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A metric structure is a many-sorted structure in which each sort is a complete metric space of finite diameter. Additionally, the structure consists of some distinguished elements as well as some functions (of several variables) (a) between sorts and (b) from sorts to bounded subsets of ℝ, and these functions are all required to be uniformly continuous. Examples arise throughout mathematics, especially in analysis and geometry. They include metric spaces themselves, measure algebras, asymptotic cones of finitely generated groups, and structures based on Banach spaces (where one takes the sorts to be 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 alternative is the logic of positive bounded formulas with an approximate semantics (see [23, 25, 24]). This was developed for structures from functional analysis that are based on Banach spaces; it is easily adapted to the more general metric structure setting that is considered here. Another successful alternative is the setting of compact abstract theories (cats; see [1, 3, 4]). A recent development is the realization that for metric structures the frameworks of positive bounded formulas and of cats are equivalent. (The full cat framework is more general.) Further, out of this discovery has come a new continuous version of first-order logic that is suitable for metric structures; it is equivalent to both the positive bounded and cat approaches, but has many advantages over them.
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