Classical descriptive set theory as a refinement of effective descriptive set theory

The (effective) Suslin-Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation. Effective Descriptive Set Theory is advertized as a refinement of the classical theory of definability (on Polish spaces) developed in the first half of the 20th century, for example in the introduction to Moschovakis (2009a). Consider the following paradigmatic case, where X = X1 × · · · ×Xn (1) is a product of copies of the natural numbers N = {0, 1, . . .} and N = (N → N), the classical Baire space of all infinite sequences from N: Suslin’s Theorem (Suslin (1917)). If A ⊆ X and X \A are both analytic, then A is Borel measurable. Suslin-Kleene Theorem. There is a recursive function u : N ×N → N , such that if α is a code of an analytic set A ⊆ X and β is a code of its complement X \A, then u(α, β) is a Borel code of A. Even without precise definitions of the notions and the codings used in these results (which will be given in the sequel), their statements suggest that the second theorem refines the first, as it provides a uniformity, an effective method to transform an “analytic-coanalytic” definition of a set A ⊆ X into a “Borel construction” of A. In fact, it is a much stronger result with wider applicability: Suslin’s Theorem is vacuous when X = N, since every set of natural numbers is (trivially) Borel measurable, while the Suslin-Kleene Theorem yields in this case (very easily) one of the most celebrated results of Kleene: The basic idea for this article was presented in Moschovakis (1971). The full paper was never written up, but I thought that it would fit well in a volume honoring Prof. N. A. Shanin. Preprint submitted to Elsevier October 14, 2010 Kleene’s Theorem (Kleene (1955a,b)). Every ∆1 set A ⊆ N is hyperarithmetical. This simple analysis, however, does not do justice to the classical theory, because it fails to take into account the “constructive bent” of the analysts who developed it: the standard proof of Suslin’s Theorem in Kuratowski (1966) or Moschovakis (2009a) is, in fact, constructive, and if we apply to it the sort of realizability analysis pioneered by Kleene, which is well-understood today, it yields the Suslin-Kleene Theorem. From this point of view, the classical work is a refinement of the modern theory, since it yields the uniformities which refine the statements of the classical results, and it also provides constructive proofs that they do. This is the main point that I want to make in this article, and it basically amounts to an observation about the work of Stephen Cole Kleene: his deepest result in what we now call effective descriptive set theory is a direct corollary of classical work and his independently developed (and to the innocent eye unrelated) work in the foundations of intuitionism. Kleene’s main technical tool is his Second Recursion Theorem, which he applies in both legs of his work: one might say that the observation we will make here simply reduces these crucial applications of the Recursion Theorem from two to one. I have included in the last section a discussion of the generality of the method in the article and whether it justifies the title. Since there are very few researchers who are familiar with both descriptive set theory and intuitionism and I would like to make these ideas accessible more broadly, I am including below precise definitions of all the notions I need, as well as (condensed) outlines of the required arguments. 1. Recursion in Baire space We summarize here the basic facts about recursive partial functions with variables ranging over N or N = (N → N) and values in N or N . To simplify notation, we reserve the Latin letters e,m, s, t, u, v, w (perhaps with subscripts) for variables over N; the Greek letters α, β, γ, δ for variables over N ; and x, y, z for variables over points, i.e., members of product spaces as in (1). By definition (X1 × · · · × Xn) × (Y1 × · · · × Ym) = X1 × · · · × Xn × Y1 × · · · × Ym, and if x = (x1, . . . , xn), y = (y1, . . . , ym), then (x, y) = x ⋆ y = (x1, . . . , xn, y1, . . . , ym). Subsets of these product spaces are called pointsets. Cf. the companion articles Moschovakis (2009b, 2010). A partial function f : X ⇀ Y is a (total) function f : Df → Y defined on some subset Df ⊆ X , its domain of convergence, and for x ∈ X , we write f(x) ↓ ⇐⇒ x ∈ Df . Partial functions compose strictly, so that f(g1(x), . . . , gm(x))↓ =⇒ g1(x)↓ , . . . , gm(x)↓ .