A presentation theorem for continuous logic and metric abstract elementary classes

In the spirit of Chang and Shelah’s presentation results (from [Cha68] and [Sh88], respectively), we prove a presentation theorem for classes of continuous structures, both those axiomatized by first-order and beyond, in terms of a class of discrete structures. The thrust of this presentation theorem is the basic analytic fact that the behavior of continuous functions is determined by their values on a dense subset of their domain. Focusing on dense subsets is key because it allows us to drop the requirement that structures be complete, which is not a property expressible by discrete (classical) logic, even in the broader contexts of Lλ,ω or Abstract Elementary Classes. The specific statements of the presentation theorems appear below (see Theorem 2.1 for continuous first-order logic and Theorem 6.1 for Metric Abstract Elementary Classes), but the general idea is the same in both cases: given a continuous language L, we define a discrete language L that allows us to approximate the values of the functions and relations by a countable dense subset of values, namely Q ∩ [0, 1]. Note that the specification that this dense set (and its completion) is standard already requires an L+ω1,ω sentence, even if we are working in continuous first-order logic. Then, given a continuous L-structure M and a nicely dense (see Definition 1.1 below) subset of it A, we can form a discrete L-structure A with universe A that encodes all of M .