Abstract: | Consider the countable semilattice T consisting of the recursivelyenumerable Turing degrees. Although T is known to be structurallyrich, a major source of frustration is that no specific, naturaldegrees in T have been discovered, except the bottom and topdegrees, 0 and 0'. In order to overcome this difficulty, weembed T into a larger degree structure which is better behaved.Namely, consider the countable distributive lattice w consistingof the weak degrees (also known as Muchnik degrees) of massproblems associated with non-empty 01 subsets of 2. It is knownthat w contains a bottom degree 0 and a top degree 1 and isstructurally rich. Moreover, w contains many specific, naturaldegrees other than 0 and 1. In particular, we show that in wone has 0 < d < r1 < f(r2, 1) < 1. Here, d is theweak degree of the diagonally non-recursive functions, and rnis the weak degree of the n-random reals. It is known that r1can be characterized as the maximum weak degree of a 01 subsetof 2 of positive measure. We now show thatf(r2, 1) can be characterizedas the maximum weak degree of a 01 subset of 2, the Turing upwardclosure of which is of positive measure. We exhibit a naturalembedding of T into w which is one-to-one, preserves the semilatticestructure of T, carries 0 to 0, and carries 0' to 1. IdentifyingT with its image in w, we show that all of the degrees in Texcept 0 and 1 are incomparable with the specific degrees d,r1, andf(r2, 1) in w. |