Gentzen's Centenary (eBook, PDF)

The Quest for Consistency

• Format: PDF

Produktbeschreibung

Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

Produktdetails

• Verlag: Springer International Publishing
• Erscheinungstermin: 2. November 2015
• Englisch
• ISBN-13: 9783319101033
• Artikelnr.: 44613300

Autorenporträt

    

Inhaltsangabe

Part I Reflections.- Gentzen’s Consistency Proof in Context, Reinhard Kahle.- Gentzen’s Anti-Formalist Views, Michael Detlefsen.- The Use of Trustworthy Principles in a Revised Hilbert’s Program, Anton Setzer.- Part II Gentzen’s Consistency Proofs.- On Gentzen’s First Consistency Proof for Arithmetic, Wilfried Buchholz.- From Hauptsatz to Hilfssatz, Jan von Plato.- A Note on How to Extend Gentzen’s Second Consistency Proof to a Proof of Normalization for First Order Arithmetic, Dag Prawitz.- A Direct Gentzen-Style Consistency Proof for Heyting Arithmetic, Annika Siders.- Gentzen’s Original Consistency Proof and the Bar Theorem, W.W. Tait.- Goodstein’s Theorem Revisited, Michael Rathjen.- Part III Results.- Cut Elimination In Situ, Sam Buss.- Spector’s Proof of the Consistency of Analysis, Fernando Ferreira.- Climbing Mount ε_0, Herman Ruge Jervell.- Semi-Formal Calculi and Their Applications, Wolfram Pohlers.- Part IV Developments.- Proof Theory for Theories of Ordinals III: Π_N-Reflection, Toshiyasu Arai.- A Proof-Theoretic Analysis of Theories for Stratified Inductive Definitions, Gerhard Jäger and Dieter Probst.- Classifying Phase Transition Thresholds for Goodstein Sequences and Hydra Games, Frederik Meskens and Andreas Weiermann.- Non-deterministic Epsilon Substitution Method for PA and ID_1, Grigori Mints.- A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis, Paulo Oliva and Thomas Powell.- Well-Ordering Principles and Bar Induction, Michael Rathjen and Pedro Francisco Valencia Vizcaíno.

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Part I Reflections.- Gentzen's Consistency Proof in Context, Reinhard Kahle.- Gentzen's Anti-Formalist Views, Michael Detlefsen.- The Use of Trustworthy Principles in a Revised Hilbert's Program, Anton Setzer.- Part II Gentzen's Consistency Proofs.- On Gentzen's First Consistency Proof for Arithmetic, Wilfried Buchholz.- From Hauptsatz to Hilfssatz, Jan von Plato.- A Note on How to Extend Gentzen's Second Consistency Proof to a Proof of Normalization for First Order Arithmetic, Dag Prawitz.- A Direct Gentzen-Style Consistency Proof for Heyting Arithmetic, Annika Siders.- Gentzen's Original Consistency Proof and the Bar Theorem, W.W. Tait.- Goodstein's Theorem Revisited, Michael Rathjen.- Part III Results.- Cut Elimination In Situ, Sam Buss.- Spector's Proof of the Consistency of Analysis, Fernando Ferreira.- Climbing Mount _0, Herman Ruge Jervell.- Semi-Formal Calculi and Their Applications, Wolfram Pohlers.- Part IV Developments.- Proof Theory for Theories of Ordinals III: _N-Reflection, Toshiyasu Arai.- A Proof-Theoretic Analysis of Theories for Stratified Inductive Definitions, Gerhard Jäger and Dieter Probst.- Classifying Phase Transition Thresholds for Goodstein Sequences and Hydra Games, Frederik Meskens and Andreas Weiermann.- Non-deterministic Epsilon Substitution Method for PA and ID_1, Grigori Mints.- A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis, Paulo Oliva and Thomas Powell.- Well-Ordering Principles and Bar Induction, Michael Rathjen and Pedro Francisco Valencia Vizcaíno.

Rezensionen

"The book collects for the first time individual studies which focus solely on this aspect of Gentzen's work. Because they admirably elucidate the historical, philosophical, and technical facets of his foundational papers in ordinal proof theory, the book can be recommended to everyone interested in these topics." (David Binder, Journal for General Philosophy of Science, Vol. 49, 2018)

"The volume is undoubtedly of a high-level technical content and thus of great interest for computer scientists and mathematical logicians. However, thanks to the numerous chapters containing philosophical reflections and historical information, the volume is also appealing for philosophers of logic." (Francesca Poggiolesi, History and Philosophy of Logic, Vol. 38 (3), March, 2017)

"The articles in this collection clearly offer a wide and lively description of Gentzen's work and of Gentzen's legacy. ... This volume will be for years to come an important key tool not only for people working in reductive and ordinal proof theory, but also for anyone more in general interested in the field of proof theory." (Enrico Moriconi, History and Philosophy of Logic, Vol. 38 (2), January, 2017)

"Gentzen's work has provided the blueprint, in many ways, for work in proof theory that has been extending its reach beyond arithmetic to subsystems of analysis and also to parts of set theory. ... This is an exciting volume: informative in many different ways about the background and current state of proof theory, in particular of those parts that have been shaped by Gentzen's work." (Wilfried Sieg, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 119, 2017)

"This book is thefollow-on to the 2009 symposium in Leeds celebrating the centenary of the great German logician Gerhard Gentzen. It contains survey as well as technical chapters by highly respected proof theorists. It is of interest to logicians and computer scientists working in theorem proving, type theory, and proof complexity." (K. Lodaya, Computing Reviews, April, 2016)