Let G be a countable group and X a Borel G-space. Then it is clear that is a Borel equivalence relation and every one of its equivalence class is countable.

Definition 4.1. A Borel equivalence relation E is countable if every equivalence class is countable.

We now have

Theorem 4.2 (Feldman-Moore [77]). The following are equivalent for each Borel equivalence relation E on a Polish space X:

1. (i) E is countable.

2. (ii) E = for some countable group G and a Borel G-space X.

Countable Borel equivalence relations have long been studied in ergodic theory and its relationship to the theory of operator algebras. One important observation is that many concepts of ergodic theory such as invariance, quasi-invariance (null set preservation), and ergodicity of measures depend only on the orbit equivalence relation and not the action inducing it. Also a countable Borel equivalence relation with an associated quasi-invariant probability measure gives rise to a canonical von Neumann algebra and this has important implications to classification problems of von Neumann algebras. See, for example, the survey Schmidt [90].

The simplest examples of countable Borel equivalence relations are those induced by Borel actions of the group of integers ℤ, i.e., by the orbits of a single Borel automorphism. These are also called hyperfinite in view of the following result.

Theorem 4.3 (Slaman-Steel [88], Weiss [84]). For each countable Borel equivalence relation E on a Polish space X the following are equivalent:

1. (i) E =, for some Borel ℤ-space X.

2. […]

We cannot end before at least briefly discussing one other spectacular result of the 90's.

We recall that the homogeneous Banach space problem (P5) is: If X is isomorphic to all Y ⊆ X, is X isomorphic to l2? This was solved by combining two beautiful pieces of work, Gowers' dichotomy theorem (Theorem 3.1) and the following theorem of Komorowski and Tomczak-Jaegermann [KT1, KT2]. A nice exposition somewhat simplifying the argument appears in [TJ1].

8.1 Theorem. If X is homogeneous and not isomorphic to l2 then X has a subspace without an unconditional basis.

It then follows that no subspace of X has an unconditional basis and so by the dichotomy theorem plus the fact that X is homogeneous we have that X must be H.I. But in view of the result of [GM1] that an H.I. space is not isomorphic to any proper subspace, this is impossible. Thus the solution of the homogeneous Banach space problem is achieved.

Komorowski and Tomczak-Jaegermann actually prove something stronger. They show

Theorem. Let X be a Banach space not containing a subspace isomorphic to l2. Then X contains a subspace without an unconditional basis.

Even more recently Komorowski and Tomczak-Jaegermann have made substantial progress on (Q4): if all subspaces of X have an unconditional basis is X isomorphic to l2? They proved [KT3].

Theorem. If every subspace of(X ⊕ X ⊕ …)l2 has an unconditional basis then X is isomorphic to l2.

The proof of Theorem 8.2 is well exposed in [TJ1] and we shall not repeat it here. It is interesting to note that spreading models enter at one part of the argument.

In this chapter we introduce the key ingredients of the logic for normed space structures that is described in this paper. These are the positive bounded formulas and the concept of approximate satisfaction of such formulas in normed space structures.

Let L be a signature for a normed space structure ℳ based on (M(s) ∣ s ∈ S). Recall that S has a distinguished element s = Sℝ for which M(s) = ℝ is the sort of real numbers.

We begin considering ℳ from the model theoretic point of view, introducing a formal language based on L and a semantics according to which this language is interpreted in ℳ. In addition to the symbols of the signature L, we also need for each element s of the sort index set S, a countable set of symbols called the variables of sort s.

We begin defining the formal language by introducing the set of terms of L, or L-terms. Each term is a finite string of symbols, each of which may be a variable or a function symbol of L, or one of the symbols (or, which are used for punctuation. In this many-sorted context, each term is associated with a unique sort which indicates its range. The formal definition is recursive.

Definition. An L-term with range of sort s is a string which can be obtained by finitely many applications of the following rules of formation:

• If x is a variable of L of sort s, then x is a term with range of sort s.

• […]