Logic for Mathematicians


Author: A. G. Hamilton
Publisher: Cambridge University Press
ISBN: 9780521368650
Category: Mathematics
Page: 228
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This is an introductory textbook which is designed to be useful not only to intending logicians but also to mathematicians in general.

Logic for Mathematicians


Author: J. Barkley Rosser
Publisher: Courier Dover Publications
ISBN: 0486468984
Category: Mathematics
Page: 574
View: 1199

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Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use.

A Course in Mathematical Logic for Mathematicians


Author: Yu. I. Manin
Publisher: Springer Science & Business Media
ISBN: 1441906150
Category: Mathematics
Page: 384
View: 1592

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1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some time. In the intervening three decades, a lot of interesting things have happened to mathematical logic: (i) Model theory has shown that insights acquired in the study of formal languages could be used fruitfully in solving old problems of conventional mathematics. (ii) Mathematics has been and is moving with growing acceleration from the set-theoretic language of structures to the language and intuition of (higher) categories, leaving behind old concerns about in?nities: a new view of foundations is now emerging. (iii) Computer science, a no-nonsense child of the abstract computability theory, has been creatively dealing with old challenges and providing new ones, such as the P/NP problem. Planning additional chapters for this second edition, I have decided to focus onmodeltheory,the conspicuousabsenceofwhichinthe ?rsteditionwasnoted in several reviews, and the theory of computation, including its categorical and quantum aspects. The whole Part IV: Model Theory, is new. I am very grateful to Boris I. Zilber, who kindly agreed to write it. It may be read directly after Chapter II. The contents of the ?rst edition are basically reproduced here as Chapters I–VIII. Section IV.7, on the cardinality of the continuum, is completed by Section IV.7.3, discussing H. Woodin’s discovery.

Mathematical Logic


Author: J.D. Monk
Publisher: Springer Science & Business Media
ISBN: 146849452X
Category: Mathematics
Page: 532
View: 7173

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From the Introduction: "We shall base our discussion on a set-theoretical foundation like that used in developing analysis, or algebra, or topology. We may consider our task as that of giving a mathematical analysis of the basic concepts of logic and mathematics themselves. Thus we treat mathematical and logical practice as given empirical data and attempt to develop a purely mathematical theory of logic abstracted from these data." There are 31 chapters in 5 parts and approximately 320 exercises marked by difficulty and whether or not they are necessary for further work in the book.

Mathematics and Logic


Author: Mark Kac,Stanislaw M. Ulam
Publisher: Courier Corporation
ISBN: 0486670856
Category: Philosophy
Page: 170
View: 619

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Fascinating study of the origin and nature of mathematical thought, including relation of mathematics and science, 20th-century developments, impact of computers, and more.Includes 34 illustrations. 1968 edition."

A Course in Mathematical Logic


Author: Yu.I. Manin
Publisher: Springer Science & Business Media
ISBN: 1475743858
Category: Mathematics
Page: 288
View: 2913

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1. This book is above all addressed to mathematicians. It is intended to be a textbook of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last ten or fifteen years. These include: the independence of the continuum hypothe sis, the Diophantine nature of enumerable sets, the impossibility of finding an algorithmic solution for one or two old problems. All the necessary preliminary material, including predicate logic and the fundamentals of recursive function theory, is presented systematically and with complete proofs. We only assume that the reader is familiar with "naive" set theoretic arguments. In this book mathematical logic is presented both as a part of mathe matics and as the result of its self-perception. Thus, the substance of the book consists of difficult proofs of subtle theorems, and the spirit of the book consists of attempts to explain what these theorems say about the mathematical way of thought. Foundational problems are for the most part passed over in silence. Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life. 2. The first two chapters are devoted to predicate logic. The presenta tion here is fairly standard, except that semantics occupies a very domi nant position, truth is introduced before deducibility, and models of speech in formal languages precede the systematic study of syntax.

Mathematical Logic for Computer Science


Author: Mordechai Ben-Ari
Publisher: Springer Science & Business Media
ISBN: 1447141296
Category: Mathematics
Page: 346
View: 6386

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Mathematical Logic for Computer Science is a mathematics textbook with theorems and proofs, but the choice of topics has been guided by the needs of students of computer science. The method of semantic tableaux provides an elegant way to teach logic that is both theoretically sound and easy to understand. The uniform use of tableaux-based techniques facilitates learning advanced logical systems based on what the student has learned from elementary systems. The logical systems presented are: propositional logic, first-order logic, resolution and its application to logic programming, Hoare logic for the verification of sequential programs, and linear temporal logic for the verification of concurrent programs. The third edition has been entirely rewritten and includes new chapters on central topics of modern computer science: SAT solvers and model checking.

