Abstract
Publisher Summary This chapter discusses the 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 ɛ0, the latter being applied only to decidable predicates. One of the strongest theories for which ordinal-theoretic bounds have been obtained is the impredicative subsystem of second-order arithmetic based on Δ 1 2 comprehension plus bar induction. The chapter also surveys 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 Π 1 2 comprehension. The role of ordinals and ordinal analysis in proof theory is also described in the chapter. The program of admissible proof theory and its achievements are reviewed. The cut-elimination procedure for Kripke–Platek set theory is also provided. Some new cut-elimination procedures for reflections higher than Π 2 are considered. Cut–elimination for Π n -reflection entails a proof-theoretic treatment of theories of non-monotone inductive definitions.
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