# Category:Axioms/Axiom of Infinity

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This category contains axioms related to Axiom of Infinity.

Related results can be found in Category:Axiom of Infinity.

### Set Theory

There exists a set containing:

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.

## Pages in category "Axioms/Axiom of Infinity"

The following 7 pages are in this category, out of 7 total.