axiom of infinity

There exists an infinite set.

The Axiom of Infinity is an axiom of Zermelo-Fraenkel set theory.At first glance, this axiom seems to be ill-defined. How are we to know whatconstitutes an infinite set when we have not yet defined the notion of afinite set? However, once we have a theory of ordinal numbers in hand, the axiom makes sense.

Meanwhile, we can give a definition of finiteness that does not rely uponthe concept of number. We do this by introducing the notion of an inductiveset. A set $S$ is said to be inductive if $\mathrm{\varnothing }\in S$and for every $x\in S$, $x\cup \{x\}\in S$. We may then state theAxiom of Infinity as follows:

There exists an inductive set.

In symbols:

$$\exists S[\mathrm{\varnothing }\in S\wedge (\forall x\in S)[x\cup \{x\}\in S]]$$

We shall then be able to prove that the following conditions are equivalent:

1. There exists an inductive set.

2. There exists an infinite set.

3. The least nonzero limit ordinal, $\omega$, is a set.