomofChoice:Everyfamilyofnonemptysets hasachoicefunction. (9) Thesystemofaxioms1-8iscalledZermelo-Fraenkelsettheory,denoted"ZF."Thesystemofaxioms1-8minustheaxiom ofreplacement(i.e.,axioms1-6plus8)iscalledZermelo settheory,denoted"Z."Thesetofaxioms1-9withtheaxiom ofchoiceisusuallydenoted"ZFC." Unfortunately,thereseemstobesomedisagreementintheliteratureaboutjustwhataxiomsconstitute"Zermelosettheory." Mendelson(1997)doesnotincludetheaxioms ofchoiceorfoundationinZermeloset theory,butdoesincludetheaxiomofreplacement. Enderton(1977)includestheaxiomsofchoiceand foundation,butdoesnotincludethe axiomofreplacement.Itôincludesan Axiomoftheemptyset,whichcanbegotten from(6)and(3),viaand . Abian(1969)provedconsistencyandindependence offouroftheZermelo-Fraenkelaxioms. SEEALSO:AxiomofChoice,AxiomofExtensionality,AxiomofFoundation, AxiomofInfinity,Axiom ofthePowerSet,AxiomofReplacement, AxiomofSubsets,Axiom oftheUnorderedPair,SetTheory,von Neumann-Bernays-GödelSetTheory,Zermelo-Fraenkel SetTheory,ZermeloSetTheory REFERENCES: Abian,A."OntheIndependenceofSetTheoreticalAxioms."Amer.Math. Monthly76,787-790,1969. Devlin,K.TheJoyofSets:FundamentalsofContemporarySetTheory,2nded.NewYork:Springer-Verlag, 1993. Enderton,H. B.Elements ofSetTheory.NewYork:AcademicPress,1977. Itô,K.(Ed.)."Zermelo-FraenkelSetTheory."§33BinEncyclopedic DictionaryofMathematics,2nded.,Vol. 1.Cambridge,MA:MITPress, pp. 146-148,1986. Iyanaga,S.andKawada,Y.(Eds.)."Zermelo-FraenkelSetTheory."§35BinEncyclopedic DictionaryofMathematics,Vol. 1.Cambridge,MA:MITPress,pp. 134-135, 1980. Jech,T.Set Theory,2nded.NewYork:Springer-Verlag,1997. Mendelson,E.Introduction toMathematicalLogic,4thed.London:Chapman&Hall,1997. Zermelo,E."ÜberGrenzzahlenundMengenbereiche."Fund.Math.16, 29-47,1930. ReferencedonWolfram|Alpha:Zermelo-FraenkelAxioms CITETHISAS: Weisstein,EricW."Zermelo-FraenkelAxioms." FromMathWorld--AWolframWebResource.https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html WolframWebResources Mathematica » The#1toolforcreatingDemonstrationsandanythingtechnical. Wolfram|Alpha » Exploreanythingwiththefirstcomputationalknowledgeengine. WolframDemonstrationsProject » Explorethousandsoffreeapplicationsacrossscience,mathematics,engineering,technology,business,art,finance,socialsciences,andmore. Computerbasedmath.org » Jointheinitiativeformodernizingmatheducation. OnlineIntegralCalculator » SolveintegralswithWolfram|Alpha. Step-by-stepSolutions » Walkthroughhomeworkproblemsstep-by-stepfrombeginningtoend.Hintshelpyoutrythenextsteponyourown. WolframProblemGenerator » Unlimitedrandompracticeproblemsandanswerswithbuilt-inStep-by-stepsolutions.Practiceonlineormakeaprintablestudysheet. WolframEducationPortal » CollectionofteachingandlearningtoolsbuiltbyWolframeducationexperts:dynamictextbook,lessonplans,widgets,interactiveDemonstrations,andmore. WolframLanguage » Knowledge-basedprogrammingforeveryone. ContacttheMathWorldTeam ©1999-2021WolframResearch,Inc.|TermsofUse THINGSTOTRY: axioms (2*3+3*4+4*5)/(10-5) circle