{
  "id": "2406.12776",
  "version": "v1",
  "published": "2024-06-18T16:44:44.000Z",
  "updated": "2024-06-18T16:44:44.000Z",
  "title": "Axiom $\\mathcal{A}$ and supercompactness",
  "authors": [
    "Alejandro Poveda"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \\cite{HMP} and provides a definitive answer to a question of Bagaria \\cite[p.19]{Bag}. This configuration is a consequence of a new axiom we introduce -- called $\\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-18T16:44:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "supercompactness",
      "strongest large-cardinal notion",
      "main identity-crises configuration",
      "supercompact cardinal",
      "incidental result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}