ZermeloFraenkel axioms
Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a for what is now called ZermeloFraenkel set theory^{}, or ZF. If this set of axioms are accepted along with the Axiom of Choice^{}, it is often denoted ZFC.

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Equality of sets: If $X$ and $Y$ are sets, and $x\in X$ iff $x\in Y$, then $X=Y$.

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

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

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: If $X$ is a set, then there exists a set $\mathcal{P}(x)$ with the property that $Y\in \mathcal{P}(x)$ iff any element $y\in Y$ is also in $X$.

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Replacement axiom: Let $F(x,y)$ be some formula^{}. 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 $b\in B$ iff there exists some $a\in A$ such that $F(a,b)$ is true.

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: 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 $y\in Y$ is $F(y)$ true.

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Existence of an infinite set^{}: There exists a nonempty set $X$ with the property that, for any $x\in X$, there is some $y\in X$ such that $x\subseteq y$ but $x\ne y$.

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: 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  ZermeloFraenkel axioms 
Canonical name  ZermeloFraenkelAxioms 
Date of creation  20130322 11:47:51 
Last modified on  20130322 11:47:51 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  20 
Author  mathcam (2727) 
Entry type  Axiom 
Classification  msc 03E99 
Synonym  ZermeloFraenkel 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 