Vaught's Two-Cardinal Theorem and Notions of Minimality in Continuous Logic

Much of the work in this thesis was motivated by an effort to prove a continuous analogue of the Baldwin-Lachlan characterization of uncountable categoricity: a theory T in a countable language is uncountably categorical (has one model up to isomorphism of size kappa for some uncountable kappa) if and only if T has no Vaughtian pairs and T is omega-stable. As in the classical setting, we approach the forward direction by proving a continuous version of Vaught's Two-Cardinal theorem. A continuous theory T has a (kappa, lambda)-model if there is M, a model of T, with density character kappa which has a definable subset with density character lambda. We show that if T has a (kappa, lambda)-model for infinite cardinals kappa>lambda, then T has an (aleph_1, aleph_0)-model. We also show that with the additional assumption that T is omega-stable, if T has an (aleph_1, aleph_0)-model, then for any uncountable kappa, T has a (kappa, aleph_0)-model. This provides us with the tools necessary to prove the forward direction of the Baldwin-Lachlan characterization of uncountable categoricity for continuous logic. Towards the reverse direction, we introduce a continuous notion of strong minimality, saying that a set is minimal if every subset which is the zero set of a definable predicate is totally bounded, or its approximate complements are all totally bounded. We show that this characterization is equivalent to the classical definition of strong minimality, and that it allows us to use algebraic closure to define a notion of dimension which determines models up to isomorphism. However, the only known examples of strongly minimal theories in the continuous setting are classical theories viewed as continuous with the discrete metric, and we see that this notion lacks the machinery necessary to prove the reverse direction of the Baldwin-Lachlan characterization of uncountable categoricity. In an effort to better understand minimality in continuous logic, we also introduce a continuous notion of dp-minimality, and provide a few equivalent characterizations. We use these to show that the theory of Infinite Dimensional Hilbert spaces is a natural example of a dp-minimal continuous theory.