Applied Proof Theory: Proof Interpretations and their Use in Mathematics

Buch | Hardcover
XX, 536 Seiten
2008 | 2008
Springer Berlin (Verlag)
978-3-540-77532-4 (ISBN)

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Applied Proof Theory: Proof Interpretations and their Use in Mathematics - Ulrich Kohlenbach
139,09 inkl. MwSt

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.

 

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". 

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.

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)

Erscheint lt. Verlag 26.5.2008
Reihe/Serie Springer Monographs in Mathematics
Zusatzinfo XX, 536 p.
Verlagsort Berlin
Sprache englisch
Maße 155 x 235 mm
Gewicht 960 g
Themenwelt Mathematik / Informatik Mathematik Allgemeines / Lexika
Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Logik / Mengenlehre
Schlagworte arithmetic • Beweis (Mathematik) • Calculus • computational mathematics • Finite • Function • Geometry • Mathematical Logic • Mathematics • Proof • Proof Interpretations • Proof Mining • Proof theory • Theorem
ISBN-10 3-540-77532-3 / 3540775323
ISBN-13 978-3-540-77532-4 / 9783540775324
Zustand Neuware
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