This is the title of a four-parts talk given at the CMU Model Theory Seminar. The talks started on Feb. 2 and ended on Feb. 23, 2015.
Fix a first-order theory T and a cardinal λ > |T|. Is the union of a chain of λ-saturated models of T λ-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 λ). We also have a theorem in stable AECs but use cardinal arithmetic assumptions on λ. 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 joint work with Will Boney.
Michael Makkai, Saharon Shelah. Categoricity of theories in Lκ,ω, with κ a compact cardinal, Annals of Pure and Applied Logic 47 (1990), 41-97.
Sebastien Vasey, Infinitary stability theory, In preparation. Draft: pdf arXiv.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 18 and 20, College Publications, 2009.