The 02 Enumeration Degrees Are Not

We show that the 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper 3] showed that the enumeration degrees (e-degrees) are not dense. He also showed that the e-degrees of the 0 2 sets are dense 2]. Since Cooper's nondensity proof constructs a 0 7 set, he posed the question: What is the least n (2 n 6) such that the e-degrees below 0 n are not dense? (We consider the Turing degrees to be a subset of the e-degrees by identifying the Turing degree of a set A with the e-degree of A A.) Cooper also conjectured that the e-degrees of the 0 2 sets are dense. Here we show that the e-degrees are not dense in the 0 2 e-degrees. Thus the answer to Cooper's question is 2 and his conjecture is false. This means that Cooper's proof that the 0 2 e-degrees are dense cannot be extended beyond the 0 2 level in the arithmetical hierarchy. 2 Description of the Construction The basic plan of the construction is to build 0 2 sets A and B such that A B is a minimal cover of B with respect to e. There are four types of requirements: