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) = m(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 development of this continuous logic for metric structures is an ongoing project being carried out as a collaboration among Itay Ben-Yaacov (Wisconsin), Alex Berenstein (Illinois), Ward Henson (Illinois), and Alex Usvyatsov (Jerusalem).

The analogy between this logic for metric space structures and the usual first order logic for ordinary structures is far reaching. Continuous logic satisfies the compactness theorem, Lowenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination and model completeness, Beth's definability theorem, Craig's interpolation theorem, the omitting types theorem, fundamental results of stability theory, and appropriate analogues of essentially all results in basic model theory of first order logic.

The theory of definable sets and functions in this continuous logic for metric structures has novel aspects and there are numerous open problems of apparent interest.

Extending important concepts from first order model theory to this continuous logic often presents a challenge.

The purpose of this tutorial is to present this logic for metric structures, and to show a few of its application areas, with emphasis on probability spaces and certain linear spaces from functional analysis.

References:

[1] I. Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Logic 3 (2003), 85--118.

[2] C. C. Chang and H. J. Keisler, Continuous Model Theory, Princeton Univ. Press (1966).

[3] C. W. Henson and J. Iovino, Ultraproducts in Analysis, in Analysis and Logic, London Mathematical Society Lecture Notes 262, Cambridge University Press (2002), 1--113.