{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThe primitive objects are points, lines and planes.\nThe primitive relations are incidence, betweenness of three points, and congruence $\\cong$ of segments and of angles.\n*Hilbert's axioms* are the following five groups of axioms on these notions.\n\n1. Incidence: two distinct points lie on exactly one [[line|0MP9BCF6DP]]. Three non-collinear points lie on exactly one [[plane|0V3XAMF3C1]]. A line with two points in a plane lies in the plane. Two planes with a common point have a second common point. Each line has at least two points, each plane has three non-collinear points, and there are four points not in one plane.\n2. Order: the [[betweenness|CE8GX6ADZS]] relation is symmetric in $A$ and $C$ and holds only for three distinct collinear points. For distinct $A$ and $C$ some point $B$ has $C$ between $A$ and $B$. Of three collinear points, at most one lies between the other two. [[Pasch's axiom|KJT98G16Y4]] holds.\n3. Congruence: a segment can be laid off on a given ray in a unique way, and congruence of segments is transitive. Congruent adjacent segments add up to congruent segments. An angle can be laid off on a given side of a given ray in a unique way. If two triangles have two sides and the included angle congruent, then the other angles are congruent too.\n4. Parallels: [[Playfair's axiom|PR7Q3QMEHZ]] holds.\n5. Continuity: the axiom of Archimedes holds for segments. The points of a line admit no extension that keeps all the other axioms (line completeness).\n\n## Remarks\n\nExample: $\\R^3$ with its usual points, lines, planes, betweenness and congruence by isometries is a model.\nFor instance, Pasch's axiom holds there because a line in a plane is the zero set of an affine function.\nUp to isomorphism, it is the only model, so the system is categorical.\nNon-example: the space $\\Q^3$ with the same notions fails the first congruence axiom.\nThe segment from $(0,0,0)$ to $(1,1,0)$ has length $\\sqrt 2$, so it cannot be laid off on the positive $x$-axis from the origin, since $(\\sqrt 2, 0, 0) \\notin \\Q^3$.\nThe plane version keeps only the axioms of groups 1 to 5 that speak of one plane.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "KM1FB8XZ2R",
    "kind": "definition",
    "msc": [
      "51M05",
      "51F05",
      "03B30"
    ],
    "name": "Hilbert's axioms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert%27s_axioms"
      }
    ],
    "tag": "YHFDV8J1KV",
    "uses": [
      "0MP9BCF6DP",
      "0V3XAMF3C1",
      "CE8GX6ADZS",
      "KJT98G16Y4",
      "PR7Q3QMEHZ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YHFDV8J1KV/",
  "links": [
    {
      "field": "defines",
      "target": "KM1FB8XZ2R"
    },
    {
      "field": "uses",
      "target": "0MP9BCF6DP"
    },
    {
      "field": "uses",
      "target": "0V3XAMF3C1"
    },
    {
      "field": "uses",
      "target": "CE8GX6ADZS"
    },
    {
      "field": "uses",
      "target": "KJT98G16Y4"
    },
    {
      "field": "uses",
      "target": "PR7Q3QMEHZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YHFDV8J1KV.md",
  "msc": [
    "51M05",
    "51F05",
    "03B30"
  ],
  "name": "Hilbert's axioms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YHF/YHFDV8J1KV.md",
  "status": null,
  "tag": "YHFDV8J1KV",
  "uri": "m2://YHFDV8J1KV",
  "url": "https://tamnd.github.io/math2/wiki/hilberts-axioms-definition/",
  "writing": {
    "tag": "YHFDV8J1KV",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}