# Non-elementary classification theory

This is the title of a talk given at the Harvard Logic Seminar on October 31, 2017.

## Abstract

The classification theory of elementary classes was started by Michael Morley in the early sixties, when he proved that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof of this result, now called Morley's categoricity theorem, led to the development of forking, a joint generalization of linear independence in vector spaces and algebraic independence of fields, which is now a central pillar of modern model theory.

In recent years, it has become apparent that the theory of forking can also be developed in several non-elementary contexts. Prime among those is the axiomatic framework of abstract elementary classes (AECs), encompassing the class of models of any *L*_{∞, ω}-theory and closely connected to the more general framework accessible categories. A test question to judge progress in this direction is the forty year old eventual categoricity conjecture of Shelah, which says that a version of Morley's categoricity theorem should hold of any AEC. I will survey recent developments, including the connections with category theory and large cardinals as well as my resolution of the eventual categoricity conjecture for classes of models axiomatized by a *universal* *L*_{∞, ω}-theory.

## 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, *Shelah's eventual categoricity conjecture in universal classes. Part I*, Annals of Pure and Applied Logic **168** (2017), no. 9, 1609–1642. Publisher version pdf arXiv.

Sebastien Vasey, *Shelah's eventual categoricity conjecture in universal classes. Part II*, Selecta Mathematica **23** (2017), no. 2, 1469–1506. Publisher version pdf arXiv.

Sebastien Vasey, *Tameness from two successive good frames*, Submitted. Preprint: pdf arXiv, 25 pages.