A non-splitting theorem for d.r.e. sets Original Research Article

A set of natural numbers is called d.r.e. (difference recursively enumerable) if it may be obtained from some recursively enumerable (r.e.) set by deleting the numbers belonging to another recursively enumerable set. Sacks showed that for each non-recursive recursively enumerable set A there are disjoint recursively enumerable sets B, C which cover A such that A is recursive in neither A ∩ B nor A ∩ C. In this paper, we construct a counterexample which shows that Sacksʹs theorem is not in general true when A is d.r.e. rather than r.e.