 We may also define a notion of completeness with respect to the levels of the arithmetical hierarchy as follows: \(A\) is \(\Sigma^0_n\)-complete if \(A\) is \(\Sigma^0_n\)-definable and for all \(\Sigma^0_n\)-definable \(B\), we have \(B \leq_m A\) (and similarly for \(\Pi^0_n\)-complete). It is not hard to show that in addition to being many-one complete, \(K\) is also \(\Sigma^0_1\)-complete. Similarly \(\overline{K}\) is \(\Pi^0_1\)-complete, \(INF\) is \(\Sigma^0_2\)-complete, and \(TOT\) is \(\Pi^0_2\)-complete. These observations can be subsumed under a more general result which relates the arithmetical hierarchy to the Turing degrees and from which  Theorem 3.10  can also be obtained as a corollary.