{
  "id": "1607.04772",
  "version": "v1",
  "published": "2016-07-16T17:27:53.000Z",
  "updated": "2016-07-16T17:27:53.000Z",
  "title": "The Approachability Ideal Without a Maximal Set",
  "authors": [
    "John Krueger"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $\\omega_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[\\omega_2]$ does not have a maximal set modulo clubs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-16T17:27:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E40",
      "03E05"
    ],
    "keywords": [
      "approachability ideal",
      "maximal set modulo clubs",
      "simultaneously adding partial square sequences",
      "multiple stationary sets",
      "side condition product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}