A Course on Mathematical Logic


Author: Shashi Mohan Srivastava
Publisher: Springer Science & Business Media
ISBN: 1461457467
Category: Mathematics
Page: 198
View: 5422

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This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.

Handbook of Mathematical Logic


Author: J. Barwise
Publisher: Elsevier
ISBN: 9780080933641
Category: Mathematics
Page: 1164
View: 8782

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The handbook is divided into four parts: model theory, set theory, recursion theory and proof theory. Each of the four parts begins with a short guide to the chapters that follow. Each chapter is written for non-specialists in the field in question. Mathematicians will find that this book provides them with a unique opportunity to apprise themselves of developments in areas other than their own.

Introduction to Logic


Author: Patrick Suppes
Publisher: Courier Corporation
ISBN: 0486138054
Category: Mathematics
Page: 336
View: 3766

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Part I of this coherent, well-organized text deals with formal principles of inference and definition. Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Ideal for undergraduates.

First-order Logic


Author: Raymond M. Smullyan
Publisher: Courier Corporation
ISBN: 9780486683706
Category: Mathematics
Page: 158
View: 5917

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Considered the best book in the field, this completely self-contained study is both an introduction to quantification theory and an exposition of new results and techniques in "analytic" or "cut free" methods. The focus in on the tableau point of view. Topics include trees, tableau method for propositional logic, Gentzen systems, more. Includes 144 illustrations.

Forcing for Mathematicians


Author: Nik Weaver
Publisher: World Scientific
ISBN: 9814566020
Category: Mathematics
Page: 152
View: 2156

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Ever since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics. Contents:Peano ArithmeticZermelo–Fraenkel Set TheoryWell-Ordered SetsOrdinalsCardinalsRelativizationReflectionForcing NotionsGeneric ExtensionsForcing EqualityThe Fundamental TheoremForcing CHForcing ¬ CHFamilies of Entire Functions*Self-Homeomorphisms of βℕ \ ℕ, I*Pure States on B(H)*The Diamond PrincipleSuslin's Problem, I*Naimark's problem*A Stronger DiamondWhitehead's Problem, I*Iterated ForcingMartin's AxiomSuslin's Problem, II*Whitehead's Problem, II*The Open Coloring AxiomSelf-Homeomorphisms of βℕ \ ℕ, II*Automorphisms of the Calkin Algebra, I*Automorphisms of the Calkin Algebra, II*The Multiverse Interpretation Readership: Graduates and researchers in logic and set theory, general mathematical audience. Keywords:Forcing;Set Theory;Consistency;Independence;C*-AlgebraKey Features:A number of features combine to make this thorough and rigorous treatment of forcing surprisingly easy to follow. First, it goes straight into the core material on forcing, avoiding Godel constructibility altogether; second, key definitions are simplified, allowing for a less technical development; and third, further care is given to the treatment of metatheoretic issuesEach chapter is limited to four pages, making the presentation very readableA unique feature of the book is its emphasis on applications to problems outside of set theory. Much of this material is currently only available in the primary literatureThe author is a pioneer in the application of set-theoretic methods to C*-algebra, having solved (together with various co-authors) Dixmier's “prime versus primitive” problem, Naimark's problem, Anderson's conjecture about pure states on B(H), and the Calkin algebra outer automorphism problemReviews: “The author presents the basics of the theory of forcing in a clear and stringent way by emphasizing important technical details and simplifying some definitions and arguments. Moreover, he presents the content in a way that should help beginners to understand the central concepts and avoid common mistakes.” Zentralblatt MATH

Logic of Mathematics

A Modern Course of Classical Logic
Author: Zofia Adamowicz,Pawel Zbierski
Publisher: John Wiley & Sons
ISBN: 1118030796
Category: Mathematics
Page: 272
View: 4970

