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Author: A. S. Troelstra
Publisher: Cambridge University Press
ISBN: 9780521779111
Category : Computers
Languages : en
Pages : 436
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Book Description
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.
Author: A. S. Troelstra
Publisher: Cambridge University Press
ISBN: 9780521779111
Category : Computers
Languages : en
Pages : 436
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Book Description
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.
Author: Paolo Mancosu
Publisher: Oxford University Press
ISBN: 0192895931
Category : Mathematics
Languages : en
Pages : 431
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Book Description
"Proof theory is a central area of mathematical logic of special interest to philosophy . It has its roots in the foundational debate of the 1920s, in particular, in Hilbert's program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, "finitary" means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a "simple" proof. This is Hilbert's central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail"--
Author: Martin Aigner
Publisher: Springer Science & Business Media
ISBN: 3662223430
Category : Mathematics
Languages : en
Pages : 194
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Book Description
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.
Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314
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Book Description
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.
Author: S.R. Buss
Publisher: Elsevier
ISBN: 0080533183
Category : Mathematics
Languages : en
Pages : 823
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Book Description
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.
Author: Helmut Schwichtenberg
Publisher: Cambridge University Press
ISBN: 1139504169
Category : Mathematics
Languages : en
Pages : 480
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Book Description
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.
Author: Peter Aczel
Publisher: Cambridge University Press
ISBN: 9780521414135
Category : Computers
Languages : en
Pages : 320
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Book Description
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.
Author: Toshiyasu Arai
Publisher: Springer Nature
ISBN: 9811564590
Category : Philosophy
Languages : en
Pages : 327
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Book Description
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.
Author: Andrew Wohlgemuth
Publisher: Courier Corporation
ISBN: 0486141683
Category : Mathematics
Languages : en
Pages : 385
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Book Description
The primary purpose of this undergraduate text is to teach students to do mathematical proofs. It enables readers to recognize the elements that constitute an acceptable proof, and it develops their ability to do proofs of routine problems as well as those requiring creative insights. The self-contained treatment features many exercises, problems, and selected answers, including worked-out solutions. Starting with sets and rules of inference, this text covers functions, relations, operation, and the integers. Additional topics include proofs in analysis, cardinality, and groups. Six appendixes offer supplemental material. Teachers will welcome the return of this long-out-of-print volume, appropriate for both one- and two-semester courses.
Author: Sara Negri
Publisher: Cambridge University Press
ISBN: 9780521068420
Category : Mathematics
Languages : en
Pages : 279
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Book Description
A concise introduction to structural proof theory, a branch of logic studying the general structure of logical and mathematical proofs.
