{
  "id": "2010.02145",
  "version": "v1",
  "published": "2020-10-05T16:48:47.000Z",
  "updated": "2020-10-05T16:48:47.000Z",
  "title": "Infinitary Logics and A.E.C",
  "authors": [
    "Saharon Shelah",
    "Andrés Villaveces"
  ],
  "comment": "13 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that every a.e.c. with LST number $\\leq \\kappa$ and vocabulary $\\tau$ of cardinality $\\leq \\kappa$ can be defined in the logic ${\\mathbb L}_{\\beth_2(\\kappa)^{+++},\\kappa^+}(\\tau)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \\emph{canonical tree} $\\mathcal S={\\mathcal S_{\\mathcal K}}$ of an a.e.c. $\\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_\\lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-05T16:48:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C75",
      "03C40"
    ],
    "keywords": [
      "infinitary logic",
      "presentation theorem",
      "lst number",
      "ec class",
      "pc class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}