The successor operation S takes a natural number n and produces the next natural number S(n). In the Peano axioms, successor is the generating operation: starting from zero and applying S repeatedly produces all natural numbers. Two of Peano’s axioms constrain S: it is injective (S(n) = S(m) implies n = m) and zero is not in its range (there is no n with S(n) = 0).
Successor is the foundation for defining arithmetic by recursion. Addition is defined as: n + 0 = n, n + S(m) = S(n + m). Multiplication is defined as: n × 0 = 0, n × S(m) = n + (n × m). Each operation is built from successor and the previous operations, making successor the single primitive from which all of arithmetic grows.
In set theory, the successor of n is S(n) = n ∪ {n}: the set n together with itself as a new element. In category theory, successor is a morphism S: N → N that, together with the zero map 0: 1 → N, characterizes the natural numbers object by its universal property.