Uniqueness of Limit Models in Classes with Amalgamation

TL;DR

This paper proves the uniqueness of limit models in certain classes with amalgamation, extending previous results and contributing to the understanding of categoricity in abstract elementary classes.

Contribution

It establishes a new uniqueness theorem for limit models under specific stability and amalgamation conditions, generalizing prior work by Shelah and others.

Findings

Uniqueness of limit models under specified conditions

Extension of Shelah's categoricity results to tame classes

Application to Shelah's categoricity conjecture

Abstract

Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the L\"owenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M. This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes.