ON d-FINITENESS IN CONTINUOUS STRUCTURES

Abstract. 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 mentioning finite tuples are valid in continuous logic when 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. The main goal of this article is to describe and study conditions under which certain results of classical model theory generalise to the model theory of metric structures, and to explain why when they do not. We start by recalling Henson’s adaptation of the Ryll-Nardzewski Theorem to metric logics (originally for the logic of positive bounded formulae, but we state and prove it for continuous first order logic). It characterises the family of countable ω-categorica