Basic Proof Theory
Basic Proof Theory

Author: A. S. Troelstra

Publisher: Cambridge University Press

Published: 2000-07-27

Total Pages: 436

ISBN-13: 9780521779111

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This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.

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.

Proofs from THE BOOK
Proofs from THE BOOK

Author: Martin Aigner

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 194

ISBN-13: 3662223430

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According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

Book of Proof
Book of Proof

Author: Richard H. Hammack

Publisher:

Published: 2016-01-01

Total Pages: 314

ISBN-13: 9780989472111

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This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Proofs and Computations
Proofs and Computations

Author: Helmut Schwichtenberg

Publisher: Cambridge University Press

Published: 2011-12-15

Total Pages: 480

ISBN-13: 1139504169

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Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Structural Proof Theory
Structural Proof Theory

Author: Sara Negri

Publisher: Cambridge University Press

Published: 2008-07-10

Total Pages: 279

ISBN-13: 9780521068420

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A concise introduction to structural proof theory, a branch of logic studying the general structure of logical and mathematical proofs.

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.

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.

Basic Simple Type Theory
Basic Simple Type Theory

Author: J. Roger Hindley

Publisher: Cambridge University Press

Published: 1997

Total Pages: 200

ISBN-13: 0521465184

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Type theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. By concentrating on the principles that make the theory work in practice, the author covers all the key ideas without getting involved in the complications of more advanced systems. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm that lies at the heart of every such system. Also featured are two other interesting algorithms that until now have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making it the first book at this level that can be used as an introduction to type theory for computer scientists.

Ordinal Analysis with an Introduction to Proof Theory
Ordinal Analysis with an Introduction to Proof Theory

Author: Toshiyasu Arai

Publisher: Springer Nature

Published: 2020-08-11

Total Pages: 327

ISBN-13: 9811564590

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This book provides readers with a guide to both ordinal analysis, and to proof theory. It mainly focuses on ordinal analysis, a research topic in proof theory that is concerned with the ordinal theoretic content of formal theories. However, the book also addresses ordinal analysis and basic materials in proof theory of first-order or omega logic, presenting some new results and new proofs of known ones.Primarily intended for graduate students and researchers in mathematics, especially in mathematical logic, the book also includes numerous exercises and answers for selected exercises, designed to help readers grasp and apply the main results and techniques discussed.