{
  "id": "1110.1584",
  "version": "v3",
  "published": "2011-10-07T16:52:41.000Z",
  "updated": "2011-10-18T08:57:26.000Z",
  "title": "Martin's Maximum and tower forcing",
  "authors": [
    "Sean Cox",
    "Matteo Viale"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "There are several examples in the literature showing that compactness-like properties of a cardinal $\\kappa$ cause poor behavior of some generic ultrapowers which have critical point $\\kappa$ (Burke \\cite{MR1472122} when $\\kappa$ is a supercompact cardinal; Foreman-Magidor \\cite{MR1359154} when $\\kappa = \\omega_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\\vec{\\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{\\omega_1}$ of $\\omega_1$-guessing, internally club sets, then $\\vec{\\mathcal{I}}$ is not presaturated (a set is $\\omega_1$-guessing iff its transitive collapse has the $\\omega_1$-approximation property as defined in Hamkins \\cite{MR2540935}). This theorem, combined with work from \\cite{VW_ISP}, shows that if $PFA^+$ or $MM$ holds and there is an inaccessible cardinal, then there is a tower with critical point $\\omega_2$ which is not presaturated; moreover this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor \\cite{MR1359154}) to exist in all models of Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at $\\omega_2$ has similar implications for towers of ideals which concentrate on the wider class $GIS_{\\omega_1}$ of $\\omega_1$-guessing, internally stationary sets. Finally, we show that the word \"presaturated\" cannot be replaced by \"precipitous\" in the theorems above: Martin's Maximum (which implies SRP and the Tree Property at $\\omega_2$) is consistent with a precipitous tower on $GIC_{\\omega_1}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-10-18T08:57:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "martins maximum",
      "tower forcing",
      "tree property",
      "critical point",
      "strong reflection principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.1584C"
    }
  }
}