{
  "id": "1906.04273",
  "version": "v1",
  "published": "2019-06-10T21:03:37.000Z",
  "updated": "2019-06-10T21:03:37.000Z",
  "title": "Independence in Arithmetic: The Method of $(\\mathcal L, n)$-Models",
  "authors": [
    "Corey Bacal Switzer"
  ],
  "comment": "17 Pages",
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "I develop in depth the machinery of $(\\mathcal L, n)$-models originally introduced by Shelah \\cite{ShelahPA} and, independently in a slightly different form by Kripke (cf \\cite{put2000}, \\cite{quin80}). This machinery allows fairly routine constructions of true but unprovable sentences in $\\mathsf{PA}$. I give three applications: 1. Shelah's alternative proof of the Paris-Harrington theorem, 2. The independence over $\\mathsf{PA}$ of a $\\Pi^0_1$ Ramsey theoretic statement about colorings of finite sequences of structures and 3. The independence over $\\mathsf{PA}$ of both a $\\Pi_2^0$-statement and a $\\Pi^0_1$-statement about choice functions for sequences of numbers and finite structures respectively. The latter is reminiscent of Shelah's example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-10T21:03:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C62",
      "03C98",
      "05D10"
    ],
    "keywords": [
      "independence",
      "arithmetic",
      "ramsey theoretic statement",
      "shelahs alternative proof",
      "paris-harrington theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}