##
**Model theory for metric structures.**
*(English)*
Zbl 1233.03045

Chatzidakis, Zoé (ed.) et al., Model theory with applications to algebra and analysis. Vol. 2. Cambridge: Cambridge University Press (ISBN 978-0-521-70908-8/pbk). London Mathematical Society Lecture Note Series 350, 315-427 (2008).

Introduction: 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 \(\mathbb R\), 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. 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). 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.

The logic for metric structures that we describe here fits into the framework of continuous logics that was studied extensively in the 1960s and then dropped. In that work, any compact Hausdorff space \(X\) was allowed as the set of truth values for a logic. This turned out to be too general for a completely successful theory.

We take the space \(X\) of truth values to be a closed, bounded interval of real numbers, with the order topology. It is sufficient to focus on the case where \(X\) is \([0, 1]\). In [C. C. Chang and H. J. Keisler, Continuous model theory. Princeton, N.J.: Princeton University Press (1966; Zbl 0149.00402)], a wide variety of quantifiers was allowed and studied. Since our truth value set carries a natural complete linear ordering, there are two canonical quantifiers that clearly deserve special attention; these are the operations sup and inf, and it happens that these are the only quantifiers we need to consider in the setting of continuous logic and metric structures.

The continuous logic developed here is strikingly parallel to the usual first-order logic, once one enlarges the set of possible truth values from \(\{0, 1\}\) to \([0, 1]\). Predicates, including the equality relation, become functions from the underlying set \(A\) of a mathematical structure into the interval \([0, 1]\). Indeed, the natural \([0, 1]\)-valued counterpart of the equality predicate is a metric \(d\) on \(A\) (of diameter at most 1, for convenience). Further, the natural counterpart of the assumption that equality is a congruence relation for the predicates and operations in a mathematical structure is the requirement that the predicates and operations in a metric structure be uniformly continuous with respect to the metric \(d\). In the \([0, 1]\)-valued continuous setting, connectives are continuous functions on \([0, 1]\) and quantifiers are sup and inf.

The analogy between this continuous version of first-order logic (CFO) for metric structures and the usual first-order logic (FOL) for ordinary structures is far reaching. In suitably phrased forms, CFO satisfies the compactness theorem, Löwenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination, Beth’s definability 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. Moreover, CFO extends FOL: indeed, each mathematical structure treated in FOL can be viewed as a metric structure by taking the underlying metric \(d\) to be discrete (\(d(a, b) = 1\) for distinct \(a\), \(b\)). All these basic results true of CFO are thus framed as generalizations of the corresponding results for FOL.

A second type of justification for focusing on this continuous logic comes from its connection to applications of model theory in analysis and geometry. These often depend on an ultraproduct construction or, equivalently, the nonstandard hull construction. This construction is widely used in functional analysis and also arises in metric space geometry. The logic of positive bounded formulas was introduced in order to provide a model-theoretic framework for the use of this ultraproduct (see [C. W. Henson and J. Iovino, “Ultraproducts in analysis”, Lond. Math. Soc. Lect. Note Ser. 262, 1–110 (2002; Zbl 1026.46007)]), which it does successfully. The continuous logic for metric structures that is presented here provides an equivalent background for this ultraproduct construction and it is easier to use. Writing positive bounded formulas to express statements from analysis and geometry is difficult and often feels unnatural; this goes much more smoothly in CFO. Indeed, continuous first-order logic provides model theorists and analysts with a common language; this is due to its being closely parallel to first-order logic while also using familiar constructs from analysis (e.g., sup and inf in place of \(\forall\) and \(\exists\)).

The purpose of this article is to present the syntax and semantics of this continuous logic for metric structures, to indicate some of its key theoretical features, and to show a few of its recent application areas.

In Sections 1 through 10 we develop the syntax and semantics of continuous logic for metric structures and present its basic properties. We have tried to make this material accessible without requiring any background beyond basic undergraduate mathematics. Sections 11 and 12 discuss imaginaries and omitting types; here our presentation is somewhat more brisk and full understanding may require some prior experience with model theory. Sections 13 and 14 sketch a treatment of quantifier elimination and stability, which are needed for the applications topics later in the paper; here we omit many proofs and depend on other articles for the details. Sections 15 through 18 indicate a few areas of mathematics to which continuous logic for metric structures has already been applied; these are taken from probability theory and functional analysis, and some background in these areas is expected of the reader.

The development of continuous logic for metric structures is very much a work in progress, and there are many open problems deserving of attention. What is presented in this article reflects work done over approximately the last three years in a series of collaborations among the authors. The material presented here was taught in two graduate topics courses offered during that time: a Fall 2004 course taught in Madison by Itaï Ben Yaacov and a Spring 2005 course taught in Urbana by Ward Henson.

