 \(K\) is clearly c.e. as it is the domain of \(\mu s T_1(x,x,s)\). On the other hand, we have seen that the characteristic function of \(K\)—i.e., the function \(\chi_K(x) = k(x)\) as defined in  Section 3.2—is  not computable. Thus \(K\) is indeed a computably enumerable set which is not computable. To see that \(\overline{K}\) is not c.e., observe that if it were, then \(K\) would be computable by  Proposition 3.4.  This in turn suggests a sense in which it is “harder” to decide membership in \(K\) than in any computable set. The hierarchies introduced in  Sections 3.5  and  Section 3.6  will provide a means of making such observations precise.