Proofs & Theories
Proofs & Theories

Author: Louise Gluck

Publisher: HarperCollins

Published: 2022-01-04

Total Pages: 154

ISBN-13: 0063117614

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Winner of the Nobel Prize in Literature Proofs and Theories, winner of the PEN/Martha Albrand Award for First Non-Fiction, is an illuminating collection of essays by Louise Glück, one of this country's most brilliant poets. Like her poems, the prose of Glück, who won the Pulitzer Prize for poetry in 1993 for The Wild Iris, is compressed, fastidious, fierce, alert, and absolutely unconsoled. The force of her thought is evident everywhere in these essays, from her explorations of other poets' work to her skeptical contemplation of current literary critical notions such as "sincerity" and "courage." Here also are Glück's revealing reflections on her own education and life as a poet, and a tribute to her teacher and mentor, Stanley Kunitz. Proofs and Theories is not a casual collection. It is the testament of a major poet.

Proof Theory
Proof Theory

Author: Peter Aczel

Publisher: Cambridge University Press

Published: 1992

Total Pages: 320

ISBN-13: 9780521414135

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The lecture courses in this work are derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles; put together in this book they form an invaluable introduction to proof theory that is aimed at both mathematicians and computer scientists.

Advances in Proof-Theoretic Semantics
Advances in Proof-Theoretic Semantics

Author: Thomas Piecha

Publisher: Springer

Published: 2015-10-24

Total Pages: 281

ISBN-13: 331922686X

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This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.

Proof Theory
Proof Theory

Author: Wolfram Pohlers

Publisher: Springer Science & Business Media

Published: 2008-10-01

Total Pages: 380

ISBN-13: 354069319X

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The kernel of this book consists of a series of lectures on in?nitary proof theory which I gave during my time at the Westfalische ̈ Wilhelms–Universitat ̈ in Munster ̈ . It was planned as a successor of Springer Lecture Notes in Mathematics 1407. H- ever, when preparing it, I decided to also include material which has not been treated in SLN 1407. Since the appearance of SLN 1407 many innovations in the area of - dinal analysis have taken place. Just to mention those of them which are addressed in this book: Buchholz simpli?ed local predicativity by the invention of operator controlled derivations (cf. Chapter 9, Chapter 11); Weiermann detected applications of methods of impredicative proof theory to the characterization of the provable recursive functions of predicative theories (cf. Chapter 10); Beckmann improved Gentzen’s boundedness theorem (which appears as Stage Theorem (Theorem 6. 6. 1) in this book) to Theorem 6. 6. 9, a theorem which is very satisfying in itself - though its real importance lies in the ordinal analysis of systems, weaker than those treated here. Besides these innovations I also decided to include the analysis of the theory (? –REF) as an example of a subtheory of set theory whose ordinal analysis only 2 0 requires a ?rst step into impredicativity. The ordinal analysis of(? –FXP) of non- 0 1 0 monotone? –de?nable inductive de?nitions in Chapter 13 is an application of the 1 analysis of(? –REF).

Handbook of Proof Theory
Handbook of Proof Theory

Author: S.R. Buss

Publisher: Elsevier

Published: 1998-07-09

Total Pages: 823

ISBN-13: 0080533183

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This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth.The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.

Subsystems of Second Order Arithmetic
Subsystems of Second Order Arithmetic

Author: Stephen George Simpson

Publisher: Cambridge University Press

Published: 2009-05-29

Total Pages: 461

ISBN-13: 052188439X

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This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.

Proof Theory
Proof Theory

Author: Vincent F. Hendricks

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 345

ISBN-13: 9401727961

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hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics.

An Introduction to Proof Theory
An Introduction to Proof Theory

Author: Paolo Mancosu

Publisher: Oxford University Press

Published: 2021-08-12

Total Pages: 336

ISBN-13: 0192649299

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An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.

Proof Theory
Proof Theory

Author: Gaisi Takeuti

Publisher: Courier Corporation

Published: 2013-10-10

Total Pages: 514

ISBN-13: 0486320677

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This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

Selected Papers
Selected Papers

Author: Calvin C. Elgot

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 475

ISBN-13: 1461381770

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Cal Elgot was a very serious and thoughtful researcher, who with great determi nation attempted to find basic explanations for certain mathematical phenomena as the selection of papers in this volume well illustrate. His approach was, for the most part, rather finitist and constructivist, and he was inevitably drawn to studies of the process of computation. It seems to me that his early work on decision problems relating automata and logic, starting with his thesis under Roger Lyndon and continuing with joint work with Biichi, Wright, Copi, Rutledge, Mezei, and then later with Rabin, set the stage for his attack on the theory of computation through the abstract treatment of the notion of a machine. This is also apparent in his joint work with A. Robinson reproduced here and in his joint papers with John Shepherdson. Of course in the light of subsequent work on decision problems by Biichi, Rabin, Shelah, and many, many others, the subject has been placed on a completely different plane from what it was when Elgot left the area. But I feel that his papers, results-and style-were very definitely influential at the time and may well have altered the course of the investigation of these problems. As Sammy Eilenberg explains, the next big influence on Elgot's thinking was category theory, which gave him a way of expressing his ideas in a sharply algebraic manner. The joint book with Eilenberg is one illustration of this influence.