# Set Theory/Zermelo-Fraenkel Axiomatic Set Theory

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## The axioms

• Extensionality, two sets with the same elements are equal.
${\displaystyle \forall x,y,z:(z\in x\iff z\in y)\Rightarrow x=y}$
• Separation, subsets exist
${\displaystyle \forall y_{1},p\,\exists y_{2}:\forall x:x\in y_{2}\iff (p\land x\in y_{1})}$
where ${\displaystyle p}$ is any proposition
• The empty set exists
${\displaystyle \exists x:\forall y:y\notin x}$
• Union, the union of all members of a set is a set.
${\displaystyle \forall x\,\exists y:\forall z:z\in y\iff (\exists u:z\in u\land u\in x)}$
• Power sets exist
${\displaystyle \forall x\,\exists y:\forall z:z\in y\iff (\forall t:t\in z\Rightarrow t\in x)}$
we denote this set ${\displaystyle y}$ by ${\displaystyle P(x)}$
• Infinity, an infinite set exists
${\displaystyle \exists x:(\varnothing \in x)\land (\forall y:y\in x\Rightarrow P(y)\in x)}$
• Foundation, no set is a member of itself
${\displaystyle \forall x:x\neq \varnothing \Rightarrow (\exists y\in x:y\cap x=\varnothing )}$