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 $\varphi$, $\forall \bar p\,\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow (z \in x \wedge \varphi(z,\bar p)))$, makes the intersection a set). On $\mathbb{N}$ we take

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

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

"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.