Overview

This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as -- via extended case studies -- carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.

Full Product Details

Author:   Ulrich Kohlenbach
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of hardcover 1st ed. 2008
Dimensions:   Width: 15.50cm , Height: 2.90cm , Length: 23.50cm
Weight:   0.842kg
ISBN:  

9783642096273

ISBN 10:   3642096271
Pages:   536
Publication Date:   21 October 2010
Audience:   Professional and scholarly ,  Professional & Vocational ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Unwinding proofs (‘Proof Mining’).- Intuitionistic and classical arithmetic in all finite types.- Representation of Polish metric spaces.- Modified realizability.- Majorizability and the fan rule.- Semi-intuitionistic systems and monotone modified realizability.- Gödel’s functional (‘Dialectica’) interpretation.- Semi-intuitionistic systems and monotone functional interpretation.- Systems based on classical logic and functional interpretation.- Functional interpretation of full classical analysis.- A non-standard principle of uniform boundedness.- Elimination of monotone Skolem functions.- The Friedman A-translation.- Applications to analysis: general metatheorems I.- Case study I: Uniqueness proofs in approximation theory.- Applications to analysis: general metatheorems II.- Case study II: Applications to the fixed point theory of nonexpansive mappings.- Final comments.

Reviews

From the reviews: This book covers ! from proof theory to a rich set of applications in areas quite distinct from mathematical logic: approximation theory and fixed point theory of nonexpansive mappings. ! Almost every chapter has a detailed ! informative final section with exercises, historical comments and references to related work. ! In summary, this book is a very welcome addition to the proof theory literature. (H. Schwichtenberg, Mathematical Reviews, Issue 2009 k)

Author Information

Ulrich Kohlenbach has been Professor of Mathematics at the Technische Universität Darmstadt since 2004. He is a managing editor of the ""Annals of Pure and Applied Logic"". 

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