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 ).

Title Zermelo-Fraenkel axioms
Canonical name ZermeloFraenkelAxioms
Date of creation 2013-03-22 11:47:51
Last modified on 2013-03-22 11:47:51
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 20
Author mathcam (2727)
Entry type Axiom
Classification msc 03E99
Synonym Zermelo-Fraenkel set theory
Synonym ZFC
Synonym ZF
Related topic AxiomOfChoice
Related topic RussellsParadox
Related topic VonNeumannOrdinal
Related topic Axiom
Related topic ContinuumHypothesis
Related topic GeneralizedContinuumHypothesis
Related topic SetTheory
Related topic VonNeumannBernausGodelSetTheory
Related topic Set