Traditional first-order logic deals with

(a) The standard sentential connectives such as ``or'', ``and'', ``not'', and ``if--then'';

(b)} A binary truth system: every sensible statement is either true or false.

One of the focuses of the workshop on the Model Theory of Metric Structures is a new type of logic called continuous logic. Although it has roots going back to the 1920s in work of Lukasiewicz, we say that continuous logic is new because it has only recently found a context and a program for its many tools.

If we think of classical logic as taking truth values 0 (for false) and 1 (for true), then it is natural to think of continuous logic as taking truth values in the entire interval [0,1]. It turns out to be convenient to flip the convention around and take 0 to denote true and 1 to denote false. Truth values strictly in between 0 and 1 should be construed as degrees of truth or falsity. It may be noted that the theory of fuzzy sets, which found its provenance in the theory of electrical engineering in the 1950s, utilizes a truth system that is quite similar to this one.

The classical sentential connectives have interesting interpretations in the language of continuous logic. As an instance, we think of sentential negation (e.g., replacing ``This hat is red.'' by ``This hat is not red.'') as the mapping x → 1 - x. If we think of the classical system, in which 0 is ``false'' and 1 is ``true'', then this makes perfect sense.

Perhaps more interesting is the continuous logic model for implication. It turns out that the right way to formulate the idea is to define

x ∼ y ≡ max (x - y, 0) .

It is important to notice that this definition of x - y is different from the ordinary arithmetic notion of subtraction. It is asymmetric in the arguments. In continuous logic, we think of x - y as corresponding to y ⇒ x. The idea here is that y is a ``weaker truth'' than x, and that is what implication really means. As a simple example, if y is false (i.e., truth value 1) and x is true (i.e., truth value 0), then (according to our definition) x - y = 0 - 1 = 0. That is of course true. And, indeed, y → x is true because y is false. An implication with false hypothesis is always true.

In fact, in continuous logic, we can take any continuous function as a connective. It turns out that the lattice version of the Stone-Weierstrass theorem allows us to show that any such connective can be approximated by very special and simple connectives which are easy to manipulate. This gives us considerable flexibility in constructing models. And that is the main point as we shall see momentarily.

It is also possible to formulate the standard quantifiers (``there exists'' and ``for all'') in the language of continuous logic. In fact ``there exists'' is simply an infimum of a graph or locus. And ``for all'' is the dual concept under negation.

In traditional logic, it is easy and natural to model algebraic structures. In continuous logic, it is easy and natural to model analytic structures (which tend to be larger).

In order to describe the utility of continuous logic in infinite dimensions, we first must describe functional analysis. A branch of mathematics that is about 100 years old, functional analysis is the study of infinite dimensional spaces. Typically these are spaces of functions.

A metric space is a set X equipped with a notion of distance. Call the distance ρ. We demand that ρ satisfy a couple of simple axioms:

(a) ρ(x,y) ≥ 0 for all x, y;
(b) ρ(x,y) = 0 if and only if x = y;
(c) ρ(x,y) ≤ ρ(x,z) + ρ(y,z) for any x,y,z ∈ X.

Of course the standard Euclidean space R^N is a metric space when equipped with the usual metric

ρ(x,y) = [∑_j=1^N (x_j - y_j)²]^1/2 .

A more interesting example of a metric space is the collection S of all closed, bounded sets in R². If S and T are two such sets then we define a distance by

d(S, T) = max { sup_{t ∈ T} [ inf_{s ∈ S} |s - t| ] , sup_{s ∈ S} [ inf_{t ∈ T} |s - t| ] } .

This is the Hausdorff metric on sets.

We are particularly interested in examples of a linear space that is equipped with a norm that makes it complete. A simple example is the space
C[0,1], which is the collection of all continuous functions with domain the interval [0,1]. What is particularly interesting in functional analysis is the topology with which we equip the space under study. A standard topology to put on C[0,1] is that of uniform convergence: If f, f_j ∈ C[0,1] then we say that f_j → f if the functions f_j converge to the function f uniformly. With this topology, the space C[0,1] is complete. This means that every Cauchy sequence (a sequence of elements that gets ever closer together) actually converges to an element of the space. [This is an abstract restatement of the fact that the uniform limit of continuous functions is continuous.] We call a complete linear space a Banach space.

Banach space theory is a big part of modern mathematics. It impinges on real analysis, complex analysis, partial differential equations, global analysis, differential geometry, and many other areas as well. The purpose of the workshop on Model Theory of Metric Structures is to explore the interface between continuous logic and Banach space theory. The workshop brings together model theorists from logic and analysts from the field of functional analysis. It is quite unusual to attempt a dialogue between two such disparate groups, but in fact these groups turn out to have a great deal in common.

An interesting construction from basic logic is the ultrafilter. An ultrafilter is a logical device for making decisions. Each ultrafilter gives rise to an equivalence relation. This device was used decisively by Abraham Robinson to construct the nonstandard real numbers---a number system that contains the classical reals as well as the infinitesimals and the infinitary numbers. Ultrafilters in turn can be used to construct a generalization of Cartesian products called ultraproducts. Ultrafilters and ultraproducts fit very naturally into the language of continuous logic. Whereas ultraproducts in algebra are useful for classical logic, ultraproducts in Banach spaces are useful in continuous logic.

The important fact for this subject area is that an ultraproduct of Banach spaces is still a Banach space. Thus ultraproducts are particularly useful for constructing examples in Banach space theory. And continuous logic provides the benchmark for what is ultimately true. This circle of ideas has been particularly effective for studying measure-based Banach spaces (such as the L^p spaces, or spaces of p^th-power integrable functions) and also the classical space C(X), the continuous functions on a compact space X.

There are actually concrete results about these Banach spaces that can be proved using methods of ultraproducts and continuous logic and for which there does not exist a classical proof. This fact is both bewildering and heartening, for it attests to the value of these new ideas. It is rewarding to see logic interact in such a positive and constructive manner with a traditional concrete area of mathematics (functional analysis is used routinely to study real analysis, partial differential equations, Fourier analysis, and other traditional parts of mathematics). The AIM workshop developed both explored the theoretical aspects of the model theory of continuous logic and also the many new connections among the areas we have described. It solidified existing bonds and created new ones.