This comprehensive monograph is a cornerstone in the area of mathematical logic and related fields. Focusing on Gentzen-type proof theory, the book presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

A previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs of Σ 2 -finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due (...) to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε -2 d 2 ) in place of exp(285ε -2 d 2 ) given by Davenport and Roth in 1955, where α is (real) algebraic of degree d ≥ 2 and $|\alpha - pq^{-1}| . (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof are polynomial of low degree (in ε -1 and log d). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), $\log \log q_\nu , where q ν are the denominators of the convergents to the continued fraction of α. (shrink)