Gentzen's proof highlights one commonly missed aspect of Gödel's second incompleteness theorem. It is sometimes claimed that the consistency of a theory can only be proved in a stronger theory. Gentzen's theory obtained by adding quantifier-free transfinite induction to primitive recursive arithmetic proves the consistency of first-order Peano arithmetic (PA) but does not contain PA. For example, it does not prove ordinary mathematical induction for all formulae, whereas PA does (since all instances of induction are axioms of PA). Gentzen's theory is not contained in PA, either, however, since it can prove a number-theoretical fact—the consistency of PA—that PA cannot. Therefore, the two theories are, in one sense, incomparable.
