Natural numbers

The natural numbers, or counting numbers, are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes zero.

If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite ordinals.

Encodings

The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. In many areas such as geometry or number theory, they are taken as primitives and not formally treated

Von Neumann ordinals

In ZFC and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \).

Zermelo ordinals

Ernst Zermelo provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals.

Frege and Russell

During the early development of foundational philosophy and logicism, Gottlob Frege and Bertrand Russell proposed defining a natural number \( n \) as the equivalence class of all sets with cardinality \( n \). This definition cannot be realized in ZFC, because the classes involved are proper classes, except for \( n = 0 \).

Church numerals

In the lambda calculus, the standard way to encode natural numbers is as Church numerals, developed by Alonzo Church. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc.

Theories of arithmetic

Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are Peano arithmetic and second-order arithmetic.

Algebraic properties of the natural numbers

The natural numbers (including zero) are closed under addition and multiplication. They satisfy commutativity and associativity of both operations, and distributivity of multiplication over addition. They form a monoid under addition, which is the free monoid with one generator. In addition, positive naturals form a monoid under multiplication -- the free monoid with countably infinite generators, which are the prime numbers.