{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "E6MB6B3YQ0"
    }
  ],
  "body": "\n## Summary\n\nRealizability is a way to read intuitionistic statements as statements about programs.\nA realizer of a formula is a program, often the code of a [[computable function|KZSB5E471K]], that carries the computational content of a proof of it.\nKleene introduced number realizability for [[Heyting arithmetic|CK8ZQRN183]] in 1945, as a precise form of the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].\nRealizability shows that intuitionistic proofs of existence yield algorithms, and it gives models in which classical principles fail.\n\n## Explanations\n\n### L1 human\n\nIn constructive mathematics, to prove \"for every $n$ there is an $m$\" one must in effect give a way to find $m$ from $n$.\nRealizability makes \"a way to find\" precise: it is a program.\nA program realizes a statement when it provides all the data the statement asks for.\nFor example, a program that sends $n$ to $2n$, together with a trivial check that $2n = n + n$, realizes \"every number has a double\".\nEvery theorem of Heyting arithmetic has a realizer, so its existence theorems come with algorithms.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03F50",
      "03F55",
      "03D80"
    ],
    "name": "Realizability",
    "same_as": {
      "wikidata": "Q3454417",
      "wikipedia": "Realizability"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Realizability"
      }
    ],
    "tag": "4H35VQ16DC",
    "uses": [
      "CK8ZQRN183",
      "KZSB5E471K",
      "7B53ZGXXEQ"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/4H35VQ16DC/",
  "links": [
    {
      "field": "uses",
      "target": "CK8ZQRN183"
    },
    {
      "field": "uses",
      "target": "KZSB5E471K"
    },
    {
      "field": "uses",
      "target": "7B53ZGXXEQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4H35VQ16DC.md",
  "msc": [
    "03F50",
    "03F55",
    "03D80"
  ],
  "name": "Realizability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4H3/4H35VQ16DC.md",
  "status": null,
  "tag": "4H35VQ16DC",
  "uri": "m2://4H35VQ16DC",
  "url": "https://math2.tamnd.com/wiki/realizability/",
  "writing": {
    "tag": "4H35VQ16DC",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}