A Survey of Proof Theory II

TLDR: In this article, the author explains recent work in proof theory from a neglected point of view, treating proofs and their representations by formal derivations as principal objects of study, not as mere tools for analyzing the consequence relation.

Abstract: This paper explains recent work in proof theory from a neglected point of view. Proofs and their representations by formal derivations are treated as principal objects of study, not as mere tools for analyzing the consequence relation. Though the paper is principally expository it also contains some material not developed in the literature. In particular, adequacy conditions on criteria for the identity of proofs (in § 1c), and a reformulation of Godeľs second theorem in terms of the notion of canonical representation (in § 1d); the use of normalization, instead of normal form, theorems for a direct proof of closure under Church's rule of the theory of species [in § 2a(ii)] and the useless-ness of bar recursive functionals for (functional) interpretations of systems containing Church's thesis [in §2b(iii)]; the use of ordinal structures in a quantifier-free formulation of transfinite induction (in § 3); the irrelevance of axioms of choice to the explicit realizability of existential theorems both for classical and for Heyting's logical rules (in § 4c) and some new uses of Heyting's rules for analyzing the indefinite cumulative hierarchy of sets (in § 4d); a semantics for equational calculi suitable when terms are interpreted as rules for computation [in Appl. Ia(iii)], and, above all, an analysis of formalist semantics and its relation to realizability interpretations (in App. Ic). A less technical account of the present point of view is in [21].