Encoding Complete Metric Structures by Classical Structures

Logica Universalis 14 (4):421-459 (2020)
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Abstract

We show how to encode, by classical structures, both the objects and the morphisms of the category of complete metric spaces and uniformly continuous maps. The result is a category of, what we call, cognate metric spaces and cognate maps. We show this category relativizes to all models of set theory. We extend this encoding to an encoding of complete metric structures by classical structures. This provide us with a general technique for translating results about infinitary logic on classical structures to the setting of infinitary continuous logic on continuous structures. Our encoding will also allow us to talk about not only the relations between complete metric structures, but also the potential relations between complete metric structures, i.e. those which are satisfied in some larger model of set theory. For example we will show that given any two complete metric structures we can determine if they are potentially isomorphic by looking at any admissible set which contains them both.

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References found in this work

Positive model theory and compact abstract theories.Itay Ben-Yaacov - 2003 - Journal of Mathematical Logic 3 (01):85-118.
Model theoretic forcing in analysis.Itaï Ben Yaacov & José Iovino - 2009 - Annals of Pure and Applied Logic 158 (3):163-174.
On perturbations of continuous structures.Itaï Ben Yaacov - 2008 - Journal of Mathematical Logic 8 (2):225-249.
Definability of groups in ℵ₀-stable metric structures.Itaï Ben Yaacov - 2010 - Journal of Symbolic Logic 75 (3):817-840.
Omitting types for infinitary [ 0, 1 ] -valued logic.Christopher J. Eagle - 2014 - Annals of Pure and Applied Logic 165 (3):913-932.

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