 It is easy to  see[3]  that models of \(\theta\) are, up to isomorphism, models \(\mm\) where \(M=U^\mm\cup\P(U^\mm)\), \(U^\mm\cap\P(U^\mm)=\emptyset\) and for all \(a,b\in M\), \(E^\mm(a,b)\leftrightarrow a\in b\). In such a model monadic second-order formulas \(\phi\) over \(U^\mm\) can be reduced to first order formulas \(\phi^*\) over M. This reduction, due to Hintikka (1955) and further developed by Montague (1963), shows that \(\Pi^1_1\)-formulas alone are enough to express any second-order property with extra predicates. The idea can be extended to third and higher order logic.