Encoding Complete Metric Structures by Classical Structures
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Logica Universalis
Encoding Complete Metric Structures by Classical Structures Nathanael Leedom Ackerman 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 (unlike the category of complete metric spaces and uniformly continuous maps). 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. Mathematics Subject Classification. 03C90, 54E50, 54E40, 03C75, 03C30. Keywords. Complete metric spaces, Uniformly continuous maps, Continuous logic, Infinitary logic.
1. Introduction First order logic is a remarkably effective tool for investigating mathematical structures. However, despite its overall success, many of the results of first order logic break down when studying structures which are from analysis or which are fundamentally based on metric spaces, for example C ∗ -algebras, Banach lattices and metric spaces themselves (see for example [25]). Over the years there have been many attempts to find a logic appropriate for the study of such structures (see for example [14,16,19]). However within the last decade and a half continuous first order logic, as introduced in [10], has emerged as one of the most promising candidates. In particular it has been
N. L. Ackerman
Log. Univers.
shown to coincide with several other such logics and in some sense has be shown to be a maximal such logic (see [13,21]). With the discovery of continuous first order logic there has been a great deal of effort and success in proving analogs of results from first order logic for classical structure. For example in [7,9] (among others) analogs various stability theoretic properties are developed for continuous logic. In addition to lifting results from first order logic to the context of continuous logic, recently there has begun to be work studying infinitary analogs of continuous first order logic and the descriptive set theory of metric structures. For example, in [11,18] omitting types theorems are proved for infinitary analogs of continuous logic (although they use slightly different notions of an infinitary continuous formula). Also [12,17] both prove a analog of the L´opez-Escobar theorem whose classical form connects Lω1 ,ω with Borel invariant subsets of the
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