Measure-theoretic construction of incomparable hyperdegrees

Abstract

Kleene-Post [10], Spector [14] and Friedberg [3, 4], have given a number of examples of functions which have incomparable degrees of recursive unsolvability satisfying various additional conditions. Our present purpose is to determine to what extent some of these incomparability theorems can be generalized by substituting for “α. is recursive in β” other relations Q(α, β) such as “α is hyperarithmetical in β” (which we shall think of as “α ≦ β”), and to determine what restrictions, if any, need be imposed on Q. As a consequence of our investigation we shall show that there are incomparable hyperdegrees as defined in [9] and [13].