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September 17, 2025Open Access

Continuous Logic in a Classical Setting

Key Points

A similar approach to continuous-valued logic extends to more general models, enriching the field of model theory.
Key elements such as elementarity and quantifier elimination are introduced to broaden understanding within compact Hausdorff spaces.
The work bridges earlier models by Henson and Iovino, adapting formulas for a wider application in analysis and logic.
This framework enhances the study of Banach space model theory, potentially offering novel insights into complex structures.

Abstract

Abstract Let {L} L be a first-order two-sorted language and consider a class of L L -structures of the form M, X ⟨ 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 et al. (Model theory for metric structures, Cambridge University Press, 2010). An earlier approach due to Henson and Iovino (Ultraproducts in analysis, analysis and logic (Mons, 1997), London Math. Soc. Lecture Note Ser. , Cambridge Univ. Press, Cambridge, 2002) is based on the notion of positive formulas and 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.

Continuous Logic in a Classical Setting

Key Points

Abstract

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