Representation theorems for transfinite computability and definability

The continuous functionals were independently introduced by Kleene [6] and Kreisel [7]. Kleenes motivation was to isolate a subclass of the total functionals of pure, finite type closed under S1-S9-computations (Kleene [5]) which could be described using countable information. He defined his functionals via the associates and the idea is that an associate for a functional Ψ contains sufficient information about Ψ to decide Ψ(F ) uniformly in any associate for F . Kleene called the functionals countable. Kreisel’s continuous functionals are defined via sets of formal neighbourhoods, and the definition is much in the spirit of what is now known as domain theory. Kreisel actually defines the hierarchy of partial continuous functionals, and then use the hereditarily total ones in his hierarchy. Kreisel’s motivation was to analyse the concept of constructivly true. It was clear already at the time of publication that the two definitions were basicly equivalent. Following Kleene’s motivation it is established that the continuous functionals support a rich computation theory, see e.g. Normann [12, 15]. In this paper we will pursue some of the ideas in Kreisel’s approach and show that transfinite versions of the continuous functionals can be used to represent complex properties. We will be more precise later. One basic result proved both by Kleene and Kreisel is the density theorem. They formulated it in different ways. Kleene showed that the set of finite number-sequences approximating an associate is primitive recursive,