# A proof of Shelah's eventual categoricity conjecture in universal classes

This is the title of a four-parts talk given at the CMU model theory seminar from March 14 to April 3, 2016. The talk presented the corresponding two papers.

## Abstract

Abstract elementary classes (AECs) are an axiomatic framework
encompassing classes of models of an *L*_{∞, ω} theory,
as well as numerous algebraic examples. They were introduced by Saharon
Shelah forty years ago. Shelah focused on generalizations of Morley’s
categoricity theorem and conjectured the following eventual version: An
AEC categorical in a high-enough cardinal is categorical on a tail of
cardinals. I will present my proof of the conjecture for universal
classes. They are a special case of AECs (studied by Shelah in a milestone
1987 paper) corresponding to classes of models of a universal *L*_{∞, ω} theory.

I will initially discuss the proof with the additional
assumption that the universal class satisfies the amalgamation
property. In this case, the argument generalizes to AECs which
have amalgamation, are tame (a locality property for orbital
types isolated by Grossberg and VanDieren), and have primes over
sets of the form *Ma*. Time permitting, I will discuss how to
use Shelah’s structure theory of universal classes to remove the
amalgamation assumption.

## References

John T. Baldwin, *Categoricity*, University Lecture Series, vol. 50, American Mathematical Society, 2009.

Saharon Shelah, *Classification Theory for Abstract Elementary Classes*, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.

Sebastien Vasey, *The lazy model theoretician's guide to Shelah's eventual categoricity conjecture in universal classes*, An expository note: pdf arXiv.

Sebastien Vasey, *Shelah's eventual categoricity conjecture in universal classes. Part I*, Preprint: pdf arXiv.

Sebastien Vasey, *Shelah's eventual categoricity conjecture in universal classes. Part II*, Preprint: pdf arXiv.

Sebastien Vasey, *Shelah's eventual categoricity conjecture in tame AECs with primes*, Preprint: pdf arXiv.

Will Boney and Sebastien Vasey, *A survey on tame abstract elementary classes*, Preprint: pdf arXiv.