On $d$-finiteness in continuous structures
We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. The notion of a $d$-finite tuple attempts to capture some aspects of the classical finite tuple behaviour. We show that many classical results involving finite tuples are valid in continuous logic upon replacing “finite” with “$d$-finite”. Other results, such as Vaught's no two models theorem and Lachlan's theorem on the number of countable models of a superstable theory are proved under the assumption of enough (uniformly) $d$-finite tuples.