{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "57PTMS2ADY"
    },
    {
      "field": "uses",
      "target": "C5E5TG8PGP"
    },
    {
      "field": "defines",
      "target": "C19Z3B6V4P"
    }
  ],
  "body": "\n## Summary\n\nThe proof-theoretic ordinal of a formal theory measures how much [[transfinite induction|0ZHQMQ2C1N]] the theory can prove.\nIt is the supremum of the order types of the computable well-orderings that the theory proves to be well-founded.\nIt is a countable [[ordinal|0YSCSX5YZC]], and it gives a scale on which theories can be compared by strength.\nFor [[Peano arithmetic|0QFT7Z76N5]] it is [[epsilon nought|KT50SMH3EC]].\n\n## Explanations\n\n### L1 human\n\nSome orderings of the natural numbers are well-orderings: every decreasing sequence in them stops.\nA theory can prove this for some such orderings and not for others.\nThe longest orderings that the theory still trusts give an ordinal, its proof-theoretic ordinal.\nA stronger theory trusts longer orderings, so it gets a larger ordinal.\nFor example, Peano arithmetic trusts each ordering shorter than $\\varepsilon_0$, but not the ordering of length $\\varepsilon_0$ itself.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03F15",
      "03F30"
    ],
    "name": "Proof-theoretic ordinal",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ordinal_analysis"
      }
    ],
    "specializes": [
      "0YSCSX5YZC"
    ],
    "tag": "C72RBH0CDR",
    "uses": [
      "0YSCSX5YZC",
      "0QFT7Z76N5",
      "0ZHQMQ2C1N",
      "KT50SMH3EC"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/C72RBH0CDR/",
  "links": [
    {
      "field": "specializes",
      "target": "0YSCSX5YZC"
    },
    {
      "field": "uses",
      "target": "0YSCSX5YZC"
    },
    {
      "field": "uses",
      "target": "0QFT7Z76N5"
    },
    {
      "field": "uses",
      "target": "0ZHQMQ2C1N"
    },
    {
      "field": "uses",
      "target": "KT50SMH3EC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/C72RBH0CDR.md",
  "msc": [
    "03F15",
    "03F30"
  ],
  "name": "Proof-theoretic ordinal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C72/C72RBH0CDR.md",
  "status": null,
  "tag": "C72RBH0CDR",
  "uri": "m2://C72RBH0CDR",
  "url": "https://math2.tamnd.com/wiki/proof-theoretic-ordinal/",
  "writing": {
    "tag": "C72RBH0CDR",
    "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
      }
    ]
  }
}