Shelah's eventual categoricity conjecture in universal classes

This is the title of a talk given at the Cornell logic seminar on March 1, 2016. The talk presents the corresponding paper.

Abstract

Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an L∞, ω sentence, as well as numerous algebraic examples. They were introduced by Saharon Shelah in the mid seventies. One of Shelah's goals was to study generalizations of Morley's categoricity theorem to the infinitary setup. Among several variations, Shelah conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals. In this talk, we will prove the conjecture for universal classes. It is an interesting type of AEC studied by Shelah in a milestone 1987 paper [Sh:300] (the work was done in 1985). They correspond to classes of models of a universal L∞, ω sentence.

The proof proceeds in two steps. First, we use Shelah's structure theory of universal classes to show that there exists an ordering on the starting universal class that makes it into an AEC with several structural properties: amalgamation, tameness (a locality property isolated by Grossberg and VanDieren which says that orbital types are determined by their small restrictions), and the existence of primes models over every set of the form Ma. The second step transfers categoricity in this AEC. This is done by showing that categoricity implies the existence of a well-behaved forking-like independence relation (a so-called good frame). A definition of a unidimensionality-like property (due to Shelah) is then proven to follow from categoricity in a single cardinal and imply categoricity on a tail of cardinals.

References