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If a nonempty set carries a relation in which every point has a successor, the axiom of dependent choice supplies an infinite sequence following that relation. Full choice implies this by selecting a successor at every point and applying the recursion theorem on the natural numbers.

The axiom of infinity asserts that an inductive set exists: some set $I$ contains $0=\varnothing$ and contains $x^{+}=x\cup \{x\}$ whenever it contains $x$. Inside one such $I$, axiom of separation definesThis intersection is itself inductive and is contained in every inductive set, by intersecting that set with $I$. It is therefore the set of natural numbers. If a property holds at $0$ and passes from $n$ to $n^{+}$, axiom of separation forms the subset of $\omega$ on which it holds. That subset is inductive, so must equal $\omega$. This proves mathematical induction, including properties with set parameters.

Recursion on $\omega$ defines the arithmetic operations byExistence and uniqueness follow from the recursion theorem on the natural numbers.

For associativity of addition fix $m,n$ and induct on $k$. At $k=0$ both sides equal $m+n$. If the equality holds at $k$, thenThusEvery step after the induction hypothesis is a defining successor equation, so commutativity has not been used prematurely.