{
  "id": "2009.14245",
  "version": "v1",
  "published": "2020-09-29T18:25:57.000Z",
  "updated": "2020-09-29T18:25:57.000Z",
  "title": "Compactness versus hugeness at successor cardinals",
  "authors": [
    "Sean Cox",
    "Monroe Eskew"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "If $\\kappa$ is regular and $2^{<\\kappa}\\leq\\kappa^+$, then the existence of a weakly presaturated ideal on $\\kappa^+$ implies $\\square^*_\\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\\omega_2$ such that $\\mathcal{P}(\\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-29T18:25:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "successor cardinal",
      "compactness",
      "ch holds",
      "tree property",
      "approachability ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}