 As a consequence, checking the validity of a second-order sentence \(\phi\) can be recursively reduced to checking the validity of the \(\Sigma^1_1\)-sentence \(\theta\to\phi^*\). Likewise checking whether a second-order sentence \(\phi\) has a model of cardinality \(\kappa\) can be reduced to asking whether the \(\Pi^1_1\)-sentence \(\theta\land\phi^*\) has a model of cardinality \(2^\kappa\). This means that the Löwenheim  number[4]  and the Hanf  number[5]  of the entire second-order logic are the same as those of the fragment \(\Pi^1_1\).