naturals

Naturals, in mathematics, refer to the natural numbers. Depending on convention, this set is {0,1,2,3,...} or {1,2,3,...}. In modern formal contexts most authors include 0, denoting the set by N and treating it as the starting point for counting and arithmetic.

The natural numbers are typically introduced via axioms such as the Peano axioms. These assert that 0 is in N, every n in N has a successor S(n) in N, 0 is not the successor of any number, distinct numbers have distinct successors, and the induction principle. From these axioms, N is closed under addition and multiplication and is totally ordered in a way compatible with these operations.

N is a countably infinite set and serves as the foundation for many mathematical theories. Core properties include closure under addition and multiplication, associativity, commutativity, and distributivity of these operations, and the well-ordering principle: every nonempty subset of N has a least element. Induction provides a rigorous method for proving statements about all natural numbers. The natural numbers underpin core results in number theory, such as prime factorization and the Euclidean algorithm for gcd.

Notation and representation vary: N or ω is standard; in set theory, numbers can be represented as von Neumann ordinals, with 0 = ∅, 1 = {0}, 2 = {0,1}, etc.

Historically, counting practices led to natural numbers, with formalization occurring in the 19th and 20th centuries, notably by Peano and later by Frege, Dedekind, and Hilbert.