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Categoricity and universal classes
Author(s) -
Hyttinen Tapani,
Kangas Kaisa
Publication year - 2018
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201700076
Subject(s) - categorical variable , mathematics , class (philosophy) , pure mathematics , transfer (computing) , discrete mathematics , computer science , artificial intelligence , statistics , parallel computing
Let ( K , ⊆ ) be a universal class withLS ( K ) = λ categorical in a regular κ > λ +with arbitrarily large models, and let K ∗ be the class of all A ∈ K > λfor which there is B ∈ K ≥ κsuch that A ⊆ B . We prove that K ∗ is totally categorical (i.e., ξ‐categorical for all ξ > LS ( K ) ) andK ≥ ℶ 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).