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

We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $\kappa>\lambda\geq \aleph_0$, a continuous theory $T$ has a model with density character $\kappa$ which has a definable subset of density character $\lambda$, then $T$ has a model with density character $\aleph_1$ which has a separable definable subset. We also show that if we assume that $T$ is $\omega$-stable, then if $T$ has a model of density character $\aleph_1$ with a separable definable set, then for any uncountable $\kappa$ we can find a model of $T$ with density character $\kappa$ which has a separable definable subset. In order to prove this, we develop an approximate notion of quasi-minimality for the continuous setting. We apply these results to show a continuous version of the forward direction of the Baldwin-Lachlan characterization of uncountable categoricity: if a continuous theory $T$ is uncountably categorical, then $T$ is $\omega$-stable and has no Vaughtian pairs.