{
  "id": "1302.4841",
  "version": "v2",
  "published": "2013-02-20T08:58:16.000Z",
  "updated": "2013-12-25T10:07:09.000Z",
  "title": "A.E.C. with not too many models",
  "authors": [
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Consider an a.e.c. (abstract elementary class), that is, a class K of models with a partial order refining inclusion (submodel) which satisfy the most basic properties of an elementary class. Our test question is trying to show that the function dot I (lambda, K), counting the number of models in K of cardinality lambda up to isomorphism, is \"nice\", not chaotic, even without assuming it is sometimes 1, i.e. categorical in some lambda's. We prove here that for some closed unbounded class C of cardinals we have (a), (b) or (c) where (a) for every lambda in C of cofinality aleph_0, dot I (lambda, K) greater than or equal to lambda, (b) for every lambda in C of cofinality aleph_0 and M belongs to K_lambda, for every cardinal kappa greater than or equal to lambda there is N_kappa of cardinality kappa extending M (in the sense of our a.e.c.), (c) mathfrak k is bounded; that is, dot I (lambda, K) = 0 for every lambda large enough (equivalently lambda greater than or equal to beth_{delta_*} where delta_* = (2^{LST(mathfrak k)})^+). Recall that an important difference of non-elementary classes from the elementary case is the possibility of having models in K, even of large cardinality, which are maximal, or just failing clause (b).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-25T10:07:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45",
      "03C48",
      "03C55",
      "03C75"
    ],
    "keywords": [
      "partial order refining inclusion",
      "abstract elementary class",
      "cardinal kappa greater",
      "large cardinality",
      "cardinality lambda"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.4841S"
    }
  }
}