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Zermelo–Fraenkel set theory

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC)…

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Axioms

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Zermelo–Fraenkel set theory

Nodes141
Edges140
Triples40
Avg. degree1.99
Density0.014184
Components1

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Zermelo–Fraenkel set theory

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related to External links · 29
Zermelo–Fraenkel set theory → Axioms, Bagaria, Edward, EMS Press, Encyclopedia, Eric, Fraenkel Set Theory, Frederic Schuller, In, In Zalta, ISSN, January, Joan, Joan Bagaria, Lec, Mathematics, MathWorld, Metamath, OCLC, Philosophy

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set zfc axiom axioms theory displaystyle sets choice zermelo fraenkel exists existence one zf also universe schema union consistency classes

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Russell's paradoxinstance ofis an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes0.80text
ZFC cannot be proved within the theory itselfinstance ofThe consistency of a theory0.80text
as shown by Gödel's second incompleteness theoreminstance ofThe consistency of a theory0.80text
Von Neumanninstance ofand that the powerset of x will be added at the next stage after α.The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and rela…0.80text
New Foundations.It is possible to change the definition of V so that at each stageinstance ofThe cumulative hierarchy is not compatible with other set theories0.80text
instead of adding all the subsets of the union of the previous stagesinstance ofThe cumulative hierarchy is not compatible with other set theories0.80text
subsets are only added if they are definable in a certain senseinstance ofThe cumulative hierarchy is not compatible with other set theories0.80text
proper classes.Many mathematical theorems can be proven in much weaker systems than ZFCinstance ofas well as for its failure to capture objects0.80text
such as Peano arithmeticinstance ofas well as for its failure to capture objects0.80text
second-order arithmeticinstance ofas well as for its failure to capture objects0.80text
Martin's axiom or large cardinal axioms to ZFCinstance ofSome of these conjectures are provable with the addition of axioms0.80text
Zermelo–Fraenkel set theoryrelated to External linksAxioms0.60section

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