Uniqueness of Limit Models in Classes with Amalgamation

TL;DR

This paper proves a uniqueness theorem for limit models in certain abstract elementary classes with amalgamation, extending prior results and providing a new approach that avoids Ehrenfeucht-Mostowski constructions.

Contribution

It extends Shelah's results on the uniqueness of limit models to classes with amalgamation, without relying on Ehrenfeucht-Mostowski constructions.

Findings

Proves that any two limit models over the same base are isomorphic.

Extends previous theorems to classes satisfying specific stability and locality conditions.

Provides an approach to uniqueness that does not depend on Ehrenfeucht-Mostowski methods.

Abstract

We prove: Main Theorem: Let $\mathcal{K}$ be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality $\mu$. Let $\mu$ be a cardinal above the the L\"owenheim-Skolem number of the class. If $\mathcal{K}$ is $\mu$-Galois-stable, has no $\mu$-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two $(\mu,\sigma_\ell)$-limits over $M$, for $\ell\in\{1,2\}$, are isomorphic over $M$. This theorem extends results of Shelah from \cite{Sh394}, \cite{Sh576}, \cite{Sh600}, Kolman and Shelah in \cite{KoSh} and Shelah and Villaveces from \cite{ShVi}. A preliminary version of our uniqueness theorem, which was circulated in 2006, was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes in \cite{GrVa2}. Preprints of this paper have also influenced the Ph.D. theses of Drueck \cite{Dr} and Zambrano \cite{Za}. This paper also serves the expository role of presenting together the arguments in \cite{Va1} and \cite{Va2} in a more natural context in which the amalgamation property holds and this work provides an approach to the uniqueness of limit models that does not rely on Ehrenfeucht-Mostowski constructions.