{
  "id": "2110.02725",
  "version": "v2",
  "published": "2021-10-06T13:18:28.000Z",
  "updated": "2023-12-29T11:32:40.000Z",
  "title": "The exact strength of generic absoluteness for the universally Baire sets",
  "authors": [
    "Grigor Sargsyan",
    "Nam Trang"
  ],
  "comment": "This is the latest version",
  "categories": [
    "math.LO"
  ],
  "abstract": "A set of reals is \\textit{universally Baire} if all of its continuous preimages in topological spaces have the Baire property. $\\sf{Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The $\\sf{Largest\\ Suslin\\ Axiom}$ ($\\sf{LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\\sf{LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\\sf{LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, $\\sf{Sealing}$ is equiconsistent with $\\sf{LSA-over-uB}$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see \\rdef{dfn:hod_pm}). As a consequence, we obtain that $\\sf{Sealing}$ is weaker than the theory $``\\sf{ZFC} + $there is a Woodin cardinal which is a limit of Woodin cardinals\". A variation of $\\sf{Sealing}$, called $\\sf{Tower \\ Sealing}$, is also shown to be equiconsistent with $\\sf{Sealing}$ over the same large cardinal theory. The result is proven via Woodin's $\\sf{Core\\ Model\\ Induction}$ technique, and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\\sf{CMI}$ as explained in the paper.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-12-29T11:32:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universally baire sets",
      "generic absoluteness",
      "exact strength",
      "exact large cardinal theory",
      "mild large cardinal theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}