Series: Penn State Logic Seminar

Date: Tuesday, January 20, 2004

Time: 2:30 - 3:45 PM

Place: 307 Boucke Building

Speaker: Rami Grossberg, Carnegie Mellon University, Mathematics


  Shelah's Categoricity Conjecture Holds for Tame Abstract Elementary

  In the seventies Saharon Shelah formulated a far reaching
  generalization of Morley's categoricity theorem to serve as a
  test-problem and a guide for the development of classification
  theory for non-first-order-theories.  Shelah's categoricity
  conjecture: If an L_{\omega_1,\omega} theory is categorical in a
  cardinal greater than the Hanf number then the theory is categorical
  in every cardinal above the Hanf number.  Despite many papers by
  Shelah and others, the conjecture is still open.  In the late
  seventies Shelah introduced the notion of abstract elementary class
  (a semantic generalization of L_{\omega_1,\omega} theory) and
  formulated a similar strong categoricity conjecture.  Recently
  Monica VanDieren and the speaker proved that Shelah's conjecture
  holds for a large family of abstract elementary classes.  Our proof
  turned to be less technical than expected, one of the surprises is
  that it is in ZFC, while previous related results of Shelah make
  heavy use of diamond-like principles. Our argument is new even when
  one specialize to first-order logic.  In the talk I will describe
  all the notions in this abstract and the general framework.  I
  intend to make my talk to be accessible also to people who did not
  take a course in model theory.