{
  "id": "2111.14413",
  "version": "v2",
  "published": "2021-11-29T09:50:44.000Z",
  "updated": "2022-01-25T11:11:37.000Z",
  "title": "Friedman-reflexivity: interpreters as consistoids",
  "authors": [
    "Albert Visser"
  ],
  "comment": "(i) some new notations were introduced to make the paper more readable, (ii) the section on interpreter logics was substantially extended, (iii) new results were added to the paper, (iv) a new both backward and forward looking last section was added",
  "categories": [
    "math.LO"
  ],
  "abstract": "Harvey Friedman shows that, over Peano Arithmetic, the consistency statement for a finitely axiomatised theory $A$ can be characterised as the weakest statement $C$ over Peano Arithmetic such that ${\\sf PA}+C$ interprets $A$. We study which base theories $U$ have the property that, for any finitely axiomatised $A$, there is a weakest $C$ such that $U+C$ interprets $A$. We call such theories Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free or Herbrand consistency statements. We prove a characterisation theorem for Friedman-reflexive sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto. The consistency-like statements provided by a Friedman-reflexive base $U$ can be used to define a provability-like notion for a finitely axiomatised $A$ that interprets $U$ via an interpretation $K$ of $U$ in $A$. We call the modal logics based on this idea \\emph{interpreter logics}. These logics satisfy the L\\\"ob Conditions. We provide conditions for when these logics extend {\\sf S}4, {\\sf K}45, and L\\\"ob's Logic. We show that, if either $U$ or $A$ is sequential, then the condition for extending L\\\"ob's Logic is fulfilled. Moreover, if our base theory $U$ is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of G\\\"odel numbers. At the end of the paper, we briefly discuss how successful the coordinate-free approach is.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-25T11:11:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F25",
      "03F30",
      "03F40"
    ],
    "keywords": [
      "peano arithmetic",
      "friedman-reflexive sequential theory",
      "interpreters",
      "consistoids",
      "base theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}