 The mathematical problem of whether a given sentence \(C\) is a logical consequence of a finite higher-order theory \(T\) coincides with the problem of whether the second-order sentence \((\neg(J \wedge T^*) \vee C^*)\) is valid. Recalling that there is a sentence \(\chi\) of \(\mathbf{IFL}\) such that \(J\) is equivalent to \(\neg \chi\), it follows that \(\neg(J \wedge T^*)\) is equivalent to a sentence of \(\mathbf{IFL}\). Consequently there is a sentence of \(\mathbf{IFL}\) which is valid iff the sentence \((\neg(J \wedge T^*) \vee C^*)\) is valid (cf. Subsect. 4.6). Mathematical problems can be understood as questions of the validity of an \(\mathbf{IFL}\) sentence. Among mathematical problems thus reconstructible using \(\mathbf{IFL}\) are the continuum hypothesis, Goldbach’s conjecture, Souslin’s conjecture, the existence of an inaccessible cardinal, and the existence of a measurable  cardinal.[81]