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单词 ZermeloFraenkelAxioms
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Zermelo-Fraenkel axioms


Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a for what is now called Zermelo-Fraenkel set theoryMathworldPlanetmath, or ZF. If this set of axioms are accepted along with the Axiom of ChoiceMathworldPlanetmath, it is often denoted ZFC.

  • Equality of sets: If X and Y are sets, and xX iff xY, then X=Y.

  • Pair set: If X and Y are sets, then there is a set Z containing only X and Y.

  • Union (http://planetmath.org/Union) over a set: If X is a set, then there exists a set that contains every element of each xX.

  • : If X is a set, then there exists a set 𝒫(x) with the property that Y𝒫(x) iff any element yY is also in X.

  • Replacement axiom: Let F(x,y) be some formulaMathworldPlanetmathPlanetmath. If, for all x, there is exactly one y such that F(x,y) is true, then for any set A there exists a set B with the property that bB iff there exists some aA such that F(a,b) is true.

  • : Let F(x) be some formula. If there is some x that makes F(x) true, then there is a set Y such that F(Y) is true, but for no yY is F(y) true.

  • Existence of an infinite setMathworldPlanetmath: There exists a non-empty set X with the property that, for any xX, there is some yX such that xy but xy.

  • : If X is a set and P is a condition on sets, there exists a set Y whose members are precisely the members of X satisfying P. (This axiom is also occasionally referred to as the ).

TitleZermelo-Fraenkel axioms
Canonical nameZermeloFraenkelAxioms
Date of creation2013-03-22 11:47:51
Last modified on2013-03-22 11:47:51
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id20
Authormathcam (2727)
Entry typeAxiom
Classificationmsc 03E99
SynonymZermelo-Fraenkel set theory
SynonymZFC
SynonymZF
Related topicAxiomOfChoice
Related topicRussellsParadox
Related topicVonNeumannOrdinal
Related topicAxiom
Related topicContinuumHypothesis
Related topicGeneralizedContinuumHypothesis
Related topicSetTheory
Related topicVonNeumannBernausGodelSetTheory
Related topicSet
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