Set Theory/Zermelo-Fraenkel Axiomatic Set Theory

The axioms

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

$\mathrm{\forall }x,y,z(z\in x\iff z\in y)\Rightarrow (x=y)$

• Separation, subsets exist

$\mathrm{\forall }{y}_1,p\mathrm{\exists }{y}_2\mathrm{\forall }xx\in y_2\iff (p\wedge x\in y_1)$
where p is any proposition

• The empty set exists

$\mathrm{\exists }x\mathrm{\forall }yy\in ̸x$

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

$\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }zz\in y\iff (\mathrm{\exists }ut\in u\wedge u\in x)$

• Power sets exist

$\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }zz\in y\iff (\mathrm{\forall }tt\in x\Rightarrow t\in y)$
we denote this set y by P(x)

• Infinity, an infinite set exists

$\mathrm{\exists }x(\mathrm{\varnothing }\in x)\wedge (\mathrm{\forall }yy\in x\Rightarrow P(y)\in x)$

• Foundation, no set is a member of itself

$\mathrm{\forall }xx\ne \mathrm{\varnothing }\Rightarrow (\mathrm{\exists }y\in xy\cap x=\mathrm{\varnothing })$