Approximate isomorphism of metric structures

Abstract We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and Yaacov, Doucha, Nies, Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences the perturbation systems former paper, such as Banach‐Mazur distance Lipschitz between metric spaces. Our is characterized syntactically mild generalization semantically certain elementary classes two‐sorted structures that witness isomorphism. As an application, show theory any ‐tree or ultrametric space finite radius stable, improving result Carlisle Henson [8].