{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "peano-arithmetic",
    "name": "Peano Arithmetic",
    "subject": "arithmetic",
    "variant": "classical",
    "description": "Axiomatic development of natural number arithmetic. Five axioms, definitions of addition and multiplication, and key theorems (associativity, commutativity, distributivity, order). Based on Landau, Foundations of Analysis.",
    "structure": {
      "axioms": 5,
      "definitions": 2,
      "theorems": 25
    }
  },
  "metadata": {
    "created": "2026-03-15",
    "lastUpdated": "2026-03-15",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Peano Arithmetic Dependency Graph. Programming Framework.",
    "keywords": [
      "Peano",
      "arithmetic",
      "natural numbers",
      "induction",
      "foundations"
    ]
  },
  "sources": [
    {
      "id": "landau",
      "type": "primary",
      "authors": "Landau, E.",
      "title": "Foundations of Analysis",
      "year": "1930",
      "publisher": "Chelsea",
      "edition": "1951",
      "notes": "Canonical development"
    },
    {
      "id": "wikipedia",
      "type": "digital",
      "title": "Peano axioms",
      "url": "https://en.wikipedia.org/wiki/Peano_axioms",
      "notes": "Overview and definitions"
    }
  ],
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      "type": "axiom",
      "label": "0 is a natural number",
      "shortLabel": "A1",
      "short": "0 ∈ N",
      "colorClass": "axiom"
    },
    {
      "id": "A2",
      "type": "axiom",
      "label": "No predecessor of 0: S(x) ≠ 0",
      "shortLabel": "A2",
      "short": "0 not a successor",
      "colorClass": "axiom"
    },
    {
      "id": "A3",
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      "label": "Successor injective: S(x)=S(y) ⇒ x=y",
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      "short": "S injective",
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      "id": "A4",
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      "label": "Closure: S(x) ∈ N for all x ∈ N",
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      "short": "N closed under S",
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      "shortLabel": "A5",
      "short": "Induction",
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      "id": "T1",
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      "shortLabel": "T1",
      "short": "Contrapositive of A3",
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      "id": "T2",
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      "label": "S(x)≠x for all x",
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      "short": "Successor ≠ identity",
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      "id": "T3",
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      "label": "If x≠0 then x=S(u) for some u",
      "shortLabel": "T3",
      "short": "Every nonzero is successor",
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      "id": "DefAdd",
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      "label": "Addition: x+0=x, x+S(y)=S(x+y)",
      "shortLabel": "DefAdd",
      "short": "Definition of +",
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      "type": "theorem",
      "label": "Addition is well-defined for all x,y",
      "shortLabel": "T4",
      "short": "Add well-defined",
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      "id": "T5",
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      "label": "(x+y)+z = x+(y+z)",
      "shortLabel": "T5",
      "short": "Associativity of +",
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    {
      "id": "T6",
      "type": "theorem",
      "label": "0+x = x",
      "shortLabel": "T6",
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      "id": "T7",
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      "label": "S(x)+y = S(x+y)",
      "shortLabel": "T7",
      "short": "Successor and add",
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      "id": "T8",
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      "label": "x+y = y+x",
      "shortLabel": "T8",
      "short": "Commutativity of +",
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    {
      "id": "T9",
      "type": "theorem",
      "label": "x+y=x+z ⇒ y=z",
      "shortLabel": "T9",
      "short": "Cancellation for +",
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      "id": "DefMul",
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      "label": "Multiplication: x·0=0, x·S(y)=x·y+x",
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      "short": "Definition of ·",
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      "id": "T10",
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      "label": "Multiplication is well-defined for all x,y",
      "shortLabel": "T10",
      "short": "Mul well-defined",
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      "id": "T11",
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      "label": "x·0 = 0",
      "shortLabel": "T11",
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      "id": "T12",
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      "id": "T13",
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      "id": "T14",
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      "id": "T15",
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      "shortLabel": "T15",
      "short": "Associativity of ·",
      "colorClass": "theorem"
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    {
      "id": "T16",
      "type": "theorem",
      "label": "x·(y+z) = x·y + x·z",
      "shortLabel": "T16",
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      "id": "T17",
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      "label": "(x+y)·z = x·z + y·z",
      "shortLabel": "T17",
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    },
    {
      "id": "T18",
      "type": "theorem",
      "label": "x≤y iff ∃z x+z=y",
      "shortLabel": "T18",
      "short": "Order definition",
      "colorClass": "theorem"
    },
    {
      "id": "T19",
      "type": "theorem",
      "label": "Trichotomy: exactly one of x<y, x=y, y<x",
      "shortLabel": "T19",
      "short": "Trichotomy",
      "colorClass": "theorem"
    },
    {
      "id": "T20",
      "type": "theorem",
      "label": "x≤y ⇒ x+z≤y+z",
      "shortLabel": "T20",
      "short": "Order + add",
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    {
      "id": "T21",
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      "shortLabel": "T21",
      "short": "Order + mul",
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      "id": "T22",
      "type": "theorem",
      "label": "1·x = x (where 1=S(0))",
      "shortLabel": "T22",
      "short": "Multiplicative identity",
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    {
      "id": "T23",
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      "label": "x·1 = x",
      "shortLabel": "T23",
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      "id": "T24",
      "type": "theorem",
      "label": "Well-ordering: every nonempty subset has least element",
      "shortLabel": "T24",
      "short": "Well-ordering",
      "colorClass": "theorem"
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      "id": "T25",
      "type": "theorem",
      "label": "Strong induction principle",
      "shortLabel": "T25",
      "short": "Strong induction",
      "colorClass": "theorem"
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