Almost recursive sets

Abstract

A set A is recursive iff (if and only if) PA is a recursive function. We shall call a set A almost recursive iff PA I A (PA, restricted to A) can be extended to a p.r. (partial recursive) function, or, intuitively, iff for every xEA we can effectively find the number of elements of A which are less than A. More generally, for any two sets A and B, we shall say that A is B-almost recursive iff PA I B can be extended to a p.r. function. In this way, a set A is almost recursive iff it is A-almost recursive. In this paper we prove some theorems on almost recursive sets, in particular, theorems relating those sets to retraceable sets (Dekker and Myhill [1]).