{
  "id": "2303.17458",
  "version": "v1",
  "published": "2023-03-30T15:33:53.000Z",
  "updated": "2023-03-30T15:33:53.000Z",
  "title": "On Indestructible Strongly Guessing Models",
  "authors": [
    "Rahman Mohammadpour",
    "Boban Velickovic"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In [14] we defined and proved the consistency of the principle ${\\rm GM}^+(\\omega_3, \\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\\omega_2$ and $\\omega_3$. In this paper we formulate a strengthening of ${\\rm GM}^+(\\omega_3, \\omega_1)$ that we call ${\\rm SGM}^+(\\omega_3, \\omega_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\\rm SGM}^+(\\omega_3, \\omega_1)$ is consistent with ZFC. In addition to all the consequences of ${\\rm GM}^+(\\omega_3, \\omega_1)$, the principle ${\\rm GSM}^+(\\omega_3, \\omega_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\\omega_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham [1] and extends a previous result of Todor\\v{c}evi\\'c [15] in this direction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-30T15:33:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E55",
      "03E57",
      "03E65"
    ],
    "keywords": [
      "indestructible strongly guessing models",
      "mild cardinal arithmetic assumptions",
      "strong forcing axioms hold",
      "supercompact cardinals",
      "consistency"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}