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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  is said to be inductive if
 and for every  ,
 . We may then state the Axiom of Infinity as follows:
There exists an inductive set.
In symbols:
We shall then be able to prove that the following conditions are equivalent:
- There exists an inductive set.
- There exists an infinite set.
- The least nonzero limit ordinal,
 , is a set.
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![$\displaystyle \exists S [\emptyset \in S \land (\forall x \in S)[x \cup \{ x \} \in S]] $](http://images.planetmath.org:8080/cache/objects/4419/l2h/img5.png "$\displaystyle \exists S [\emptyset \in S \land (\forall x \in S)[x \cup \{ x \} \in S]] $"){kind=small}