Recursive Equivalence Types and Combinatorial Functions

Abstract

This chapter provides information on the Recursion theory that is originated as an ancillary of Hilbertian proof-theory. It is developed into an independent branch of mathematics in the last decade with no philosophical presuppositions. It is the theory of those properties of the sets of natural numbers that are preserved under recursive permutations, where a recursive permutation is a one-one recursive function mapping the set of all natural numbers onto itself. Dekker has introduced the conception of recursive equivalence type—which belongs to two sets α and β (of natural numbers) that are called “recursively equivalent” if there is a one-one partial recursive function defined at least on α—and mapping it onto β. The class of all sets recursively equivalent to a given set α is called the “recursive equivalence type” (R.E.T.) of α, and is denoted by Req (α). The collection of all R.E.T.'s is denoted by Ω. The relation between recursive equivalence and recursive isomorphism is given by a theorem that completely reduces the study of recursive isomorphism types to the study of recursive equivalence types.