Categoricity and universal classes

Abstract

Let (K,⊆) be a universal class with LS (K)=λ categorical in a regular κ>λ+ with arbitrarily large models, and let K∗ be the class of all A∈scriptK>λ for which there is B∈scriptK≥κ such that A⊆B. We prove that K∗ is totally categorical (i.e., ξ‐categorical for all ξ> LS (K)) and scriptK≥ℶμ+⊆scriptK∗ for μ=2λ+. This result is partially stronger and partially weaker than a related result due to Vasey. In addition to small differences in our categoricity transfer results, we provide a shorter and simpler proof. In the end we prove the main theorem of this paper: the models of K>λ+∗ are essentially vector spaces (or trivial, i.e., disintegrated).

UNIVERSAL CLASSES NEAR ${\aleph _1}$

“…The reader may wonder: are there any interesting examples of eventually categorical universal classes? After the initial submission of this article, Hyttinen and Kangas [10] showed that the answer is no: in any universal class categorical in a high-enough regular cardinal, any big-enough model will eventually look like either a set or a vector space (the methods are geometric in nature and also eventual, hence completely different from the tools used in this article). Thus a reader wanting a nontrivial example illustrating, e.g., Theorem 3.5 is out of luck: the statement of Theorem 3.5 combined with the Hyttinen-Kangas result implies that any example will eventually look like a class of vector spaces or a class of sets!…”

Categoricity in multiuniversal classes

“…), the restrictions on the definition of universal classes make it difficult to have universal classes that are both mathematically interesting and model-theoretically well-behaved. Recent work of Hyttinen and Kangas [HK18] have given some confirmation of this by building on the third author's work to show that in any universal class that is categorical in a high-enough successor, the models must eventually look like either vector spaces or be disintegrated (i.e. look like sets).…”