Subsystems of Second Order Arithmetic (Perspectives in Mathematical Logic) - Hardcover

Stephen G. Simpson

Hardcover

ISBN 10: 3540648828 ISBN 13: 9783540648826

Publisher: Springer, 1999, 1999

Synopsis

This book is an original contribution to the foundations of mathematics, with emphasis on the role of set existence axioms. Part A demonstrates that many familiar theorems of algebra, analysis, functional analysis,and combinatorics are logically equivalent to the axioms needed to prove them. This phenomenon is known as Reverse Mathematics. Subsystems of second order arithmetic based on such axioms correspond to several well known foundational programs: finitistic reductionism (Hilbert), constructivism (Bishop), predicativism (Weyl), and predicative reductionism (Feferman/Friedman). Part B is a thorough study of models of these and other systems. The book includes an extensive bibliography and a detailed index.

About this title

Book Description

What are the appropriate axioms for mathematics? Through a series of case studies, this volumes examines these axioms to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics.

From the Back Cover

"From the point of view of the foundations of mathematics, this definitive work by Simpson is the most anxiously awaited monograph for over a decade. The "subsystems of second order arithmetic" provide the basic formal systems normally used in our current understanding of the logical structure of classical mathematics. Simpson provides an encyclopedic treatment of these systems with an emphasis on *Hilbert's program* (where infinitary mathematics is to be secured or reinterpreted by finitary mathematics), and the emerging *reverse mathematics* (where axioms necessary for providing theorems are determined by deriving axioms from theorems). The classical mathematical topics treated in these axiomatic terms are very diverse, and include standard topics in complete separable metric spaces and Banach spaces, countable groups, rings, fields, and vector spaces, ordinary differential equations, fixed points, infinite games, Ramsey theory, and many others. The material, with its many open problems and detailed references to the literature, is particularly valuable for proof theorists and recursion theorists. The book is both suitable for the beginning graduate student in mathematical logic, and encyclopedic for the expert." Harvey Friedman, Ohio State University

Edition details

• Publisher: Springer, 1999
• Publication date: 1999
• ISBN 10: 3540648828
• ISBN 13: 9783540648826
• Binding: Hardcover
• Language: English
• Edition number: 1
• Number of pages: 444