{
  "id": "2007.05885",
  "version": "v1",
  "published": "2020-07-12T01:43:03.000Z",
  "updated": "2020-07-12T01:43:03.000Z",
  "title": "On Non-standard Models of Arithmetic with Uncountable Standard Systems",
  "authors": [
    "Wei Wang"
  ],
  "comment": "6 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "In 1960s, Dana Scott gave a recursion theoretic characterization of standard systems of countable non-standard models of arithmetic, i.e., collections of sets of standard natural numbers coded in non-standard models. Later, Knight and Nadel proved that Scott's characterization also applies to non-standard models of arithmetic with cardinality $\\aleph_1$. But the question, whether the limit on cardinality can be removed from the above characterization, remains a long standing question, known as the Scott Set Problem. This article presents two constructions of non-standard models of arithmetic with non-trivial uncountable standard systems. The first one leads to a new proof of the above theorem of Knight and Nadel, and the second proves the existence of models with non-trivial standard systems of cardinality the continuum. A partial answer to the Scott Set Problem under certain set theoretic hypothesis also follows from the second construction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-12T01:43:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C62",
      "03H15",
      "03D28",
      "03E50"
    ],
    "keywords": [
      "arithmetic",
      "scott set problem",
      "non-trivial standard systems",
      "standard natural numbers",
      "cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}