# Axiom:Axiom of Infinity/Class Theory

## Axiom

### Formulation 1

Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction:

 $\ds 0$ $:=$ $\ds \O$ $\ds 1$ $:=$ $\ds 0 \cup \set 0$ $\ds 2$ $:=$ $\ds 1 \cup \set 1$ $\ds 3$ $:=$ $\ds 2 \cup \set 2$ $\ds$ $\vdots$ $\ds$ $\ds n + 1$ $:=$ $\ds n \cup \set n$ $\ds$ $\vdots$ $\ds$

Then $\omega$ is a set.

### Formulation 2

There exists an inductive set.

### Formulation 3

Not every set is a natural number.