Set Theory/Zermelo-Fraenkel Axiomatic Set Theory

The axioms

• Extensionality, two sets with the same elements are equal.

$\forall x,y,z \ (z \in x \Leftrightarrow z \in y) \Rightarrow (x=y)$

• Separation, subsets exist

$\forall y_1,p \ \exists y_2 \ \forall x\ x\in y_2 \Leftrightarrow (p \wedge x\in y_1)$
where p is any proposition

• The empty set exists

$\exists x \ \forall y\ y\not\in x$

• Union, the union of all members of a set is a set.

$\forall x\ \exists y\ \forall z \ z\in y \Leftrightarrow (\exists u \ z\in u \wedge u\in x)$

• Power sets exist

$\forall x \ \exists y \ \forall z \ z\in y \Leftrightarrow (\forall t \ t \in z \Rightarrow t \in x)$
we denote this set y by P(x)

• Infinity, an infinite set exists

$\exists x \ (\empty \in x) \wedge ( \forall y \ y\in x \Rightarrow P(y)\in x)$

• Foundation, no set is a member of itself

$\forall x \ x \ne \empty \Rightarrow (\exists y \in x \ y \cap x = \empty)$