# Natural Number

**Codex:** Mathematics Codex · thecodex.expert/mathematics/  
**Domain:** 02 — Number Systems & Number Theory  
**Type:** Concept  
**URL:** https://thecodex.expert/mathematics/number-theory/natural-number/  
**Last reviewed:** 2026-05-27 · **Confidence:** High

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## One-line summary

The counting numbers (1, 2, 3, …) — the oldest mathematical objects in human history, independently invented by every known civilisation.

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## Definition

A **natural number** is a member of the set ℕ used in counting and ordering. Formally characterised by Peano's axioms (1889) and constructible from the empty set in ZFC set theory. **Convention:** ISO 80000-2 (2019) defines ℕ = {0, 1, 2, 3, …} (including zero). Traditional number theory often uses ℕ = {1, 2, 3, …}.

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## Curious

Humans were counting before we had language. The Lebombo bone (c. 43,000 years ago, Eswatini) and the Ishango bone (c. 20,000 years ago, Congo) are among the earliest physical evidence. Every known human civilisation independently developed counting: Babylonian (base 60), Egyptian (decimal), Indian (Vedic Sanskrit number words), Chinese (rod numerals), Maya (base 20 with independent zero). Giuseppe Peano wrote down five formal axioms for the natural numbers in 1889. Whether zero is a natural number remains a genuine convention disagreement — the ISO standard says yes; many number theorists say no.

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## Exploring

**Peano's axioms (1889):** 0 is a natural number. Every natural number n has a unique successor S(n). 0 is not a successor. The successor function is injective. Induction: if a property holds for 0 and passes from n to S(n), it holds for all natural numbers. From these, addition and multiplication are defined recursively.

**Civilisations:** Babylonian (base 60, positional), Egyptian (Rhind Papyrus, additive decimal), Indian Vedic (decimal, enormous numbers named), Chinese (Jiuzhang Suanshu), Maya (base 20, independent zero).

**Structure:** ℕ forms a commutative semiring under + and ×. Not a ring (no additive inverses). Cardinality: ℵ₀ (countably infinite).

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## Deep Dive

**Von Neumann ordinals in ZFC:** 0 = ∅, S(n) = n ∪ {n}. So 1 = {∅}, 2 = {∅, {∅}}, etc. The Axiom of Infinity guarantees ω = ℕ exists.

**Peano Arithmetic (PA):** First-order theory, incomplete by Gödel's theorem. Paris-Harrington (1977) gives a combinatorial true statement about ℕ not provable in PA.

**Categoricity:** The second-order Peano axioms characterise ℕ up to isomorphism (Dedekind, 1888).

**Euclid's proof of infinitely many primes (c. 300 BCE):** Suppose finitely many: p₁,…,pₙ. Let N = (p₁×…×pₙ) + 1. N has a prime factor not in the list. Contradiction. (Euclid, *Elements*, Book IX, Prop. 20; Heath trans., CUP, 1908.)

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## Sources

1. [Tier 1] Peano, G. *Arithmetices principia, nova methodo exposita*. Turin, 1889.
2. [Tier 1] Dedekind, R. *Was sind und was sollen die Zahlen?* Brunswick, 1888.
3. [Tier 1] ISO 80000-2: Quantities and Units — Mathematics. ISO, Geneva, 2019.

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**Cite as:** "Natural Number", Mathematics Codex, https://thecodex.expert/mathematics/number-theory/natural-number/, last updated 2026-05-27.

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