Gaisi Takeuti Livres

Focusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics. The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen's cut-elimination theorem and the theory of natural numbers in terms of Gödel's incompleteness theorem and Gentzen's consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.

The book explores the application of mathematical logic through two distinct approaches: set theory and proof theory. In the first part, Gaisi Takeuti utilizes Scott-Solovay's Boolean-valued models to enhance analysis with complete Boolean algebras of projections. The second part focuses on classical and complex analysis within Peano's arithmetic, demonstrating that any theorem from analytic number theory can also be derived in Peano's framework, employing Gentzen's cut elimination theorem to strengthen the connections between these mathematical disciplines.