 Post (1944) demonstrated part vii by showing that there exist so-called simple sets—i.e., sets \(A\) which are c.e. and such that \(\overline{A}\) is infinite but does not contain an infinite c.e. subset. It is easy to see that a simple set cannot be computable. But on the other hand, Post also showed that a simple set cannot be \(m\)-complete. And it thus follows that if \(A\) is simple \(\mathbf{a} =_{\textrm{df}} \textrm{deg}_m(A) \neq \mathbf{0}_m\) but \(A \not\equiv_m K\) and thus \(\mathbf{a} &lt; \mathbf{0}'_m\). Suppose we now understand “degrees of unsolvability” in the passage quoted at the beginning of this section as corresponding to the c.e. \(m\)-degrees. It thus follows from part v of  Proposition 3.6 that \(\mathbf{0}'_m\) is indeed a “highest” such degree and also from part vii that there is a lower but still “unsolvable” (i.e., non-computable) degree.