  In  (1936),  Gentzen published a consistency proof of first-order Peano Arithmetic (\(\PA\)). As Gödel had shown was necessary, Gentzen’s proof used methods that could not be formalized in \(\PA\) itself, namely, transfinite induction along the ordinal \(\varepsilon_0\). Gentzen’s work marks the beginning of post-Gödelian proof theory and work on Relativized Hilbert Programs. Proof theory in the tradition of Gentzen has analyzed axiomatic systems according to what extensions of the finitary standpoint are necessary to prove their consistency. Usually, the consistency strength of systems has been measured by the system’s proof-theoretic ordinal, i.e., the ordinal transfinite induction along which suffices to prove consistency. In the case of \(\PA\), that ordinal is \(\varepsilon_0\). (For further discussion, see the entry on  the development of proof theory.)