Strongly Minimal Sets and Categoricity in Continuous Logic

TL;DR

This paper develops the theory of strongly minimal sets in continuous logic, introduces new concepts like dictionaric theories and indiscernible subspaces, and provides counterexamples to classical categoricity results.

Contribution

It extends the understanding of minimal sets and categoricity in continuous logic, introducing new notions and analyzing their implications for Banach space theories.

Findings

Not every inseparably categorical theory has a strongly minimal set.

Introduction of dictionaric theories and indiscernible subspaces.

Counterexamples showing limitations of classical categoricity in continuous logic.

Abstract

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 $\omega$-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.