Ordinal analysis of theories is a core area of proof theory whose origins can be traced back to Hilbert’s programme – the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. The central theme of ordinal analysis is the classification of theories by means of transfinite ordinals that measure their ‘consistency strength’ and ‘computational power’. The so-called proof-theoretic ordinal of a theory also serves to characterize its provably recursive functions and can yield both conservation and combinatorial independence results.
This paper intends to survey the development of “ordinally informative” proof theory from the work of Gentzen up to more recent advances in determining the proof-theoretic ordinals of strong subsystems of second order arithmetic.