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A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic Logic of Mathematics combines a full-scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems: * Gödel's theorems of completeness and incompleteness * The independence of Goodstein's theorem from Peano arithmetic * Tarski's theorem on real closed fields * Matiyasevich's theorem on diophantine formulas Logic of Mathematics also features: * Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types * Clear, concise explanations of all key concepts, from Boolean algebras to Skolem-Löwenheim constructions and other topics * Carefully chosen exercises for each chapter, plus helpful solution hints At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic-requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms. Part I contains a thorough introduction to mathematical logic and model theory-including a full discussion of terms, formulas, and other fundamentals, plus detailed coverage of relational structures and Boolean algebras, Gödel's completeness theorem, models of Peano arithmetic, and much more. Part II focuses on a number of advanced theorems that are central to the field, such as Gödel's first and second theorems of incompleteness, the independence proof of Goodstein's theorem from Peano arithmetic, Tarski's theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems. With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use-the perfect textbook for advanced students of mathematics, computer science, and logic.

Sets for Mathematics


Author: F. William Lawvere,Robert Rosebrugh
Publisher: Cambridge University Press
ISBN: 9780521010603
Category: Mathematics
Page: 261
View: 9212

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In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.

First Course in Mathematical Logic


Author: Patrick Suppes,Shirley Hill
Publisher: Courier Corporation
ISBN: 0486150941
Category: Mathematics
Page: 288
View: 8247

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Rigorous introduction is simple enough in presentation and context for wide range of students. Symbolizing sentences; logical inference; truth and validity; truth tables; terms, predicates, universal quantifiers; universal specification and laws of identity; more.

Algebraic Methods of Mathematical Logic


Author: Ladislav Rieger
Publisher: Elsevier
ISBN: 1483270521
Category: Mathematics
Page: 210
View: 1163

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Algebraic Methods of Mathematical Logic focuses on the algebraic methods of mathematical logic, including Boolean algebra, mathematical language, and arithmetization. The book first offers information on the dialectic of the relation between mathematical and metamathematical aspects; metamathematico-mathematical parallelism and its natural limits; practical applications of methods of mathematical logic; and principal mathematical tools of mathematical logic. The text then elaborates on the language of mathematics and its symbolization and recursive construction of the relation of consequence. Discussions focus on recursive construction of the relation of consequence, fundamental descriptively-semantic rules, mathematical logic and mathematical language as a material system of signs, and the substance and purpose of symbolization of mathematical language. The publication examines expressive possibilities of symbolization; intuitive and mathematical notions of an idealized axiomatic mathematical theory; and the algebraic theory of elementary predicate logic. Topics include the notion of Boolean algebra based on joins, meets, and complementation, logical frame of a language and mathematical theory, and arithmetization and algebraization. The manuscript is a valuable reference for mathematicians and researchers interested in the algebraic methods of mathematical logic.

The Art of Proof

Basic Training for Deeper Mathematics
Author: Matthias Beck,Ross Geoghegan
Publisher: Springer Science & Business Media
ISBN: 9781441970237
Category: Mathematics
Page: 182
View: 7751

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The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Creative Mathematics


Author: H. S. Wall
Publisher: MAA
ISBN: 9780883857502
Category: Mathematics
Page: 195
View: 9356

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"Professor H.S. Wall wrote Creative Mathematics with the intention of leading students to develop their mathematical abilities, helping them learn the art of mathematics, and teaching them to create mathematical ideas. Creative Mathematics, according to Wall, "is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over paintings. It is, instead, a sketchbook in which readers try their hands at mathematical discovery." The book is self-contained and assumes little formal mathematical background on the part of the reader. Wall is earnest about developing mathematical creativity and independence in students. Wall developed Creative Mathematics over a period of many years of working with students at the University of Texas at Austin. In fewer than 200 pages, he takes the reader on a stimulating tour starting with numbers and then moving on to simple graphs, the integral, simple surfaces, successive approximations, linear spaces of simple graphs, and concluding with mechanical systems. The student who has worked through Creative Mathematics will come away with heightened mathematical maturity."--P. [4] of cover.

Introduction to Mathematical Logic, Fourth Edition


Author: Elliott Mendelson
Publisher: CRC Press
ISBN: 9780412808302
Category: Mathematics
Page: 440
View: 2738

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The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them. Introduction to Mathematical Logic includes: propositional logic first-order logic first-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarski axiomatic set theory theory of computability The study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.