Toward the Definability of the Array Noncomputable Degrees

by Stephen M. Walk We focus on a particular class of computably enumerable (c.e.) degrees, the array noncomputable degrees defined by Downey, Jockusch, and Stob, to answer questions related to array noncomputable degrees and definability in the partial ordering (R,≤) of c.e. degrees. First we demonstrate that the lattice M5 cannot be embedded into the c.e. degrees below every array noncomputable degree, or even below every nonlow array noncomputable degree. As Downey and Shore have proved that M5 can be embedded below every nonlow2 degree, our result is the best possible in terms of array noncomputable degrees and jump classes. Further, it shows that the array noncomputable degrees are definably different from the nonlow2 degrees. We demonstrate also that there are embeddings of M5 in which all five degrees are array noncomputable, and in which the bottom degree is the computable degree 0 but the other four are array noncomputable. Next we turn to the relativization of some important concepts in computability theory. These include array noncomputability, which we have conjectured to be definable in (R,≤), and other properties, such as promptness and contiguity, that are known to be definable in (R,≤). We find that if d is a c.e. degree that is not