{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "TQTDKMBJ0F"
    }
  ],
  "body": "\n## Summary\n\nReverse mathematics is a program in mathematical logic that asks which axioms are needed to prove the theorems of ordinary mathematics.\nIt works in subsystems of [[second-order arithmetic|DC60FYCSPM]] over a weak base theory, and proves theorems equivalent to set existence axioms.\nA striking finding is that many classical theorems are equivalent to one of five subsystems, often called the big five.\n\n## Explanations\n\n### L1 human\n\nUsually one goes from axioms to theorems.\nReverse mathematics goes backwards: it starts from a theorem and finds the weakest axioms that prove it.\nOften the theorem turns out to be equivalent to those axioms, so it needs exactly them.\nFor example, the Bolzano–Weierstrass theorem, that every bounded sequence of reals has a convergent subsequence, is equivalent to the axiom system called $\\mathsf{ACA}_0$ over the base theory.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03B30",
      "03F35"
    ],
    "name": "Reverse mathematics",
    "same_as": {
      "wikidata": "Q2005236",
      "wikipedia": "Reverse mathematics"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reverse_mathematics"
      }
    ],
    "tag": "7SQS69T17R"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/7SQS69T17R/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/7SQS69T17R.md",
  "msc": [
    "03B30",
    "03F35"
  ],
  "name": "Reverse mathematics",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7SQ/7SQS69T17R.md",
  "status": null,
  "tag": "7SQS69T17R",
  "uri": "m2://7SQS69T17R",
  "url": "https://math2.tamnd.com/wiki/reverse-mathematics/",
  "writing": {
    "tag": "7SQS69T17R",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}