Sebastien Vasey: Chains of saturated models in Abstract Elementary Classes

Model Theory Seminar
 
Sebastien Vasey (joint work with Will Boney)
 
CMU

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”.
Date: Monday, February 2, 2015

Time: 5:00 – 6:30 PM
Location: Wean 8220

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Model Theory Seminar
 
Sebastien Vasey (joint work with Will Boney)
 
CMU

Title:   Chains of saturated models in Abstract Elementary Classes, Part 2

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”.
Date: Monday, February 9, 2015

Time: 5:00 – 6:30 PM
Location: Wean 8220

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