{
  "id": "2006.10539",
  "version": "v1",
  "published": "2020-06-18T13:56:20.000Z",
  "updated": "2020-06-18T13:56:20.000Z",
  "title": "Provability and interpretability logics with restricted realizations",
  "authors": [
    "Thomas F. Icard",
    "Joost J. Joosten"
  ],
  "journal": "Notre Dame Journal of Formal Logic, 53 (2), 133-154, 2012",
  "categories": [
    "math.LO"
  ],
  "abstract": "The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\\Gamma$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $\\Gamma$, where each sentence in $\\Gamma$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$\\Sigma_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $\\subset$ ILM.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-18T13:56:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "restricted realizations",
      "upper bound",
      "linear frames",
      "arithmetical realization",
      "major open question"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}