Theorem 4.6 (Kent and Hodgson 1982) A set \(X\) belongs to \(\Sigma^P_i\) if and only \(X\) is definable by a \(\Sigma^b_i\)-formula \(\phi(x)\) in the standard model – i.e. \(X = \{n \in \mathbb{N} \mid \mathcal{N} \models \phi(\overline{n}) \}\).