 Theorem 3.13 (Kleene 1955a): For all \(n \geq 1\), there exists a set \(A \subseteq \mathbb{N}\) which is \(\Pi^1_n\)-definable but not \(\Sigma^1_n\)-definable and hence also neither \(\Sigma^1_m\)- nor \(\Pi^1_m\)-definable for any \(m &lt; n\). It thus follows that \(\overline{A}\) is \(\Sigma^1_n\)-definable but not \(\Pi^1_n\)-definable and also neither \(\Sigma^1_m\)- nor \(\Pi^1_m\)-definable for any \(m &lt; n\).