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Author : Rob Nederpelt,Herman Geuvers
Publisher : Cambridge University Press
Page : 465 pages
File Size : 52,5 Mb
Release : 2014-11-06
Category : Computers
ISBN : 9781107036505
A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.
Author : Paolo Mancosu,Sergio Galvan,Richard Zach
Publisher : Oxford University Press
Page : 431 pages
File Size : 44,9 Mb
Release : 2021
Category : Philosophy
ISBN : 9780192895936
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.
Author : K. Schütte
Publisher : Springer Science & Business Media
Page : 309 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642664731
This book was originally intended to be the second edition of the book "Beweis theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic. We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie."
Author : S.R. Buss
Publisher : Elsevier
Page : 823 pages
File Size : 42,6 Mb
Release : 1998-07-09
Category : Mathematics
ISBN : 9780080533186
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 : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 49,6 Mb
Release : 2024-07-04
Category : Electronic
ISBN : 8210379456XXX
Author : Peter B. Andrews
Publisher : Springer Science & Business Media
Page : 404 pages
File Size : 54,9 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401599344
In case you are considering to adopt this book for courses with over 50 students, please contact [email protected] for more information. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.
Author : Mariangiola Dezani-Ciancaglini
Publisher : Unknown
Page : 398 pages
File Size : 40,9 Mb
Release : 2002
Category : Proof theory
ISBN : OCLC:249252552
Author : Takahashi, Masako
Publisher : Unknown
Page : 314 pages
File Size : 46,9 Mb
Release : 1998-12
Category : Applied mathematics
ISBN : UOM:39015050808990
Author : Ulrich Kohlenbach
Publisher : Springer Science & Business Media
Page : 539 pages
File Size : 55,6 Mb
Release : 2008-05-23
Category : Mathematics
ISBN : 9783540775331
This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.
Author : Per Martin-Löf,Giovanni Sambin
Publisher : Unknown
Page : 116 pages
File Size : 44,8 Mb
Release : 1984
Category : Mathematics
ISBN : STANFORD:36105021234930
Author : Morten Heine Sørensen,Pawel Urzyczyn
Publisher : Elsevier
Page : 457 pages
File Size : 42,7 Mb
Release : 2006-07-04
Category : Mathematics
ISBN : 9780080478920
The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.The isomorphism has many aspects, even at the syntactic level:formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transformsproofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features- The Curry-Howard Isomorphism treated as common theme- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics- Thorough study of the connection between calculi and logics- Elaborate study of classical logics and control operators- Account of dialogue games for classical and intuitionistic logic- Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme.· Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics.· Elaborate study of classical logics and control operators.· Account of dialogue games for classical and intuitionistic logic.· Theoretical foundations of computer-assisted reasoning
Author : Gaisi Takeuti
Publisher : Courier Corporation
Page : 514 pages
File Size : 53,8 Mb
Release : 2013-10-10
Category : Mathematics
ISBN : 9780486320670
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.
Author : A. S. Troelstra,H. Schwichtenberg
Publisher : Cambridge University Press
Page : 436 pages
File Size : 52,8 Mb
Release : 2000-07-27
Category : Computers
ISBN : 0521779111
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 : Alfred North Whitehead,Bertrand Russell
Publisher : Cambridge University Press
Page : 524 pages
File Size : 54,8 Mb
Release : 1927
Category : Mathematics
ISBN : 052106791X
The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.
Author : Sara Negri,Jan von Plato
Publisher : Cambridge University Press
Page : 279 pages
File Size : 54,8 Mb
Release : 2008-07-10
Category : Mathematics
ISBN : 0521068428
A concise introduction to structural proof theory, a branch of logic studying the general structure of logical and mathematical proofs.