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 $\mathrm{\varnothing }\in S$ 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:

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

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.

Title	axiom of infinity
Canonical name	AxiomOfInfinity
Date of creation	2013-03-22 13:43:52
Last modified on	2013-03-22 13:43:52
Owner	Sabean (2546)
Last modified by	Sabean (2546)
Numerical id	6
Author	Sabean (2546)
Entry type	Axiom
Classification	msc 03E30
Synonym	infinity