axiom of infinity
There exists an infinite set^{}.
The Axiom of Infinity^{} is an axiom of ZermeloFraenkel set theory^{}. At first glance, this axiom seems to be illdefined. 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  20130322 13:43:52 
Last modified on  20130322 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 