Second Vaught’s conjecture workshop, Berkeley, June 2015 (Berkeley slides)
Abstract: Vaught’s conjecture has numerous model-theoretic and topological variations. One key connection between the model-theoretic and topological sides is provided by López-Escobar’s theorem, which states that any Borel class of countable structures can be axiomatized using a sentence of infinitary logic (with countable conjunctions and disjunctions). We present a variant of López-Escobar’s theorem for metric structures, which implies that any Borel class of separable metric structures can be axiomatized using a sentence of an appropriate infinitary continuous logic. As a consequence we obtain a new implication between the topological Vaught conjecture and a model-theoretic version for metric structures. This is joint work with Martino Lupini.