Proof Theory: An Introduction: 1407 (Lecture Notes in Mathematics, 1407) - Softcover

Pohlers, Wolfram

Softcover

ISBN 10: 3540518428 ISBN 13: 9783540518426

Publisher: Springer, 1989

Synopsis

Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.

About this title

Review

From the reviews:

"Proof Theory takes various axiom systems ... that treat induction in different ways and analyzes them from the ordinal viewpoint to gauge their relative strengths. ... This new version includes several developments in the field that have occurred over the twenty years since the original. Although the current book, appearing in the Universitext series, claims to be ‘pitched at undergraduate/graduate level,’ an undergraduate course out of Proof theory would be ambitious indeed." (Leon Harkleroad, The Mathematical Association of America, March, 2009)

"The book is addressed primarily to students of mathematical logic interested in the basics of proof theory, and it can be used both for introductory and advanced courses in proof theory. ... this book may be recommended to a larger circle of readers interested in proof theory." (Branislav Boricic, Zentrablatt MATH, Vol. 1153, 2009)

“This is a textbook―an excellent one―on proof theory, starting from the very elementary (heuristic accounts of sets, ordinals, logic, etc.), and going into a sophisticated area (impredicativity). ... The author’s main tool is enquiry into truth complexity and ordinal analysis.” (M. Yasuhara, Mathematical Reviews, Issue 2010 a)

From the Back Cover

This book verifies with compelling evidence the author’s intent to "write a book on proof theory that needs no previous knowledge of proof theory". Avoiding the cryptic terminology of proof theory as far as possible, the book starts at an elementary level and displays the connections between infinitary proof theory and generalized recursion theory, especially the theory of inductive definitions. As a "warm up" Gentzen's classical analysis of pure number theory is presented in a more modern terminology, followed by an explanation and proof of the famous result of Feferman and Schütte on the limits of predicativity. The author also provides an introduction to ordinal arithmetic, introduces the Veblen hierarchy and employs these functions to design an ordinal notation system for the ordinals below Epsilon 0 and Gamma 0, while emphasizing the first step into impredicativity, that is, the first step beyond Gamma 0. This is first done by an analysis of the theory of non-iterated inductive definitions using Buchholz’s improvement of local predicativity, followed by Weiermann's observation that Buchholz’s method can also be used for predicative theories to characterize their provably recursive functions. A second example presents an ordinal analysis of the theory of $/Pi_2$ reflection, a subsystem of set theory that is proof-theoretically equivalent to Kripke-Platek set.

The book is pitched at undergraduate/graduate level, and thus addressed to students of mathematical logic interested in the basics of proof theory. It can be used for introductory as well as more advanced courses in proof theory.

An earlier version of this book was published in 1989 as volume 1407 of the "Lecture Notes in Mathematics" (ISBN 978-3-540-51842-6).

Edition details

• Publisher: Springer
• Publication date: 1989
• Language: English
• ISBN 10: 3540518428
• ISBN 13: 9783540518426
• Binding: Paperback
• Edition number: 1
• Number of pages: 228