Axiom:Axiom of Infinity

Axiom

There exists a set containing a set with no elements and the successor of each of its elements.

$\exists x: \left({\left({\exists y: y \in x \land \forall z: \neg \left({z \in y}\right)}\right) \land \forall u: u \in x \implies \left({u \cup \left\{{u}\right\} \in x}\right)}\right)$

In this context, the successor of the set $u$ is defined as $u \cup \left\{{u}\right\}$.

Note that the symbols $\cup$ and $\left\{\right\}$ are used here, whereas a strict presentation of this axiom would not use them, as they have not strictly speaking been defined.

Also see

• Infinite successor set

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html