Natural Numbers

The natural numbers ℕ = {0, 1, 2, 3, …} are the numbers generated from zero by repeated application of the successor operation. They are characterized by the Peano axioms: zero is a natural number, every natural number has a successor, zero is not a successor, the successor function is injective, and induction holds.

The natural numbers support addition (defined recursively: n + 0 = n, n + S(m) = S(n + m)) and multiplication (defined recursively: n × 0 = 0, n × S(m) = n + (n × m)). Under these operations, ℕ forms a commutative semiring: addition and multiplication are associative and commutative, multiplication distributes over addition, 0 is the additive identity, and 1 is the multiplicative identity. There are no additive inverses — subtraction is not always possible in ℕ.

In set theory, the natural numbers are constructed as sets: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, and each successor is n ∪ {n}. In category theory, the natural numbers object is characterized by a universal property: an object N with maps 0: 1 → N and S: N → N such that any other such structure factors through N uniquely. This universal property captures recursion — it says that recursive definitions over ℕ always exist and are unique.