# Zermelo-Fraenkel axioms

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

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

• : 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 $X$.

• 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.

• : 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.

• Existence of an infinite set: There exists a non-empty set $X$ with the property that, for any $x\in X$, there is some $y\in X$ such that $x\subseteq y$ but $x\neq y$.

• : 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