{
  "schema_version": "1.0",
  "codex": {"name": "Mathematics Codex", "url": "https://thecodex.expert/mathematics/", "domain": "mathematics", "llms_txt": "https://thecodex.expert/mathematics/llms.txt"},
  "entry": {
    "id": "natural-number", "status": "active",
    "url_html": "https://thecodex.expert/mathematics/number-theory/natural-number/",
    "url_json": "https://thecodex.expert/mathematics/number-theory/natural-number/index.json",
    "url_markdown": "https://thecodex.expert/mathematics/number-theory/natural-number/index.md",
    "name": "Natural Number", "type": "concept",
    "aliases": ["counting number", "positive integer", "whole number"],
    "domain": "02", "domain_name": "Number Systems & Number Theory",
    "msc2020": ["11-XX", "03Exx"],
    "one_line": "The counting numbers (1, 2, 3, …) — the oldest mathematical objects in human history, independently invented by every known civilisation.",
    "definition": {
      "canonical": "A natural number is a member of the set ℕ of non-negative (or positive) integers used in counting and ordering. Natural numbers are formally characterised by Peano's axioms (1889) and constructible from the empty set in ZFC via the von Neumann ordinal construction.",
      "summary": "The counting numbers — 1, 2, 3, … — the oldest mathematical objects in human history.",
      "is_contested": true,
      "contested_positions": [
        {"position": "ℕ = {0, 1, 2, 3, …} (includes zero)", "source": "ISO 80000-2, 2019"},
        {"position": "ℕ = {1, 2, 3, …} (excludes zero)", "source": "traditional number theory convention"}
      ]
    },
    "type_specific": {
      "definition_formal": "The second-order Peano axioms: 0 ∈ ℕ; S: ℕ → ℕ (successor); ∀n: S(n) ≠ 0; S is injective; ∀X: (0∈X ∧ ∀n∈X: S(n)∈X) → X=ℕ.",
      "definition_formal_latex": "\\mathbb{N} = \\{0, 1, 2, 3, \\ldots\\}",
      "examples": ["1, 2, 3, 4, 5 (basic counting)", "The Ishango bone tally marks (c. 20,000 BCE)", "The set {1, 2, 3, …, 100} has cardinality 100"],
      "notation": "ℕ (from German Natürliche Zahlen). |ℕ| = ℵ₀ (countably infinite)."
    },
    "prerequisites": [
      {"slug": "set", "canonical_name": "Set", "rel": "prerequisite", "note": "In ZFC, the natural numbers are constructed from sets via the von Neumann ordinal construction.", "url": "/mathematics/foundations-logic/set/"}
    ],
    "consequences": [
      {"slug": "integer", "canonical_name": "Integer", "rel": "consequence", "note": "The integers extend ℕ to include additive inverses, producing ℤ.", "url": "/mathematics/number-theory/integer/"},
      {"slug": "prime-number", "canonical_name": "Prime Number", "rel": "consequence", "note": "Primes are natural numbers greater than 1 with no divisors other than 1 and themselves.", "url": "/mathematics/number-theory/prime-number/"}
    ],
    "lineage": {
      "predecessors": [{"slug": "set", "canonical_name": "Set (foundational prerequisite)", "rel": "lineage_predecessor", "note": "The empty set is the set-theoretic starting point for the construction of natural numbers in ZFC.", "url": "/mathematics/foundations-logic/set/"}],
      "successors": [{"slug": "integer", "canonical_name": "Integer", "rel": "lineage_successor", "note": "Integers are the natural extension of ℕ to include negative numbers and zero.", "url": "/mathematics/number-theory/integer/"}]
    },
    "civilisations": ["Universal — Babylonian, Egyptian, Indian (Vedic), Chinese, Greek, Mayan, and all others"],
    "era": "Prehistoric (tallying) through 19th century (formal axiomatisation)",
    "sources": [
      {"id": "peano-1889", "tier": 1, "type": "book", "author": "Giuseppe Peano", "title": "Arithmetices principia, nova methodo exposita", "publisher": "Fratelli Bocca, Turin", "year": 1889},
      {"id": "dedekind-1888", "tier": 1, "type": "book", "author": "Richard Dedekind", "title": "Was sind und was sollen die Zahlen?", "publisher": "Vieweg, Brunswick", "year": 1888},
      {"id": "iso-80000-2-2019", "tier": 1, "type": "standards_body", "author": "International Organization for Standardization", "title": "ISO 80000-2: Quantities and Units — Part 2: Mathematics", "publisher": "ISO, Geneva", "year": 2019}
    ],
    "confidence": "high",
    "published": "2026-05-27", "last_reviewed": "2026-05-27", "review_due": "2027-05-27",
    "build_status": "complete",
    "citation": "\"Natural Number\", Mathematics Codex, https://thecodex.expert/mathematics/number-theory/natural-number/, last updated 2026-05-27."
  }
}