 It is a consequence of the Prenex Normal Form Theorem for first-order logic that every \(\mathcal{L}_a\)-formula \(\varphi(\vec{y})\) is provably equivalent to one of the form \(\mathsf{Q}_1 x_1 \mathsf{Q}_2 x_2 \ldots \mathsf{Q}_{n} \varphi(\vec{x},\vec{y})\) for \(\mathsf{Q}_i \equiv \exists\) or \(\forall\) (e.g., Boolos, Jeffrey, &amp; Burgess 2007, ch. 19.1). It thus follows that up to provable equivalence, every such formula is \(\Sigma^0_n\) or \(\Pi^0_n\) for some \(n \in \mathbb{N}\). Since it is conventional to allow that blocks of quantifiers may be empty in the  Definition 3.15,  it follows that