 Theorem 3.10 (Kleene 1943): For all \(n \geq 1\), there exists a set \(A \subseteq \mathbb{N}\) which is \(\Pi^0_n\)-definable but not \(\Sigma^0_n\)-definable and hence also neither \(\Sigma^0_m\)- nor \(\Pi^0_m\)-definable for any \(m &lt; n\). It thus follows that \(\overline{A}\) is \(\Sigma^0_n\)-definable but not \(\Pi^0_n\)-definable and hence also neither \(\Sigma^0_m\)- nor \(\Pi^0_m\)-definable for any \(m &lt; n\).