{
  "id": "2108.06727",
  "version": "v1",
  "published": "2021-08-15T12:16:17.000Z",
  "updated": "2021-08-15T12:16:17.000Z",
  "title": "Universality: new criterion for non-existence",
  "authors": [
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We find new \"reasons\" for a class of models for not having a universal model in a cardinal $\\lambda$. This work, though has consequences in model theory, is really in combinatorial (set theory). We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\\rm ceq}$ (essentially another representation of $T_{\\rm feq}$ which was already considered but the proof with $T_{\\rm ceq}$ is more transparent). Models of $T_{\\rm ceq}$ consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in $\\lambda$. This work is continued in [Sh:F2071].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-15T12:16:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E45"
    ],
    "keywords": [
      "non-existence",
      "universality",
      "universal model",
      "model theoretic sense",
      "set theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}