Ways Of Proof Theory

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Ways of Proof Theory

Ralf Schindler
Ways of Proof Theory Book PDF

Author : Ralf Schindler
Publisher : Walter de Gruyter
Page : 498 pages
File Size : 50,5 Mb
Release : 2010
Category : Proof theory
ISBN : 3110324911

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Ways of Proof Theory by Ralf Schindler Pdf

On the occasion of the retirement of Wolfram Pohlers the Institut fur Mathematische Logik und Grundlagenforschung of the University of Munster organized a colloquium and a workshop which took place July 17 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory."

Ways of Proof Theory

Ralf Schindler
Ways of Proof Theory Book PDF

Author : Ralf Schindler
Publisher : Walter de Gruyter
Page : 495 pages
File Size : 46,8 Mb
Release : 2013-05-02
Category : Philosophy
ISBN : 9783110324907

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Ways of Proof Theory by Ralf Schindler Pdf

On the occasion of the retirement of Wolfram Pohlers the Institut für Mathematische Logik und Grundlagenforschung of the University of Münster organized a colloquium and a workshop which took place July 17 – 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory.

Proof Theory

Wolfram Pohlers
Proof Theory Book PDF

Author : Wolfram Pohlers
Publisher : Springer Science & Business Media
Page : 380 pages
File Size : 43,5 Mb
Release : 2008-10-01
Category : Mathematics
ISBN : 9783540693192

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Proof Theory by Wolfram Pohlers Pdf

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).

Proof Theory

Wolfram Pohlers
Proof Theory Book PDF

Author : Wolfram Pohlers
Publisher : Springer
Page : 220 pages
File Size : 48,6 Mb
Release : 2009-06-10
Category : Mathematics
ISBN : 9783540468257

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Proof Theory by Wolfram Pohlers Pdf

Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.

How to Prove It

Daniel J. Velleman
How to Prove It Book PDF

Author : Daniel J. Velleman
Publisher : Cambridge University Press
Page : 401 pages
File Size : 53,6 Mb
Release : 2006-01-16
Category : Mathematics
ISBN : 9780521861243

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How to Prove It by Daniel J. Velleman Pdf

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

Proof Theory

Katalin Bimbo
Proof Theory Book PDF

Author : Katalin Bimbo
Publisher : CRC Press
Page : 386 pages
File Size : 50,8 Mb
Release : 2014-08-20
Category : Mathematics
ISBN : 9781466564688

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Proof Theory by Katalin Bimbo Pdf

Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi

Proof Theory

Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen
Proof Theory Book PDF

Author : Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 41,5 Mb
Release : 2013-03-09
Category : Philosophy
ISBN : 9789401727969

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Proof Theory by Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jørgensen Pdf

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.

Proofs from THE BOOK

Martin Aigner,Günter M. Ziegler
Proofs from THE BOOK Book PDF

Author : Martin Aigner,Günter M. Ziegler
Publisher : Springer Science & Business Media
Page : 194 pages
File Size : 43,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662223437

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Proofs from THE BOOK by Martin Aigner,Günter M. Ziegler Pdf

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.

Advances in Proof-Theoretic Semantics

Thomas Piecha,Peter Schroeder-Heister
Advances in Proof Theoretic Semantics Book PDF

Author : Thomas Piecha,Peter Schroeder-Heister
Publisher : Springer
Page : 283 pages
File Size : 55,8 Mb
Release : 2015-10-24
Category : Philosophy
ISBN : 9783319226866

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Advances in Proof-Theoretic Semantics by Thomas Piecha,Peter Schroeder-Heister Pdf

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.

Principia Mathematica

Alfred North Whitehead,Bertrand Russell
Principia Mathematica Book PDF

Author : Alfred North Whitehead,Bertrand Russell
Publisher : Cambridge University Press
Page : 524 pages
File Size : 54,5 Mb
Release : 1927
Category : Mathematics
ISBN : 052106791X

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Principia Mathematica by Alfred North Whitehead,Bertrand Russell Pdf

The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.

Subsystems of Second Order Arithmetic

Stephen George Simpson
Subsystems of Second Order Arithmetic Book PDF

Author : Stephen George Simpson
Publisher : Cambridge University Press
Page : 461 pages
File Size : 40,9 Mb
Release : 2009-05-29
Category : Mathematics
ISBN : 9780521884396

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Subsystems of Second Order Arithmetic by Stephen George Simpson Pdf

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

The Semantics and Proof Theory of the Logic of Bunched Implications

David J. Pym
The Semantics and Proof Theory of the Logic of Bunched Implications Book PDF

Author : David J. Pym
Publisher : Springer Science & Business Media
Page : 323 pages
File Size : 41,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401700917

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The Semantics and Proof Theory of the Logic of Bunched Implications by David J. Pym Pdf

This is a monograph about logic. Specifically, it presents the mathe matical theory of the logic of bunched implications, BI: I consider Bl's proof theory, model theory and computation theory. However, the mono graph is also about informatics in a sense which I explain. Specifically, it is about mathematical models of resources and logics for reasoning about resources. I begin with an introduction which presents my (background) view of logic from the point of view of informatics, paying particular attention to three logical topics which have arisen from the development of logic within informatics: • Resources as a basis for semantics; • Proof-search as a basis for reasoning; and • The theory of representation of object-logics in a meta-logic. The ensuing development represents a logical theory which draws upon the mathematical, philosophical and computational aspects of logic. Part I presents the logical theory of propositional BI, together with a computational interpretation. Part II presents a corresponding devel opment for predicate BI. In both parts, I develop proof-, model- and type-theoretic analyses. I also provide semantically-motivated compu tational perspectives, so beginning a mathematical theory of resources. I have not included any analysis, beyond conjecture, of properties such as decidability, finite models, games or complexity. I prefer to leave these matters to other occasions, perhaps in broader contexts.

An Introduction to Proof Theory

Paolo Mancosu,Sergio Galvan,Richard Zach
An Introduction to Proof Theory Book PDF

Author : Paolo Mancosu,Sergio Galvan,Richard Zach
Publisher : Oxford University Press
Page : 431 pages
File Size : 40,9 Mb
Release : 2021
Category : Philosophy
ISBN : 9780192895936

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An Introduction to Proof Theory by Paolo Mancosu,Sergio Galvan,Richard Zach Pdf

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.

Handbook of Proof Theory

S.R. Buss
Handbook of Proof Theory Book PDF

Author : S.R. Buss
Publisher : Elsevier
Page : 810 pages
File Size : 49,5 Mb
Release : 1998-07-09
Category : Mathematics
ISBN : 0080533183

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Handbook of Proof Theory by S.R. Buss Pdf

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.

Hybrid Logic and its Proof-Theory

Torben Braüner
Hybrid Logic and its Proof Theory Book PDF

Author : Torben Braüner
Publisher : Springer Science & Business Media
Page : 231 pages
File Size : 52,9 Mb
Release : 2010-11-17
Category : Philosophy
ISBN : 9789400700024

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Hybrid Logic and its Proof-Theory by Torben Braüner Pdf

This is the first book-length treatment of hybrid logic and its proof-theory. Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model (where the points represent times, possible worlds, states in a computer, or something else). This is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about proof-theory for ordinary modal logic. Many modal-logical proof systems lack important properties and the relationships between proof systems for different modal logics are often unclear. In the present book we demonstrate that hybrid-logical proof-theory remedies these deficiencies by giving a spectrum of well-behaved proof systems (natural deduction, Gentzen, tableau, and axiom systems) for a spectrum of different hybrid logics (propositional, first-order, intensional first-order, and intuitionistic).