Peano Axiomatic Construction of N (Natural Numbers)

$0$ is a natural number

For every natural number $x$, $x=x$. That is, equality is reflexive.

For all natural numbers $x$ and $y$, if $x=y$, then $y=x$. That is, equality is symmetric.

For all natural numbers $x$, $y$, and $z$, if $x=y$ and $y=z$, then $x=z$. That is, equality is transitive.

For all $a$ and $b$, if $a$ is a natural number and $a=b$, then $b$ is also a natural number. that is, the natural numbers are closed under equality.

Define a successor function $\mathcal{S}$ in the natural numbers. For every natural number $n$, $\mathcal{S}(n)$ is a natural number.

For every natural number $n$, $\mathcal{S}(n)=0$ is false. That is, there exists no natural number whose successor is $0$.

For all natural numbers $m$ and $n$, if $\mathcal{S}(m)=\mathcal{S}(n)$, then $m=n$. That is, $\mathcal{S}$ is an injection.

The Inductive Axiom: If $\mathcal{K}$ is a set such that $0 \in \mathcal{K}$ and, for all natural numbers $n$, if $n \in \mathcal{K}$, then $\mathcal{S}(n)\in \mathcal{K}$, then $\mathcal{K}$ contains every natural number.