{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "JMFHAMX0JG"
    }
  ],
  "body": "\n## Summary\n\nTarski's axioms are an axiom system for plane [[Euclidean geometry|KRZD96GSVC]] in first-order logic, due to Alfred Tarski.\nThey speak only of points and of two relations: [[betweenness|CE8GX6ADZS]] of three points and congruence of two segments.\nTarski proved that the resulting [[theory|92SWXST1ZM]] is [[complete|655K8Y7VS8]] and decidable: an algorithm decides, for each sentence, whether it is a theorem.\n\n## Explanations\n\n### L1 human\n\nTarski's axioms describe plane geometry with only points, no lines or circles as basic objects.\nTwo relations are enough: \"$y$ is between $x$ and $z$\" and \"the segment $xy$ is as long as the segment $zw$\".\nA line, for example, is the set of points $y$ such that one of $x$, $y$, $z$ is between the other two, for two fixed points $x \\ne z$.\nBecause the theory is decidable, every elementary question about the plane can be decided by a computer in principle.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "51M05",
      "03B30",
      "03B25"
    ],
    "name": "Tarski's axioms",
    "same_as": {
      "wikidata": "Q7686757",
      "wikipedia": "Tarski's axioms"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tarski%27s_axioms"
      }
    ],
    "tag": "14DWEDXWCC"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/14DWEDXWCC/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/14DWEDXWCC.md",
  "msc": [
    "51M05",
    "03B30",
    "03B25"
  ],
  "name": "Tarski's axioms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/14D/14DWEDXWCC.md",
  "status": null,
  "tag": "14DWEDXWCC",
  "uri": "m2://14DWEDXWCC",
  "url": "https://tamnd.github.io/math2/wiki/tarskis-axioms/",
  "writing": {
    "tag": "14DWEDXWCC",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}