A minimal pair of recursively enumerable degrees

Abstract

Our principal result is that there exist two incomparable recursively enumerable degrees whose greatest lower bound in the upper semilattice of degrees is 0. This was conjectured by Sacks [5]. As a secondary result, we prove that on the other hand there exists a recursively enumerable degree a < 0(1) such that for no non-zero recursively enumerable degree b is 0 the greatest lower bound of a and b.The proof of the main theorem involves a method that we have developed elsewhere [8] to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority.