Herbrand's Theorem, in the form of $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\exists } $$ -inversion lemmata for finitary and infinitary sequent calculi, is the crucial tool for the determination of the provably total function(al)s of a variety of theories. The theories are (second order extensions of) fragments of classical arithmetic; the classes of provably total functions include the elements of the Polynomial Hierarchy, the Grzegorczyk Hierarchy, and the extended Grzegorczyk Hierarchy $\mathfrak{E}^\alpha $ , α < ε0. A subsidiary aim of the paper is to show (...) that the proof theoretic methods used here are distinguished by technical elegance, conceptual clarity, and wide-ranging applicability. (shrink)

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (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)