 Although the other parts of  Proposition 3.6 have straightforward proofs, they provide some insight into the fact that \(\mathcal{D}_m\) is itself a highly complex structure (see, e.g., Odifreddi 1999b, 1). Nonetheless the first two parts of this theorem are often taken to illustrate awkward features of the many-one degrees as an abstract representation of computational difficulty—i.e., the exceptional behavior of \(\textrm{deg}_m(\emptyset)\) and \(\textrm{deg}_m(\mathbb{N})\) and the fact a set and its complement may inhabit different degrees (as is easy to see is exemplified by \(K\) and \(\overline{K}\)). It is partly in light of these features that the Turing degrees \(\mathcal{D}_T\) are the structure which are now most widely studied in computability theory. But as Post also alludes, it is relative to \(\mathcal{D}_T\) for which he was originally unable to demonstrate the existence of a c.e. set of an intermediate degree of unsolvability.