{
  "id": "math/9201240",
  "version": "v1",
  "published": "1990-01-15T00:00:00.000Z",
  "updated": "1990-01-15T00:00:00.000Z",
  "title": "Categoricity over P for first order T or categoricity for phi in L_{omega_1 omega} can stop at aleph_k while holding for aleph_0, ..., aleph_{k-1}",
  "authors": [
    "Bradd Hart",
    "Saharon Shelah"
  ],
  "journal": "Israel J. Math. 70 (1990), 219--235",
  "categories": [
    "math.LO"
  ],
  "abstract": "Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then there is an isomorphism i:M-> N which is the identity on P(M). T is relatively categorical if it is relatively lambda-categorical for every lambda. The question arises whether the relative lambda-categoricity of T for some lambda >|T| implies that T is relatively categorical. In this paper, we provide an example, for every k>0, of a theory T_k and an L_{omega_1 omega} sentence varphi_k so that T_k is relatively aleph_n-categorical for n < k and varphi_k is aleph_n-categorical for n<k but T_k is not relatively beth_k-categorical and varphi_k is not beth_k-categorical.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1990-01-15T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "categoricity",
      "complete first order theory",
      "l-structure",
      "unary predicate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}