ZF Set Theory Axiom of Infinity

Question

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.

Answer 1

The axiom of infinity says that there exists a set $A$ such that $\varnothing\in A$, that is the empty set is an element of $A$, and for every $x\in A$ the set $x\cup\{x\}$ is also an element of $A$.

The definable function $f(x)=x\cup\{x\}$ is an injection from $A$ into itself, and since $f(x)\neq\varnothing$ for every $x$, it follows that $f$ is not surjective. Therefore $A$ must be infinite.

Do note that $\{\varnothing\}$ is not the empty set, and so $\varnothing\cup\{\varnothing\}=\{\varnothing\}\neq\varnothing$.

Answer 2

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.