{
  "id": "math/9312203",
  "version": "v1",
  "published": "1993-12-03T00:00:00.000Z",
  "updated": "1993-12-03T00:00:00.000Z",
  "title": "Any behaviour of the Mitchell Ordering of Normal Measures Is Possible",
  "authors": [
    "Jiří Witzany"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $U_0,U_1$ be two normal measures on $\\kappa .$ We say that $U_0$ is in the Mitchell ordering less then $U_1,$ $U_0\\vartriangleleft U_1,$ if $U_0 \\in Ult(V,U_1) .$ The ordering is well-known to be transitive and well-founded. It has been an open problem to find a model where the Mitchell ordering embeds the four-element poset $|\\; | .$ We show that in the Kunen-Paris extension all well-founded posets are embeddable. Hence there is no structural restriction on the Mitchell ordering. Moreover we show that it is possible to have two $vartriangleleft$-incomparable measures that extend in a generic extension into two $\\vartriangleleft$-comparable measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1993-12-03T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normal measures",
      "generic extension",
      "open problem",
      "four-element poset",
      "kunen-paris extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1993math.....12203W"
    }
  }
}