Publication date:
1 October 2017
Source:Advances in Mathematics, Volume 318
Author(s): Itaï Ben Yaacov, Michal Doucha, André Nies, Todor Tsankov
We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the López-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below ${\omega }_1$. Finally, we apply our methods to study the Gromov–Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance 0 to a fixed space is a Borel set.