Applied proof theory: proof interpretations and their use in mathematics

This book presents an applied form of proof theory that has led to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others) in recent years . This applied approach isbased 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 (independence of solutions from certain parameters, generalizations of proofs by elimination of premises and others). 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. First book on the subject at all INDICE: Preface.- Introduction.- Unwinding of 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 Skolemfunctions.- The Friedman-Dragalin 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.- References.- Index.

• ISBN: 978-3-540-77532-4
• Editorial: Springer
• Encuadernacion: Cartoné
• Páginas: 550
• Fecha Publicación: 01/04/2008
• Nº Volúmenes: 1
• Idioma: Inglés