Continuity, proof systems and the theory of transfinite computations

The purpose of this paper is to show how the concept of transfinite computations relative to certain functionals of type 3 can be used to construct topologies and transfinite proof systems adding extra structure to some sets transfinitly definable over the continuum. Our aim is to initiate a fine-structure analysis of sets of the form Lκ(HC). We will motivate this below. Before doing so, let us make a few remarks on the choice of terminology. Accepting to a large extent the argumentation of Soare [42] and realizing that the raw material of most of the constructions of the paper are transfinite analogues of computations, we will use the expression ’theory of transfinite computations’ to cover at least the part of ’higher recursion theory’ that deals with generalisations of computations. We will also use the expressions ’computable’ and ’computable relative to’. When we use the word ’recursion’ or any of its derivatives it will be in a context where the use has a well-known technical interpretation, like in ’recursivly inaccessible ordinal’, or in the case where we use the recursion theorem or the fixpoint theorem for domains.