 Definition 3.14: A formula \(\varphi(\vec{x})\) of \(\mathcal{L}_a\) is said to be in the class \(\Delta^0_0\) if it contains only bounded first-order quantifiers—i.e., those of the form \(\exists x(x &lt; t \wedge \ldots)\) and \(\forall x(x &lt; t \rightarrow \ldots)\) for \(t\) an \(\mathcal{L}_a\)-term not containing \(x\). A formula is said to be in the class \(\Sigma^0_1\) if it is of the form \(\exists \vec{y} \varphi(\vec{x},\vec{y})\) for \(\varphi(\vec{x},\vec{y}) \in \Delta^0_0\) and to be in the class \(\Pi^0_1\) if it is of the form \(\forall \vec{y} \varphi(\vec{x},\vec{y})\) for \(\varphi(\vec{x},\vec{y}) \in \Delta^0_0\). Finally, a formula \(\varphi(\vec{x})\) is said to be in the class \(\Delta^0_1\) if it is semantically equivalent to both a \(\Sigma^0_1\)-formula \(\psi(\vec{x})\) and a \(\Pi^0_1\)-formula \(\chi(\vec{x})\)—i.e., \(\mathfrak{N} \models \varphi(\vec{n})\) iff \(\mathfrak{N} \models \psi(\vec{n})\) iff \(\mathfrak{N} \models \chi(\vec{n})\) for all \(\vec{n} \in \mathbb{N}^k\).