 Giving a rigorous proof of the second theorem in a more general form that covers all such sentences, however, has turned out to be very complicated. The basic reason for this is that, unlike in the first theorem, not just any, merely extensionally adequate provability predicate works for the formalization of the consistency claim. The manner of presentation makes all the difference. For example, Rosser’s provability predicate mentioned above would not do; one can prove the “consistency” of \(F\) in \(F\), if consistency is expressed in terms of Rosser’s provability predicate. One must thus add some further conditions for the provability predicate in order for the proof of the second incompleteness theorem to go through. Following Feferman (1960), it is customary to say that whereas the first theorem and its relatives are extensional results, the second theorem is intensional: it must be possible to think that \(\Cons(F)\) in some sense expresses the consistency of \(F\)—that it really means that \(F\) is consistent.