{
  "id": "2403.09020",
  "version": "v1",
  "published": "2024-03-14T01:00:30.000Z",
  "updated": "2024-03-14T01:00:30.000Z",
  "title": "Forcing \"$\\mathrm{NS}_{ω_1}$ is $ω_1$-dense\" From Large Cardinals",
  "authors": [
    "Andreas Lietz"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We answer a question of Woodin by showing that assuming an inaccessible cardinal $\\kappa$ which is a limit of ${<}\\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\\mathbb{P}$ so that $V^{\\mathbb P}\\models``\\mathrm{NS}_{\\omega_1}\\text{ is }\\omega_1\\text{-dense}\"$. We also introduce a new forcing axiom $\\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\\mathbb{Q}_{\\mathrm{max}}\\text{-}(*)$. Consequently, $\\mathrm{QM}$ implies ``$\\mathrm{NS}_{\\omega_1}$ is $\\omega_1$-dense\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-14T01:00:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E57",
      "03E55",
      "03E50",
      "03E35"
    ],
    "keywords": [
      "large cardinals",
      "supercompact cardinals",
      "supercompact limit",
      "inaccessible cardinal",
      "stationary set preserving forcing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}