Recursion Theory on Baire’s Space

In this chapter we lay the basis for a unified Type 2 recursion theory. Our investigations from Chapter 3.1 and especially Theorem 3.1.14 show that it suffices to study continuity and computability on Baire’s space B . We shall choose the set B = ℕℕ as our standard set on which we develop a theory of continuity (w.r.t. Baire’s topology) and of computability. The theory turns out to be formally similar to Type 1 recursion theory. Most definitions and theorems will appear in a topological and in a computational version in accordance with our previously explained program for Type 2 theory. First we introduce a representation $$ \hat \varphi $$ : B →[B → B_o] of the continuous functions from B to B_o which is effective: it satisfies the Type 2 utm-theorem and smn-theorem. From $$ \hat \varphi $$ we derive a representation φ : B →[B → B] where [B →B] is the set of continuous functions г: B → B the domains of which are the G_δ-subsets of B , and we derive a representation x: B→[B→ℕ] of the set of the continuous functions г: B → ℕ the domains of which are the open subsets of B . The representations φ and x satisfy the utm-theorem and the smn-theorem. As a counterpart of the recursively enumerable subsets in Type 1 theory we study the open and the computably open subsets of B. This Type 2 theory of open subsets of B is formally very similar to the Type 1 theory of the recursively enumerable sets.