If Cons(F) were provable in F, then the Gödel sentence G_F would also be provable in F by simple logic

where \(G_F\) is the Gödel sentence for \(F\) provided by the first theorem. If \(\Cons(F)\) were provable in \(F\), so would be \(G_F\), by simple logic. This would contradict Gödel’s first theorem. Consequently, \(\Cons(F)\) cannot be provable in \(F\) either.