Continuous vs classical structures (1) I am moving my first steps in continuous model theory (tl;dr). This is one of two soft questions on the relation between a continuous and a classical structure.
What can be said about continuous structures with the discrete metric (but real valued predicates). Are these "essentially" classical first order structures? (I guess not but, as the metric is trivial, in what other sense are they trivial?)
 A: Here is a simple example that I would say shows that even uniformly discrete metric structures are not really fully captured by discrete first-order logic (at least in terms of the things that model theorists care about):
Consider the structure $(M,d,P)$ where $M = [0,1]$, $d$ is the discrete metric on $[0,1]$, and $P$ is a predicate satisfying $P(x) = x$ for all $x \in [0,1]$.
This is a very simple example in the sense that it is weakly minimal with trivial geometry, but its theory $T$ is strictly superstable, because $S_1(T)$ has uncountable density character (relative to the $d$-metric). In particular, $S_1(T)$ is homeomorphic to $[0,1]$ (and has a discrete $d$-metric).
Despite the fact that $T$ is strictly superstable and is interpretable in a discrete superstable theory, one can prove that  $T$ only interprets $\omega$-stable discrete theories. In this sense, it's hard to say that $T$ is really morally equivalent to any particular discrete theory.
For a significantly less trivial examples: Hrushovski has a recent paper on some heavy model theory involving uniformly discrete metric structures that are not equivalent to discrete structures. There's also a recent paper by Gomez and Pillay on discrete abelian groups equipped with homomorphisms to a compact Hausdorff group, which one can encode with $[0,1]$-valued predicates.
