 Definition 3.15: In order to simplify notation, the classes \(\Sigma^0_0\) and \(\Pi^0_0\) are both used as alternative names for the class \(\Delta^0_0\). A formula is said to be in the class \(\Sigma^0_{n+1}\) if it is of the form \(\exists \vec{y} \varphi(\vec{x},\vec{y})\) for \(\varphi(\vec{x},\vec{y}) \in \Pi^0_n\) and to be in the class \(\Pi_{n+1}\) if it is of the form \(\forall \vec{y} \varphi(\vec{x},\vec{y})\) for \(\varphi(\vec{x},\vec{y}) \in \Sigma^0_n\). A formula \(\varphi(\vec{x})\) is \(\Delta^0_{n+1}\) if it is semantically equivalent to both a \(\Sigma^0_{n+1}\)-formula \(\psi(\vec{x})\) and a \(\Pi^0_{n+1}\)-formula \(\chi(\vec{x})\).