Continuous logic for the classical logician

Let L be a first-order 2-sorted language. Let X be some fixed structure. A standard structure is an L-structure of the form ⟨M,X⟩. When X is a compact topological space (and L meets a few additional requirements) it is possible to adapt a significant part of model theory to the class of standard structures. This has been noticed by Henson and Iovino in the case of Banach spaces (and metric structures in general). However, in the last 20 years the most popular approach to the model theory of metric structures uses real-valued logic (Ben Yaacov, Berenstein, Henson, Usvyatsov). Arguably, this is neither natural nor general enough. We show that a few adaptations of Henson and Iovino's approach suffices for a natural and powerful theory. This is based on three facts: - every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent. - in a sufficiently saturated structure, the negation of a positive formula is an infinite disjunction of positive formulas. - there is a pure model theoretic notion that corresponds to Cauchy completeness. To exemplify how this setting applies to model theory we discuss ω-categoricity and (local) stability.