We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.

We show how to encode, by classical structures, both the objects and the morphisms of the category of complete metric spaces and uniformly continuous maps. The result is a category of, what we call, cognate metric spaces and cognate maps. We show this category relativizes to all models of set theory. We extend this encoding to an encoding of complete metric structures by classical structures. This provide us with a general technique for translating results about infinitary logic on classical structures (...) to the setting of infinitary continuous logic on continuous structures. Our encoding will also allow us to talk about not only the relations between complete metric structures, but also the potential relations between complete metric structures, i.e. those which are satisfied in some larger model of set theory. For example we will show that given any two complete metric structures we can determine if they are potentially isomorphic by looking at any admissible set which contains them both. (shrink)

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result (...) is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic. (shrink)

We describe locally compact groups which are separably categorical metric structures.