Continuous logic in a classical setting
Claudio Agostini, Stefano Baratella, Silvia Barbina, Luca Motto Ros, Domenico Zambella
TL;DR
The paper develops a robust two-sorted classical model-theoretic framework for structures $\langle M,X\rangle$ with a fixed compact Hausdorff space $X$, aiming to capture continuous-like phenomena while preserving classical truth values. It introduces the $\EuScript F$-fragment of formulas, a notion of approximation and strong negation, and a standard-part construction via nonstandard analysis to achieve a compactness theorem and saturated models. It then builds a monster model, proves approximate elimination of sort-$S$ quantifiers, provides a Tarski–Vaught type test and downward Löwenheim–Skolem in separable languages, and develops a theory of Cauchy complete models, including a concrete bounded-metric-spaces example. The work bridges continuous-valued logic and classical model theory, yielding tools for analyzing two-sorted, fixed-topology structures and paving the way for applications to metric- or Banach-space-like settings within a rigorous model-theoretic framework.
Abstract
Let $\mathcal{L}$ be a first-order two-sorted language and consider a class of $\mathcal{L}$-structures of the form $\langle M, X \rangle$ 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.