The countably based functionals

Abstract

In [5], Kleene extended previous notions of computations to objects of higher finite type in the maximal type-structure of functionals given by:Tp(0) = N = the natural numbers,Tp(n + 1) = NTp(n) = the set of total maps from Tp(n) to N.He gave nine schemata, S1–S9, for describing algorithms for computations from a finite list of functionals, and indices to denote these algorithms. It is generally agreed that S1-S9 give a natural concept of computations in higher types.The type-structure of countable functions, Ct(n) for n ϵ N, was first developed by Kleene [6] and Kreisel [7]. It arises from the notions of ‘constructivity’, and has been extensively studied as a domain for higher type recursion theory. Each countable functional is globally described by a countable amount of information coded in its associate, and it is determined locally by a finite amount of information about its argument. The countable functionals are well summarised in Normann [9], and treated in detail in Normann [8].The purpose of this paper is to discuss a generalisation of the countable functionals, which we shall call the countably based functions, Cb(n) for n ϵ N. It is suggested by the notions of ‘predicativity’, in which we view N as a completed totality, and build higher types on it in a constructive manner. So we shall allow quantification over N and include application of 2E in our schemata. Each functional is determined locally by a countable amount of information about its argument, and so is globally described by a continuum of information coded in its associate, which will now be a type-2 object. This generalisation, obtained via associates, was proposed by Wainer, and seems to reflect topological properties of the countable functionals.