Mathematical Logic Quarterly

Categoricity and universal classes

Journal Article

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 K > λ for which there is B K κ such that A B . We prove that K is totally categorical (i.e., ξ‐categorical for all ξ > LS ( K ) ) and K μ + K for μ = 2 λ + . 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).

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