{
  "id": "2212.00489",
  "version": "v1",
  "published": "2022-12-01T13:33:17.000Z",
  "updated": "2022-12-01T13:33:17.000Z",
  "title": "The Biggest Five of Reverse Mathematics",
  "authors": [
    "Dag Normann",
    "Sam Sanders"
  ],
  "comment": "37 pages + Appendix + References",
  "categories": [
    "math.LO"
  ],
  "abstract": "The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the 'Big Five'. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations of the aforementioned third-order theorems fall far outside of the Big Five.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-01T13:33:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B30",
      "03F35"
    ],
    "keywords": [
      "reverse mathematics",
      "third-order theorems fall far outside",
      "base theory",
      "ordinary mathematics",
      "minimal axioms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}