Definition
Peano arithmetic (PA) is the standard first-order axiomatization of the natural numbers, due to Giuseppe Peano in 1889 (building on Dedekind). Its axioms: 0 is a natural number, every natural has a successor, 0 is not a successor, the successor function is injective, and induction. From these, all of elementary arithmetic — addition, multiplication, divisibility, primality — can be derived.
Why it matters
How it works
The Peano axioms in first-order form: (1) 0 is a natural number. (2) For every n, S(n) is a natural number. (3) For all n, S(n) ≠ 0. (4) For all m, n, if S(m) = S(n) then m = n. (5) Induction: for every property P, if P(0) and ∀n: (P(n) ⊃ P(S(n))), then ∀n: P(n). Addition and multiplication are defined recursively: n + 0 = n, n + S(m) = S(n + m); n × 0 = 0, n × S(m) = n × m + n. Everything else follows.