{
  "id": "1410.1224",
  "version": "v1",
  "published": "2014-10-05T23:23:16.000Z",
  "updated": "2014-10-05T23:23:16.000Z",
  "title": "Forcing a countable structure to belong to the ground model",
  "authors": [
    "Itay Kaplan",
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\\dot{\\tau}$ a $P$-name such that $P\\Vdash$ \"$\\dot{\\tau}$ is a countable $L$-structure\". In the product $P\\times P$, there are names $\\dot{\\tau_{1}},\\dot{\\tau_{2}}$ such that for any generic filter $G=G_{1}\\times G_{2}$ over $P\\times P$, $\\dot{\\tau}_{1}[G]=\\dot{\\tau}[G_{1}]$ and $\\dot{\\tau}_{2}[G]=\\dot{\\tau}[G_{2}]$. Zapletal asked whether or not $P \\times P \\Vdash \\dot{\\tau}_{1}\\cong\\dot{\\tau}_{2}$ implies that there is some $M\\in V$ such that $P \\Vdash \\dot{\\tau}\\cong\\check{M}$. We answer this negatively and discuss related issues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-05T23:23:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C95",
      "03C55",
      "03C45"
    ],
    "keywords": [
      "ground model",
      "countable structure",
      "generic filter",
      "forcing notion",
      "related issues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1224K"
    }
  }
}