SOME FUNDAMENTAL ISSUES CONCERNING DEGREES OF UNSOLVABILITY

Recall that is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning . The first issue is to find a specific, natural, recursively enumerable Turing degree which is > 0 and < 0'. The second issue is to find a “smallness property” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree is > 0 and < 0'. In order to address these issues, we embed into a slightly larger degree structure, , which is much better behaved. Namely, is the lattice of weak degrees of mass problems associated with nonempty subsets of 2^ω. We define a specific, natural embedding of into , and we present some recent and new research results.