Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an ordinal number, based on the structure of the proof, with each of these ordinals less than ε0. He then proves by transfinite induction on these ordinals that no proof can conclude in a contradiction. The method used in this proof can also be used to prove a cut elimination result for Peano arithmetic in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of primitive recursive arithmetic and a transfinite induction principle. Tait (2005) gives a game-theoretic interpretation of Gentzen's method.