 By restricting what kind of quantifiers the sequence \(Q_1X_1\ldots Q_nX_n\) of the Prenex Normal Form (\ref{qqq}) can contain we obtain the classes of \(\Sigma^1_n\)-, \(\Pi^1_n\)- and \(\Delta^1_n\)-formulas, as indicated in  Table 1.  For example, the second-order formulas (1) and (2) are \(\Pi^1_1\)-formulas.