# Axiom:Axiom of Infinity

## Axiom

### Set Theory

There exists a set containing:

$(1): \quad$ a set with no elements
$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$

### Class Theory

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.