{
  "id": "2401.01979",
  "version": "v1",
  "published": "2024-01-03T20:58:48.000Z",
  "updated": "2024-01-03T20:58:48.000Z",
  "title": "Low level definability above large cardinals",
  "authors": [
    "Farmer Schlutzenberg"
  ],
  "comment": "22 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study some connections between definability in generalized descriptive set theory and large cardinals, particularly measurable cardinals and limits thereof, working in ZFC. We show that if $\\kappa$ is a limit of measurable cardinals then there is no $\\Sigma_1(H_\\kappa\\cup\\mathrm{OR})$ wellorder of a subset of $P(\\kappa)$ of length $\\geq\\kappa^+$; this answers a question of L\\\"ucke and M\\\"uller. However, in $M_1$, the minimal proper class mouse with a Woodin cardinal, for every uncountable cardinal $\\kappa$ which is not a limit of measurables, there is a $\\Sigma_1(H_\\kappa\\cup\\{\\kappa\\})$ good wellorder of $H_{\\kappa^+}$. If $\\kappa$ is a limit of measurables then there is no $\\Sigma_1(H_\\kappa\\cup\\mathrm{OR})$ mad family $F\\subseteq P(\\kappa)$ of cardinality $>\\kappa$, and if also $\\mathrm{cof}(\\kappa)>\\omega$ then there is no $\\Sigma_1(H_\\kappa\\cup\\mathrm{OR})$ almost disjoint family $F\\subseteq P(\\kappa)$ of cardinality $>\\kappa$. However, relative to the consistency of large cardinals, $\\Pi_1(\\{\\kappa\\})$ mad families and maximal independent families $F\\subseteq P(\\kappa)$ can exist, when $\\kappa$ is a limit of measurables, and even more. We also examine some of the features of $L[U]$, and answer another question of L\\\"ucke and M\\\"uller, showing that if $\\kappa$ is a weakly compact cardinal such that every $\\Sigma_1(H_\\kappa\\cup\\{\\kappa\\})$ subset of $P(\\kappa)$ of cardinality $>\\kappa$ has a subset which is the range of a perfect function, then there is an inner model satisfying \"there is a weakly compact limit of measurable cardinals\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-03T20:58:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55",
      "03E45",
      "03E15",
      "03E05"
    ],
    "keywords": [
      "large cardinals",
      "low level definability",
      "measurable cardinals",
      "minimal proper class mouse",
      "maximal independent families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}