Mathematical Logic seminar

  • Monica VanDieren

Categoricity and Stability in Abstract Elementary Classes, Part 1

One of the most influential conjectures in model theory was Los' Conjecture, which generalizes Steinitz's Theoremabout algebraically closed fields. Shelah in the mid-seventies conjectured that a categoricity theorem, similar to Los' conjecture,should hold for the infinitiary logic, $L_{omega_1,omega}$, and more generally for abstract elementary classes (AEC). Moreover heforesaw that work in this direction would lead to a stability theory for non-first-order logics.I will present the work of Shelah-Villaveces towards a categoricity theorem in AEC under the assumption of no maximal models andrelate it to another paper by Shelah (Sh600). I will present the proof that under some mild set-theoretic assumptions, categoricityimplies density of amalgamation bases, which is necessary to develop a theory of types. In all the proofs of Los' Conjecture for first-order theories, saturated models played a pivotal role. I will present some ideas fromShelah-Villaveces that may lead to a good substitution for saturated models in the AEC case. Finally, I will present Shelah-Villaveces' result that categorcity implies that there does not exist an $omega$-chain of splittingtypes. In first order logic, this is a consequence of superstability. Proving this theorem under a superstability assumption would be agood test question for an AEC stability theory.
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