The decision problem for recursively enumerable degrees

TLDR: In this paper, the authors discuss the problem of determining whether any given number is in a given set A is recursive in a set B if there is an algorithm by which we can decide whether a given number x is in A, provided we are supplied with answers to all questions we choose to ask of the form Ts y in BT.

Abstract: If I have any message today for mathematicians in general, it is that consideration of difficult problems can be useful even when the problem is at present beyond solution. The problem I will discuss is unlikely to be solved in the near future, but I hope to show how the study of it leads to many more accessible problems. In order to state the problem, we need some definitions. To save words, we agree that number means natural number (nonnegative integer) and set means set of numbers. A set A is recursive if there is an algorithm for determining whether any given number is in A. A set A is recursive in a set B if there is an algorithm by which we can decide whether any given number x is in A, provided we are supplied with answers to all questions we choose to ask of the form Ts y in BT. As an example, let A ={2x : x e B}. Then B is recursive in A ; for x e B iff 2x € A. Also A is recursive in B ; for x e A iff x is even and \\x e B. (All this is independent of the choice of B.) Writing A^RB for A is recursive in B, we easily see that (1) A ^ R A ,