Definition ($Sigma1$, $Pi1$). A formula is called ${\mathrm{\Sigma }}_1$ if it is of the form $\exists x_1,\dots \exists x_n,\phi$ where $\phi$ is ${\mathrm{\Delta }}_0$.

It is called ${\mathrm{\Pi }}_1$ if it is of the form $\forall x_1,\dots ,\forall x_n,\phi$ where $\phi$ is ${\mathrm{\Delta }}_0$.