The Peano axioms are five axioms that characterize the natural numbers. They state: (1) zero is a natural number, (2) every natural number has a successor, (3) zero is not the successor of any natural number, (4) the successor function is injective (distinct numbers have distinct successors), and (5) if a property holds for zero and is preserved by successor, it holds for all natural numbers (induction).
The first four axioms establish the basic structure: a starting point and a way to generate new numbers from old. The fifth — the induction axiom — is the most powerful: it guarantees that the natural numbers contain nothing beyond what zero and successor generate. Without induction, models could contain extra elements (“nonstandard” natural numbers) unreachable from zero.
The Peano axioms support recursion: any function on ℕ can be defined by specifying its value at zero and its behavior on successors, and the induction axiom guarantees this definition covers all natural numbers. In category theory, the natural numbers object encodes the Peano axioms through a universal property: any object with a “zero” and a “successor” receives a unique morphism from ℕ.