{
  "id": "2308.13478",
  "version": "v1",
  "published": "2023-08-25T16:32:37.000Z",
  "updated": "2023-08-25T16:32:37.000Z",
  "title": "Generalized cardinal invariants for an inaccessible $κ$ with compactness at $κ^{++}$",
  "authors": [
    "Radek Honzik",
    "Sarka Stejskalova"
  ],
  "comment": "19 pages, submitted",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal $\\kappa$ and compactness principles at $\\kappa^{++}$. We show that if the existence of a supercompact cardinal $\\kappa$ with a weakly compact cardinal $\\lambda$ above $\\kappa$ is consistent, then the following are consistent as well (where $\\mathfrak{t}(\\kappa)$ and $\\mathfrak{u}(\\kappa)$ are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal $\\kappa$ such that $\\kappa^+ < \\mathfrak{t}(\\kappa)= \\mathfrak{u}(\\kappa)< 2^\\kappa$ and stationary reflection and the disjoint stationary sequence property at $\\kappa^{++}$ hold. (ii) There is an inaccessible cardinal $\\kappa$ such that $\\kappa^+ = \\mathfrak{t}(\\kappa) < \\mathfrak{u}(\\kappa)< 2^\\kappa$ and the principles in (i) hold plus the tree property at $\\kappa^{++}$ and the negation of the weak Kurepa Hypothesis at $\\kappa^+$. The cardinals $\\mathfrak{u}(\\kappa)$ and $2^\\kappa$ can have any reasonable values in these models. We compute several other values of cardinal invariants, such $\\mathfrak{a}(\\kappa)$ and $\\mathfrak{b}(\\kappa)$, and the invariants of the meager ideal (they are all equal to $\\mathfrak{u}(\\kappa)$). In (ii), we compute $\\mathfrak{p}(\\kappa) = \\mathfrak{t}(\\kappa) = \\kappa^+$ by observing that the $\\kappa^+$-distributive quotient of the Mitchell forcing adds a tower of size $\\kappa^+$. As a corollary of the construction, we obtain that (i) and (ii) are also true for $\\kappa = \\omega$ (starting with a weakly compact cardinal in the ground model).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-25T16:32:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55",
      "03E35",
      "03E17",
      "03E05"
    ],
    "keywords": [
      "generalized cardinal invariants",
      "compactness",
      "weakly compact cardinal",
      "inaccessible cardinal",
      "disjoint stationary sequence property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}