On the classification of Polish metric spaces up to isometry

(A) A Polish metric space is a complete separable metric space (X, d). Our first goal in this paper is to determine the exact complexity of the classification problem of Polish metric spaces up to isometry. Our work was motivated by a recent paper of Vershik [1998], where the author remarks (in the beginning of Section 2): “The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” The first main theorem (Theorem 1 below) quantifies precisely the enormity of this task. We will first summarize the basic ideas of a theory of complexity of classification problems, which will help to put our results in perspective. Detailed expositions can be found, e.g., in Hjorth [2000], Kechris [1999], [2000]. In mathematics one frequently deals with problems of classification of various objects up to some notion of equivalence by invariants. Quite often these objects can be viewed as forming a definable (Borel, analytic, etc.) subset X of a standard Borel space X̂ (i.e., a Polish space with its associated σ-algebra of Borel sets), and the equivalence relation as a definable (Borel, analytic, etc.) equivalence relation E on X. A complete classification of X up to E consists then of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). For this to be of interest both I and c must be as simple and concrete as possible. For our purposes, the simplest case is when the invariants are concrete enough so that they can be represented as elements of a standard Borel space (and the map c is fairly explicitly definable). More precisely let us call E (and the classification problem it represents) concretely classifiable (or smooth or tame) if there is a standard Borel space Y and a Borel (measurable) map c : X → Y such that xEy ⇔ c(x) = c(y).