  Since the induction principle can be expressed but not proved in  ordinary arithmetic, a formula unprovable in Peano arithmetic is  found. An easy consequence of Gentzen's version of the incompleteness theorem is the consistency of Peano arithmetic, because anything  would be provable in an inconsistent system. Contrary to Gödel's “artificial” unprovable formula that was obtained through a coding of the arithmetized provability predicate, Gentzen's  transfinite induction principle is a principle of  “ordinary” mathematics.