Submission date: 2 February 2024

Abstract:

Let ℒ be a first-order two-sorted language and consider a class of ℒ-structures of the form < M, X > where M varies among structures of the first sort, while X is fixed in the second sort, and it is assumed to be a compact Hausdorff space. When X is a compact subset of the real line, one way to treat classes of this kind model-theoretically is via continuous-valued logic, as in [Ben Yaacov-Berenstein-Henson-Usvyatsov 2010]. Prior to that, Henson and Iovino proposed an approach based on the notion of positive formulas [Henson-Iovino 2002]. Their work is tailored to the model theory of Banach spaces. Here we show that a similar approach is possible for a more general class of models. We introduce suitable versions of elementarity, compactness, saturation, quantifier elimination and other basic tools, and we develop basic model theory.

Mathematics Subject Classification: 03C66

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Full text arXiv 2402.01245: pdf, ps.