 Cut elimination fails for first-order arithmetic (i.e., PA), not even partial cut elimination is possible since the induction axioms have unbounded complexity. Gentzen, however, found an ingenious way of dealing with purported contradictions in arithmetic. In Gentzen 1938b he showed how to effectively transform an alleged PA-proof of an inconsistency (the empty sequent) in his sequent calculus into another proof of the empty sequent such that the latter gets assigned a smaller ordinal than the former. Ordinals are a central concept in set theory as well as in proof theory. To present Gentzen’s work we shall first discuss the notion of ordinal from a proof-theoretic point of view.