Model theory for metric structures

Abstract

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally there are functions (of several variables) between sorts, assumed to be uniformly continuous. Examples include metric spaces themselves, measure algebras (with the metric d(A, B) = µ(A∆B), where ∆ is symmetric difference), and structures based on Banach spaces (where one interprets the sorts as balls), including Banach lattices, C*algebras, etc. The usual first-order logic does not work very well for such structures, and several good alternatives have been developed. One is the logic of positive bounded formulas with an approximate semantics (see [3]). Another is the setting of compact abstract theories (cats) (see [1]). A recent development, which will be emphasized in this course, is the convergence of these two points of view and the realization that they are also connected to the [0,1]-valued continuous logic that was studied extensively in the 1960s and then dropped (see [2]). This leads to a new formalism for the logic of metric structures in which the operations of sup and inf play a central role. The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. This new logic satisfies the compactness theorem, Löwenheim-Skolem