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# 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. There exists an inductive set.

2. There exists an infinite set.

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

## Mathematics Subject Classification

03E30*no label found*

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