Inductive set

Working in ZFC, for a set $x$ write $x^{+}:=x\cup \{x\}$ for its successor (a set, by the axioms of Pairing and Union, The Axiom of Pairing: $\mathrm{\forall }x\mathrm{\forall }y\mathrm{\exists }z\mathrm{\forall }t(t\in z\leftrightarrow (t=x\vee t=y))$ and The Axiom of Union: $\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y\leftrightarrow \mathrm{\exists }s(s\in x\wedge z\in s))$). A set $I$ is inductive when

$\mathrm{\varnothing }\in I\text{and}(x\in I⟹x^{+}\in I)\text{ for every }x\mathrm{.}$

That is, $I$ contains the empty set and is closed under the successor operation.

The Axiom of Infinity (The Axiom of Infinity: there is a set containing a set with no elements and closed under $y\mapsto y\cup \{y\}$) is precisely the assertion that an inductive set exists. Inductive sets can be large and are far from unique (if $I$ is inductive so is $I\cup \{\text{anything}\}$ once closed off), so an inductive set is not yet a good definition of $\mathbb{N}$. The natural numbers are carved out as the smallest inductive set, the intersection of all of them (The natural numbers exist: a smallest inductive set, The natural numbers $\mathbb{N}$ (von Neumann)); minimality is what delivers the induction principle.

Here $0:=\mathrm{\varnothing }$ and the successor of $n$ is $\sigma (n):=n^{+}=n\cup \{n\}$, so $1=\{0\}$, $2=\{0,1\}$, $3=\{0,1,2\}$: each natural number is the set of all smaller natural numbers.