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

Title:

Shelah's Categoricity Conjecture Holds for Tame Abstract Elementary
Classes

Abstract:

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.