The axioms characterizing the natural numbers — zero is a natural number, every natural number has a successor, zero is not the successor of any number, different numbers have different successors, and induction — providing the standard foundation for arithmetic.
The second-order Dedekind-Peano axioms characterize the natural numbers up to isomorphism (categorically); this is philosophically significant because it means there is essentially one structure satisfying the axioms — the natural numbers.
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