 In set theory there is the Levy-hierarchy of \(\Sigma_n\)- and \(\Pi_n\)-formulas (Lévy 1965). Formulas where all quantifiers are bounded, i.e., of the form \(\forall x\in y\) or of the form \(\exists x\in y\) for some x and y, are called \(\Sigma_0\) or equivalently \(\Pi_0\). Formulas of the form \(\forall x\,\phi\), where \(\phi\) is \(\Sigma_n\), are called \(\Pi_{n+1}\). Formulas of the form \(\exists x\,\phi\), where \(\phi\) is \(\Pi_n\), are called \(\Sigma_{n+1}\). This is a strict hierarchy roughly for the same reason why the hierarchy of second-order \(\Sigma^1_n\)- and \(\Pi^1_n\)-formulas is strict. But there is no known method to reduce the truth of an arbitrary formula to the truth of a formula on the \(\Sigma_1\cup\Pi_1\) level. In fact, if the decision problem, Löwenheim-Skolem number and Hanf number are suitably defined for \(\Sigma_n\cup \Pi_n\)-formulas of set theory, the decision problem gets more and more complicated, and the numbers obtained get bigger and bigger as n increases (see  Theorem 4;  Väänänen 1979).