Definition 2.2.7 (Delta0-formula, Sigma1-formula). A ${\mathrm{\Delta }}_0$-formula of PA is one whose quantifiers are bounded, i.e. $\exists x<t.\phi (x)$ or $\forall x<t.\phi (x)$, where $t$ is not free in $\phi$ and $\phi$ is quantifier free.

We say $\phi (\overline{x})$ is a ${\mathrm{\Sigma }}_1$-formula if there is a ${\mathrm{\Delta }}_0$-formula $\psi (\overline{x},\overline{y})$ such that

$$\mathrm{PA}⊢\phi (\overline{x})\leftrightarrow \exists \overline{y}.\psi (\overline{x},\overline{y}).$$

It is a ${\mathrm{\Pi }}_1$-formula if there is a ${\mathrm{\Delta }}_0$-formula $\psi (\overline{x},\overline{y})$ such that

$$\mathrm{PA}⊢\phi (\overline{x})⟺\forall \overline{y}.\psi (\overline{x},\overline{y}).$$