A Course in Mathematical Logic

Author: Yu.I. Manin
Publisher: Springer Science & Business Media
ISBN: 9781475743852
Release Date: 2013-06-29
Genre: Mathematics

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.

A Course in Mathematical Logic for Mathematicians

Author: Yu. I. Manin
Publisher: Springer Science & Business Media
ISBN: 9781441906151
Release Date: 2009-10-13
Genre: Mathematics

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.

A Course in Mathematical Logic

Author: John Lane Bell
Publisher: Elsevier
ISBN: 9780080934747
Release Date: 1977-01-01
Genre: Logic, Symbolic and mathematical

A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

A First Course in Mathematical Logic and Set Theory

Author: Michael L. O'Leary
Publisher: John Wiley & Sons
ISBN: 9780470905883
Release Date: 2015-09-08
Genre: Mathematics

Rather than teach mathematics and the structure of proofssimultaneously, this book first introduces logic as the foundationof proofs and then demonstrates how logic applies to mathematicaltopics. This method ensures that readers gain a firmunderstanding of how logic interacts with mathematics and empowersthem to solve more complex problems. The study of logic andapplications is used throughout to prepare readers for further workin proof writing. Readers are first introduced tomathematical proof-writing, and then the book provides anoverview of symbolic logic that includes two-column logicproofs. Readers are then transitioned to set theory andinduction, and applications of number theory, relations, functions,groups, and topology are provided to further aid incomprehension. Topical coverage includes propositional logic,predicate logic, set theory, mathematical induction, number theory,relations, functions, group theory, and topology.

First Course in Mathematical Logic

Author: Patrick Suppes
Publisher: Courier Corporation
ISBN: 9780486150949
Release Date: 2012-04-30
Genre: Mathematics

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.

A Course on Mathematical Logic

Author: Shashi Mohan Srivastava
Publisher: Springer Science & Business Media
ISBN: 9781461457466
Release Date: 2013-01-16
Genre: Mathematics

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.

A Course in Mathematical Logic

Author: John Lane Bell
Publisher: North-Holland
ISBN: UOM:39015040409156
Release Date: 1977-01-01
Genre: Computers

A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

Introduction to Mathematical Logic Fourth Edition

Author: Elliott Mendelson
Publisher: CRC Press
ISBN: 0412808307
Release Date: 1997-06-01
Genre: Mathematics

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.

A Course in Model Theory

Author: Bruno Poizat
Publisher: Springer Science & Business Media
ISBN: 9781441986221
Release Date: 2012-12-06
Genre: Mathematics

Translated from the French, this book is an introduction to first-order model theory. Starting from scratch, it quickly reaches the essentials, namely, the back-and-forth method and compactness, which are illustrated with examples taken from algebra. It also introduces logic via the study of the models of arithmetic, and it gives complete but accessible exposition of stability theory.

Course of Mathematical Logic

Author: R. Fraïssé
Publisher: Springer Science & Business Media
ISBN: 9789401020978
Release Date: 2012-12-06
Genre: Philosophy

This book is addressed primarily to researchers specializing in mathemat ical logic. It may also be of interest to students completing a Masters Degree in mathematics and desiring to embark on research in logic, as well as to teachers at universities and high schools, mathematicians in general, or philosophers wishing to gain a more rigorous conception of deductive reasoning. The material stems from lectures read from 1962 to 1968 at the Faculte des Sciences de Paris and since 1969 at the Universities of Provence and Paris-VI. The only prerequisites demanded of the reader are elementary combinatorial theory and set theory. We lay emphasis on the semantic aspect of logic rather than on syntax; in other words, we are concerned with the connection between formulas and the multirelations, or models, which satisfy them. In this context considerable importance attaches to the theory of relations, which yields a novel approach and algebraization of many concepts of logic. The present two-volume edition considerably widens the scope of the original [French] one-volume edition (1967: Relation, Formule logique, Compacite, Completude). The new Volume 1 (1971: Relation et Formule logique) reproduces the old Chapters 1, 2, 3, 4, 5 and 8, redivided as follows: Word, formula (Chapter 1), Connection (Chapter 2), Relation, operator (Chapter 3), Free formula (Chapter 4), Logicalformula,denumer able-model theorem (L6wenheim-Skolem) (Chapter 5), Completeness theorem (G6del-Herbrand) and Interpolation theorem (Craig-Lyndon) (Chapter 6), Interpretability of relations (Chapter 7).

A Beginner s Guide to Mathematical Logic

Author: Raymond M. Smullyan
Publisher: Courier Corporation
ISBN: 9780486492377
Release Date: 2014-07-23
Genre: Mathematics

Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate courses, this book will also amuse and enlighten mathematically minded readers. Dover (2014) original publication. See every Dover book in print at www.doverpublications.com

Logic of Mathematics

Author: Zofia Adamowicz
Publisher: John Wiley & Sons
ISBN: 9781118030790
Release Date: 2011-09-26
Genre: Mathematics

A thorough, accessible, and rigorous presentation of the centraltheorems of mathematical logic . . . ideal for advanced students ofmathematics, computer science, and logic Logic of Mathematics combines a full-scale introductory course inmathematical logic and model theory with a range of speciallyselected, more advanced theorems. Using a strict mathematicalapproach, this is the only book available that contains completeand precise proofs of all of these important theorems: * Gödel's theorems of completeness and incompleteness * The independence of Goodstein's theorem from Peanoarithmetic * 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 oftypes * Clear, concise explanations of all key concepts, from Booleanalgebras to Skolem-Löwenheim constructions and othertopics * Carefully chosen exercises for each chapter, plus helpfulsolution hints At last, here is a refreshingly clear, concise, and mathematicallyrigorous presentation of the basic concepts of mathematicallogic-requiring only a standard familiarity with abstract algebra.Employing a strict mathematical approach that emphasizes relationalstructures over logical language, this carefully organized text isdivided into two parts, which explain the essentials of the subjectin specific and straightforward terms. Part I contains a thorough introduction to mathematical logic andmodel theory-including a full discussion of terms, formulas, andother fundamentals, plus detailed coverage of relational structuresand Boolean algebras, Gödel's completeness theorem, models ofPeano arithmetic, and much more. Part II focuses on a number of advanced theorems that are centralto the field, such as Gödel's first and second theorems ofincompleteness, the independence proof of Goodstein's theorem fromPeano arithmetic, Tarski's theorem on real closed fields, andothers. No other text contains complete and precise proofs of allof these theorems. With a solid and comprehensive program of exercises and selectedsolution hints, Logic of Mathematics is ideal for classroom use-theperfect textbook for advanced students of mathematics, computerscience, and logic.

Mathematical Logic

Author: Ian Chiswell
Publisher: OUP Oxford
ISBN: 0198571003
Release Date: 2007-05-17
Genre: Mathematics

Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in Logic, Mathematics, Philosophy, and Computer Science.