# Axiom:Axiom of Infinity/Class Theory

< Axiom:Axiom of Infinity(Redirected from Axiom:Axiom of Infinity (Class Theory))

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## 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.