Axiom:Axiom of Infinity

Axiom
There exists a set containing:
 * $(1): \quad$ a set with no elements
 * $(2): \quad$ the successor of each of its elements.

That is:
 * $\exists x: \left({\left({\exists y: y \in x \land \forall z: \neg \left({z \in y}\right)}\right) \land \forall u: u \in x \implies u^+ \in x}\right)$

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

The symbol $\cup$ represents set union and $\left\{{u}\right\}$ represents the singleton containing $u$.

In an axiomatization of set theory that includes the Axiom of the Empty Set, the above can be abbreviated to:


 * $\exists x: \varnothing \in x \land \forall u: \left({u \in x \implies u^+ \in x}\right)$

Also see

 * Definition:Infinite Successor Set