Peano axioms

The Peano axioms characterize the natural numbers using zero, a successor operation, and induction. First-order and second-order versions differ in expressive strength and categoricity.

Natural number: A number used for counting or ordering, conventionally beginning with zero or one. The axioms characterize this number system.
Peano arithmetic: A first-order formal theory of natural numbers with addition, multiplication, and induction. It develops the first-order Peano axioms into an arithmetic theory.
Giuseppe Peano: An Italian mathematician who formalized foundational principles for arithmetic in the late nineteenth century. He published the influential axiom system that bears his name.
Zermelo–Fraenkel set theory: An axiomatic set theory used as a foundation for much of modern mathematics. It can construct the natural numbers as sets rather than taking them as primitive objects.
Successor function: A function that assigns each natural number its next number. The axioms use successor to generate the numbers and distinguish them.
Categoricity: The property of a theory having exactly one model up to isomorphism. Second-order Peano axioms are categorical under full second-order semantics; first-order axioms are not.
Richard Dedekind: A German mathematician who developed foundational accounts of number and continuity. His account of the natural numbers anticipated the structural ideas formalized by Peano.
Second-order arithmetic: A formal system combining arithmetic with variables for sets or properties of natural numbers. It makes explicit the stronger resources used by second-order formulations.
Mathematical induction: A proof method showing a property holds for all natural numbers from a base case and a successor step. The induction principle is the axioms’ key rule for extending properties across the numbers.
Nonstandard model: A model of arithmetic containing elements that are not any standard natural number. First-order Peano arithmetic has models beyond the ordinary natural numbers.

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