{
  "id": "1806.11310",
  "version": "v1",
  "published": "2018-06-29T09:19:20.000Z",
  "updated": "2018-06-29T09:19:20.000Z",
  "title": "Self-similarity in the Foundations",
  "authors": [
    "Paul K. Gorbow"
  ],
  "comment": "Ph.D. thesis",
  "categories": [
    "math.LO"
  ],
  "abstract": "This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on the existence of embeddings between countable non-standard models of a fragment of ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal the original model. The second part of the work on set theory consists in combining these two results into a technical machinery, yielding several results about non-standard models of set theory relating such notions as self-embeddings, their sets of fixed points, strong rank-cuts, and set theories of different strengths. The work in foundational category theory consists in the formulation of a novel algebraic set theory which is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic or classical NF, with or without atoms. A key axiom of this theory expresses that its structures have an endofunctor with natural properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-29T09:19:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03G30"
    ],
    "keywords": [
      "set theory consists",
      "foundations",
      "novel algebraic set theory",
      "foundational category theory consists",
      "self-similarity"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}