ZF Set Theory Axiom of Infinity Could someone please state and explain the axiom of infinity in ZF set theory?  This isn't homework, it's just something that has interested me for awhile.

In ZF-Infinity, $\omega=\{\alpha\mid\alpha\text{ and its elements are either }0\text{ or successor}\}=\mathbb{N}$ can be either a set or not a set (proper class). The Axiom of Infinity basically  postulate that $\omega$ is indeed a set, which enable set theory to deal with all sorts of infinite sets. Without it, one can only prove finite sets exist.
