The natural numbers $\mathbb{N}$ (von Neumann)

The set of natural numbers is the smallest inductive set (Inductive set),

$\mathbb{N}=\omega :=\bigcap \{I:I\text{ is inductive}\},$

which exists and is itself inductive by The natural numbers exist: a smallest inductive set (the Axiom of Infinity, The Axiom of Infinity: there is a set containing a set with no elements and closed under $y\mapsto y\cup \{y\}$, supplies one inductive set to intersect within, and Separation, The Axiom Schema of Separation: for each formula $\phi$, $\mathrm{\forall }\bar{p}\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow (z\in x\wedge \phi (z,\bar{p})))$, makes the intersection a set). On $\mathbb{N}$ we take

$0:=\mathrm{\varnothing },\sigma (n):=n\cup \{n\},$

the distinguished element and the successor function. Thus $0=\mathrm{\varnothing }$, $1=\{0\}$, $2=\{0,1\}$, $3=\{0,1,2\}$, and in general $n=\{0,1,\dots ,n-1\}$ is the set of its predecessors.

Inductive set

The natural numbers exist: a smallest inductive set

The Axiom of Infinity: there is a set containing a set with no elements and closed under $y\mapsto y\cup \{y\}$

The Axiom Schema of Separation: for each formula $\phi$, $\mathrm{\forall }\bar{p}\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow (z\in x\wedge \phi (z,\bar{p})))$

"Smallest" means $\omega \subseteq I$ for every inductive set $I$. This minimality is exactly the induction principle (The principle of mathematical induction): a subset of $\mathbb{N}$ that contains $0$ and is closed under $\sigma$ is itself inductive, hence contains $\omega =\mathbb{N}$, hence equals $\mathbb{N}$.

With $0$ and $\sigma$ so defined, $(\mathbb{N},0,\sigma )$ satisfies the Peano axioms (Peano system, proved in The von Neumann naturals form a Peano system), so it is a model of the abstract natural numbers. By categoricity (Categoricity: the natural numbers are unique up to unique isomorphism) any other model is uniquely isomorphic to it, so the particular set-theoretic encoding chosen here is immaterial to every arithmetic and order property that follows: those are developed from the Peano axioms, not from the sets themselves.