Axiom:Axiom of Infinity

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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:

\(\displaystyle 0\) \(:=\) \(\displaystyle \O\)
\(\displaystyle 1\) \(:=\) \(\displaystyle 0 \cup \set 0\)
\(\displaystyle 2\) \(:=\) \(\displaystyle 1 \cup \set 1\)
\(\displaystyle 3\) \(:=\) \(\displaystyle 2 \cup \set 2\)
\(\displaystyle \) \(\vdots\) \(\displaystyle \)
\(\displaystyle n + 1\) \(:=\) \(\displaystyle n \cup \set n\)
\(\displaystyle \) \(\vdots\) \(\displaystyle \)

Then $\omega$ is a set.