For the entire collection see [Zbl 1152.03006].

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. 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). 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.

The logic for metric structures that we describe here fits into the framework of continuous logics that was studied extensively in the 1960s and then dropped. In that work, any compact Hausdorff space \(X\) was allowed as the set of truth values for a logic. This turned out to be too general for a completely successful theory.

We take the space \(X\) of truth values to be a closed, bounded interval of real numbers, with the order topology. It is sufficient to focus on the case where \(X\) is \([0, 1]\). In [C. C. Chang and H. J. Keisler, Continuous model theory. Princeton, N.J.: Princeton University Press (1966; Zbl 0149.00402)], a wide variety of quantifiers was allowed and studied. Since our truth value set carries a natural complete linear ordering, there are two canonical quantifiers that clearly deserve special attention; these are the operations sup and inf, and it happens that these are the only quantifiers we need to consider in the setting of continuous logic and metric structures.

The continuous logic developed here is strikingly parallel to the usual first-order logic, once one enlarges the set of possible truth values from \(\{0, 1\}\) to \([0, 1]\). Predicates, including the equality relation, become functions from the underlying set \(A\) of a mathematical structure into the interval \([0, 1]\). Indeed, the natural \([0, 1]\)-valued counterpart of the equality predicate is a metric \(d\) on \(A\) (of diameter at most 1, for convenience). Further, the natural counterpart of the assumption that equality is a congruence relation for the predicates and operations in a mathematical structure is the requirement that the predicates and operations in a metric structure be uniformly continuous with respect to the metric \(d\). In the \([0, 1]\)-valued continuous setting, connectives are continuous functions on \([0, 1]\) and quantifiers are sup and inf.

The analogy between this continuous version of first-order logic (CFO) for metric structures and the usual first-order logic (FOL) for ordinary structures is far reaching. In suitably phrased forms, CFO satisfies the compactness theorem, Löwenheim-Skolem theorems, diagram arguments, existence of saturated and homogeneous models, characterizations of quantifier elimination, Beth’s definability 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. Moreover, CFO extends FOL: indeed, each mathematical structure treated in FOL can be viewed as a metric structure by taking the underlying metric \(d\) to be discrete (\(d(a, b) = 1\) for distinct \(a\), \(b\)). All these basic results true of CFO are thus framed as generalizations of the corresponding results for FOL.

A second type of justification for focusing on this continuous logic comes from its connection to applications of model theory in analysis and geometry. These often depend on an ultraproduct construction or, equivalently, the nonstandard hull construction. This construction is widely used in functional analysis and also arises in metric space geometry. The logic of positive bounded formulas was introduced in order to provide a model-theoretic framework for the use of this ultraproduct (see [C. W. Henson and J. Iovino, “Ultraproducts in analysis”, Lond. Math. Soc. Lect. Note Ser. 262, 1–110 (2002; Zbl 1026.46007)]), which it does successfully. The continuous logic for metric structures that is presented here provides an equivalent background for this ultraproduct construction and it is easier to use. Writing positive bounded formulas to express statements from analysis and geometry is difficult and often feels unnatural; this goes much more smoothly in CFO. Indeed, continuous first-order logic provides model theorists and analysts with a common language; this is due to its being closely parallel to first-order logic while also using familiar constructs from analysis (e.g., sup and inf in place of \(\forall\) and \(\exists\)).

The purpose of this article is to present the syntax and semantics of this continuous logic for metric structures, to indicate some of its key theoretical features, and to show a few of its recent application areas.

In Sections 1 through 10 we develop the syntax and semantics of continuous logic for metric structures and present its basic properties. We have tried to make this material accessible without requiring any background beyond basic undergraduate mathematics. Sections 11 and 12 discuss imaginaries and omitting types; here our presentation is somewhat more brisk and full understanding may require some prior experience with model theory. Sections 13 and 14 sketch a treatment of quantifier elimination and stability, which are needed for the applications topics later in the paper; here we omit many proofs and depend on other articles for the details. Sections 15 through 18 indicate a few areas of mathematics to which continuous logic for metric structures has already been applied; these are taken from probability theory and functional analysis, and some background in these areas is expected of the reader.

The development of continuous logic for metric structures is very much a work in progress, and there are many open problems deserving of attention. What is presented in this article reflects work done over approximately the last three years in a series of collaborations among the authors. The material presented here was taught in two graduate topics courses offered during that time: a Fall 2004 course taught in Madison by Itaï Ben Yaacov and a Spring 2005 course taught in Urbana by Ward Henson.

For the entire collection see [Zbl 1152.03006].

### MSC:

03C65 | Models of other mathematical theories |

03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |

28A99 | Classical measure theory |

46B99 | Normed linear spaces and Banach spaces; Banach lattices |

54E35 | Metric spaces, metrizability |