2020/11/01 by James Hanson, Hanson, James · 2 citations
Computer Science · Mathematics · #03C35 #03C45 (Primary) #03C66 #46B04 (Secondary) #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge
paper · pdf · doi:10.48550/arxiv.2011.00610
openalex publication_date 2020/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Banach theories, the notion of an 'indiscernible subspace,' which we use to improve a result of Shelah and Usvyatsov. Both of these notions are applicable to continuous logic outside of the context of inseparably categorical theories. Finally, we construct or present a slew of counterexamples, including an ω-stable theory with no Vaughtian pairs which fails to be inseparably categorical and an inseparably categorical theory with strongly minimal sets in its home sort only over models of sufficiently high dimension.