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Model Theory Seminar
Sebastien Vasey
Carnegie Mellon University
Title: Chains of saturated models in Abstract Elementary Classes, Part 1

Abstract: Fix a first-order theory $T$ and a cardinal $\lambda > |T|$. Is the union of a chain of $\lambda$-saturated models of $T$ $\lambda$-saturated? By a classical result of Saharon Shelah, the answer is positive when $T$ is superstable. When $T$ is only stable, this also holds provided that the length of the chain has cofinality at least $|T|^+$. In both cases, the proofs depend on the heavy machinery of forking and averages.

We prove versions of these two results in the general framework of tame abstract elementary classes. For a suitable definition of superstability, we manage to fully generalize Shelah's result (for high-enough $\lambda$). We also have a theorem in stable AECs but use cardinal arithmetic assumptions on $\lambda$. Our main tool is a generalization of averages to abstract elementary classes. The starting point is Shelah's work on averages in the framework of "stability theory inside a model".

This is a joint work with Will Boney.

Date: Monday, February 2, 2015
Time: 5:00 pm
Location: Wean Hall 8220
Submitted by:  Grossberg