Logic And Foundations Of Mathematics

Author : William S. Hatcher
Publisher : Elsevier
Page : 331 pages
File Size : 46,5 Mb
Release : 2014-05-09
Category : Mathematics
ISBN : 9781483189635

Get Book

The Logical Foundations of Mathematics by William S. Hatcher Pdf

The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

Author : G. T. Kneebone
Publisher : Dover Publications
Page : 0 pages
File Size : 49,5 Mb
Release : 2001
Category : Logic, Symbolic and mathematical
ISBN : 0486417123

Get Book

Mathematical Logic and the Foundations of Mathematics by G. T. Kneebone Pdf

Ideal for students intending to specialize in the topic. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics. Part III focuses on the philosophy of mathematics.

Author : Yves Nievergelt
Publisher : Springer Science & Business Media
Page : 425 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461201250

Get Book

Foundations of Logic and Mathematics by Yves Nievergelt Pdf

This modern introduction to the foundations of logic and mathematics not only takes theory into account, but also treats in some detail applications that have a substantial impact on everyday life (loans and mortgages, bar codes, public-key cryptography). A first college-level introduction to logic, proofs, sets, number theory, and graph theory, and an excellent self-study reference and resource for instructors.

Author : Pavel Pudlák
Publisher : Springer Science & Business Media
Page : 699 pages
File Size : 50,7 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9783319001197

Get Book

Logical Foundations of Mathematics and Computational Complexity by Pavel Pudlák Pdf

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

Author : Haskell Brooks Curry
Publisher : Courier Corporation
Page : 420 pages
File Size : 41,6 Mb
Release : 1977-01-01
Category : Mathematics
ISBN : 0486634620

Get Book

Foundations of Mathematical Logic by Haskell Brooks Curry Pdf

Written by a pioneer of mathematical logic, this comprehensive graduate-level text explores the constructive theory of first-order predicate calculus. It covers formal methods — including algorithms and epitheory — and offers a brief treatment of Markov's approach to algorithms. It also explains elementary facts about lattices and similar algebraic systems. 1963 edition.

Author : H.-D. Ebbinghaus,J. Flum,Wolfgang Thomas
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 47,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475723557

Get Book

Mathematical Logic by H.-D. Ebbinghaus,J. Flum,Wolfgang Thomas Pdf

This introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.

Author : Richard E. Hodel
Publisher : Courier Corporation
Page : 514 pages
File Size : 43,7 Mb
Release : 2013-01-01
Category : Mathematics
ISBN : 9780486497853

Get Book

An Introduction to Mathematical Logic by Richard E. Hodel Pdf

This comprehensive overview ofmathematical logic is designedprimarily for advanced undergraduatesand graduate studentsof mathematics. The treatmentalso contains much of interest toadvanced students in computerscience and philosophy. Topics include propositional logic;first-order languages and logic; incompleteness, undecidability,and indefinability; recursive functions; computability;and Hilbert’s Tenth Problem.Reprint of the PWS Publishing Company, Boston, 1995edition.

Author : Morten Heine Sørensen,Pawel Urzyczyn
Publisher : Elsevier
Page : 457 pages
File Size : 43,8 Mb
Release : 2006-07-04
Category : Mathematics
ISBN : 9780080478920

Get Book

Lectures on the Curry-Howard Isomorphism by Morten Heine Sørensen,Pawel Urzyczyn Pdf

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 : Douglas Cenzer,Jean Larson,Christopher Porter,Jindrich Zapletal
Publisher : World Scientific
Page : 222 pages
File Size : 50,5 Mb
Release : 2020-04-04
Category : Mathematics
ISBN : 9789811201943

Get Book

Set Theory And Foundations Of Mathematics: An Introduction To Mathematical Logic - Volume I: Set Theory by Douglas Cenzer,Jean Larson,Christopher Porter,Jindrich Zapletal Pdf

This book provides an introduction to axiomatic set theory and descriptive set theory. It is written for the upper level undergraduate or beginning graduate students to help them prepare for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra.The book is designed as a flexible and accessible text for a one-semester introductory course in set theory, where the existing alternatives may be more demanding or specialized. Readers will learn the universally accepted basis of the field, with several popular topics added as an option. Pointers to more advanced study are scattered throughout the text.

Author : Y. Gauthier
Publisher : Springer Science & Business Media
Page : 276 pages
File Size : 45,8 Mb
Release : 2002-06-30
Category : Mathematics
ISBN : 1402006896

Get Book

Internal Logic by Y. Gauthier Pdf

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.

Author : Jaap van Oosten
Publisher : Elsevier
Page : 327 pages
File Size : 49,8 Mb
Release : 2008-04-10
Category : Mathematics
ISBN : 9780080560069

Get Book

Realizability by Jaap van Oosten Pdf

Aimed at starting researchers in the field, Realizability gives a rigorous, yet reasonable introduction to the basic concepts of a field which has passed several successive phases of abstraction. Material from previously unpublished sources such as Ph.D. theses, unpublished papers, etc. has been molded into one comprehensive presentation of the subject area. - The first book to date on this subject area- Provides an clear introduction to Realizability with a comprehensive bibliography- Easy to read and mathematically rigorous- Written by an expert in the field

Author : Kenneth Kunen
Publisher : Unknown
Page : 251 pages
File Size : 49,5 Mb
Release : 2009
Category : Mathematics
ISBN : 1904987141

Get Book

The Foundations of Mathematics by Kenneth Kunen Pdf

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.

Author : Wei Li
Publisher : Unknown
Page : 316 pages
File Size : 51,6 Mb
Release : 2014-11-30
Category : Electronic
ISBN : 3034808631

Get Book

Mathematical Logic by Wei Li Pdf

Author : Robert S. Wolf
Publisher : American Mathematical Soc.
Page : 397 pages
File Size : 51,9 Mb
Release : 2005-12-31
Category : Algebra, Abstract
ISBN : 9781614440284

Get Book

A Tour Through Mathematical Logic by Robert S. Wolf Pdf

A Tour Through Mathematical Logic provides a tour through the main branches of the foundations of mathematics. It contains chapters covering elementary logic, basic set theory, recursion theory, Gödel's (and others') incompleteness theorems, model theory, independence results in set theory, nonstandard analysis, and constructive mathematics. In addition, this monograph discusses several topics not normally found in books of this type, such as fuzzy logic, nonmonotonic logic, and complexity theory.

Author : Georg Kreisel,Jean Louis Krivine
Publisher : Elsevier
Page : 222 pages
File Size : 53,8 Mb
Release : 1967
Category : Electronic books
ISBN : 0444534121

Get Book

Elements of Mathematical Logic by Georg Kreisel,Jean Louis Krivine Pdf