We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of (...) the set instead of the cardinality, is ${\aleph_0}$ . In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs. (shrink)

We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators K by calculating them in the (...) finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle $. (shrink)

In recent years, mathematical logic has developed in many directions, the initial unity of its subject matter giving way to a myriad of seemingly unrelated areas. The articles collected here, which range from historical scholarship to recent research in geometric model theory, squarely address this development. These articles also connect to the diverse work of Väänänen, whose ecumenical approach to logic reflects the unity of the discipline.