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Axiom of infinity

The integers are defined by an axiom that asserts the existence of a set $\omega$ that contains all the integers. $\omega$ is defined as the set containing $0$ and having the property that if $n$ is in $\omega$ then $n+1$ is in $\omega$. To write this compactly we define some notation. We use the integer 0 to represent the empty set. If we have some set $x$ then we can construct a set containing $x$ that is written as $\{x\}$.

$$\exists x \, 0 \in x \wedge [\forall y \, (y \in x) \rightarrow (y \cup \{y\} \in x)]$$

This says there exists a set $x$ that contains the empty set 0 and for every set $y$ that belongs to $x$ the set $y+1$ constructed as $y \cup \{y\}$ also belongs to $x$.