A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes

Abstract

In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $\mu$ in $\lambda$-categorical abstract elementary classes with no maximal models, where $\lambda$ is some cardinal larger than $\mu$. In \cite{Va} and \cite{Va-errata} we identified several gaps in the approach outlined in \cite{ShVi}, and we added the assumption that the union of an increasing chain of limit models is a limit model. Here we replace this assumption with the seemingly weaker statement that the union of an increasing and continuous chain of limit models is an amalgamation base. Moreover, we prove that this assumption is not only sufficient but is necessary to settle the uniqueness of limit models problem attempted in \cite{ShVi} for $\lambda ={\mu }^{+n}$ when $0<n<\omega$.

this article generalizes the results from arXiv:1507.01990, arXiv:1507.01991, arXiv:1507.01989 to a context in which the amalgamation property is not assumed

References in corpus (3)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Transferring Symmetry

Cited by in corpus (2)

• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Superstability and Symmetry