# 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 what constitutes an infinite set when we have not yet defined the notion of a finite 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 upon the concept of number. We do this by introducing the notion of an inductive set. A set $S$ is said to be inductive if $\emptyset\in S$ and for every $x\in S$, $x\cup\{x\}\in S$. We may then state the Axiom of Infinity as follows:

There exists an inductive set.

In symbols:

 $\exists S[\emptyset\in S\land(\forall x\in S)[x\cup\{x\}\in S]]$

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

1. 1.

There exists an inductive set.

2. 2.

There exists an infinite set.

3. 3.

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

Title axiom of infinity AxiomOfInfinity 2013-03-22 13:43:52 2013-03-22 13:43:52 Sabean (2546) Sabean (2546) 6 Sabean (2546) Axiom msc 03E30 infinity