Applied Proof Theory: Proof Interpretations and their Use in Mathematics
Authors: Kohlenbach, Ulrich
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- About this book
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Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises.
The book first develops the necessary logical machinery emphasizing novel forms of Gödel's famous functional ('Dialectica') interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.
- About the authors
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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".
- Reviews
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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)
- Table of contents (19 chapters)
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Introduction
Pages 1-11
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Unwinding proofs (‘Proof Mining’)
Pages 13-40
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Intuitionistic and classical arithmetic in all finite types
Pages 41-76
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Representation of Polish metric spaces
Pages 77-95
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Modified realizability
Pages 97-107
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Table of contents (19 chapters)
- Download Preface 1 PDF (149.2 KB)
- Download Sample pages 2 PDF (371.7 KB)
- Download Table of contents PDF (192.5 KB)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Applied Proof Theory: Proof Interpretations and their Use in Mathematics
- Authors
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- Ulrich Kohlenbach
- Series Title
- Springer Monographs in Mathematics
- Copyright
- 2008
- Publisher
- Springer-Verlag Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- eBook ISBN
- 978-3-540-77533-1
- DOI
- 10.1007/978-3-540-77533-1
- Hardcover ISBN
- 978-3-540-77532-4
- Softcover ISBN
- 978-3-642-09627-3
- Series ISSN
- 1439-7382
- Edition Number
- 1
- Number of Pages
- XX, 536
- Topics
