{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThe provability logic $\\mathrm{GL}$ is the smallest normal [[modal logic|C3RZY20ETK]] that contains Löb's axiom\n$$\\square(\\square p \\to p) \\to \\square p.$$\nIts intended reading interprets $\\square$ as provability in [[Peano arithmetic|0QFT7Z76N5]].\nAn arithmetical realization $*$ maps each propositional variable to a sentence of arithmetic and commutes with the connectives.\nIt sets $(\\square\\varphi)^* = \\mathrm{Prov}(\\ulcorner \\varphi^* \\urcorner)$, where $\\mathrm{Prov}$ is the provability predicate of Peano arithmetic and $\\ulcorner \\cdot \\urcorner$ is a [[Gödel numbering|MFJXPAF0R0]].\n\n## Remarks\n\nSolovay's theorem (1976) says: a modal formula $\\varphi$ is in $\\mathrm{GL}$ if and only if Peano arithmetic proves $\\varphi^*$ for every arithmetical realization $*$.\n\nExample: $\\square p \\to \\square\\square p$ is in $\\mathrm{GL}$.\nIts realizations are instances of the fact that Peano arithmetic proves \"if $\\sigma$ is provable, then it is provable that $\\sigma$ is provable\".\n\nNon-example: $\\square p \\to p$ is not in $\\mathrm{GL}$.\nUnder a realization with $p^* = (0 = 1)$, it becomes the consistency statement of Peano arithmetic, which Peano arithmetic does not prove by Gödel's second incompleteness theorem.\n\nOn Kripke frames, $\\mathrm{GL}$ is the logic of the finite transitive irreflexive frames.\nThe name \"provability logic\" also covers the study of related systems for other theories and other provability notions.\n\n## Explanations\n\n### L1 human\n\nThe system $\\mathrm{GL}$ is named after Gödel and Löb.\nIts one extra axiom is the modal form of [[Löb's theorem|3HD30YG3DG]].\nIt captures every general law of provability that Peano arithmetic can prove about itself.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "WJK0YA4GP2",
    "kind": "definition",
    "msc": [
      "03F45",
      "03B45"
    ],
    "name": "Provability logic",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Provability_logic"
      }
    ],
    "tag": "61THBNS8WE",
    "uses": [
      "C3RZY20ETK",
      "0QFT7Z76N5",
      "MFJXPAF0R0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/61THBNS8WE/",
  "links": [
    {
      "field": "defines",
      "target": "WJK0YA4GP2"
    },
    {
      "field": "uses",
      "target": "C3RZY20ETK"
    },
    {
      "field": "uses",
      "target": "0QFT7Z76N5"
    },
    {
      "field": "uses",
      "target": "MFJXPAF0R0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/61THBNS8WE.md",
  "msc": [
    "03F45",
    "03B45"
  ],
  "name": "Provability logic",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/61T/61THBNS8WE.md",
  "status": null,
  "tag": "61THBNS8WE",
  "uri": "m2://61THBNS8WE",
  "url": "https://math2.tamnd.com/wiki/provability-logic-definition/",
  "writing": {
    "tag": "61THBNS8WE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}