 Part iv of  Theorem 3.11  can also be understood as generalizing  Proposition 3.4 (i.e., Post’s Theorem). In particular, it characterizes the \(\Delta^0_{n+1}\)-definable sets as those sets \(B\) such that both \(B\) and \(\overline{B}\) are computably enumerable in some \(\Sigma^0_n\)-complete set such as \(\emptyset^{(n)}\). Restricting to the case \(n = 1\), this observation can also be used to provide an independent computational characterization of the \(\Delta^0_2\)-definable sets, extending those given for the \(\Sigma^0_1\)-definable and \(\Delta^0_1\)-definable sets by  Proposition 3.8.