Axiom:Axiom of Infinity

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

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

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.