Admissible Proof Theory and Beyond

Ordinals made their entrance in proof theory through Gentzen’s second consistency proof for Peano Arithmetic by transfinite induction up to ε0, the latter being applied only to decidable predicates (cf. Gentzen [1938]). Gentzen’s constructive use of ordinals as a method of analyzing formal theories has come to be a paradigm for much of proof theory from then on, particularly as exemplified in the work of Schütte, Takeuti and their schools. One of the strongest theories for which ordinal–theoretic bounds have been obtained is the impredicative subsystem of second order arithmetic based on ∆2 comprehension plus bar induction. The latter result was achieved by employing the most advanced techniques in this area of research: cut elimination for infinitary calculi of ramified set theory with Π2–reflection rules. This gathering of tools was entitled “Admissible Proof Theory” (cf. Pohlers [1982]), yet another appropriate title could have been “Proof Theory of Π2– Reflection”. Unfortunately, these methods are not strong enough for carrying through an ordinal analysis of Π2 comprehension, let alone for second order arithmetic. This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of Π2 comprehension. Although a great deal of ordinally informative proof theory has been pursuing an extension of Hilbert’s program, that is sought-for consistency proofs, I shall only indulge very little in this issue. Even those who wish to detach themselves from consistency matters may benefit from ordinal analyses. Ordinal analysis has proved to be an important tool in reductive proof theory and also for the determination of the provably total functions of various complexities of a variety of theories. Putting things into a broader perspective, a leit–motif for ordinal analysis could have been Kreisel’s question: