{
  "id": "1912.02130",
  "version": "v1",
  "published": "2019-12-04T17:28:03.000Z",
  "updated": "2019-12-04T17:28:03.000Z",
  "title": "The $κ$-Strongly Proper Forcing Axiom",
  "authors": [
    "David Asperó",
    "Asaf Karagila"
  ],
  "comment": "6 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study methods with which we can obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first prove that the consistency of a supercompact cardinal $\\theta>\\kappa$ implies the consistency of a forcing axiom for $\\kappa$-strongly proper forcing notions which are also $\\kappa$-lattice, and then eliminate the need for the supercompact cardinal. The proof goes through a natural reflection property for $\\kappa$-strongly proper forcings and through the fact that every $\\kappa$-sequence of ordinals added by a $\\kappa$-lattice and $\\kappa$-strongly proper forcing is in a $\\kappa$-Cohen extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-04T17:28:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E57",
      "03E55",
      "03E35"
    ],
    "keywords": [
      "strongly proper forcing axiom",
      "supercompact cardinal",
      "consistency",
      "natural reflection property",
      "cohen extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}