Ernest Schimmerling

### Mathematical logic seminar - September 18, 2002

 Speaker: Andrés Villaveces Visiting Assistant Professor Department of Mathematical Sciences Carnegie Mellon University Assistant Professor Departmento de Matemáticas Universidad Nacional de Colombia Title: Shelah's categoricity conjecture and excellent classes Abstract:

Morley's theorem says that if T is a countable first order theory and T is categorical in some uncountable cardinality, then T is categorical in all uncountable cardinalities. Shelah has conjectured a certain generalization of Morley's theorem for countable theories of the language $L_{\omega_1,\omega}$. Shelah proved his conjecture under the additional assumption that the class of models of the theory is a so-called excellent class. In response to a question of Harrington, Hart and Shelah produced a counterexample in the non-excellent case.

I will review the background needed to understand the statements of the results to which I referred above, highlight aspects of Shelah's proof, and describe the counterexample of Hart and Shelah.