UC Berkeley

We investigate structures of size at most continuum using various techniques originating from computable structure theory and continuous logic. Our approach, which we are naming ``computable continuous structure theory", allows the fine-grained tools of computable structure theory to be generalized to apply to a wide class of separable completely-metrizable structures, such as Hilbert spaces, the p-adic integers, and many others. We can generalize many ideas, such as effective Scott families and effective type-omitting, to this wider class of structures. Since our logic respects the underlying topology of the space under consideration, it is in some sense more natural for structures with a metrizable topology which is not discrete.