Einführung in die mathematische Logik

Author: Heinz-Dieter Ebbinghaus,Jörg Flum,Wolfgang Thomas

Publisher: Springer Spektrum

ISBN: 9783662580288

Category: Mathematics

Page: 367

View: 6832

Was ist ein mathematischer Beweis? Wie lassen sich Beweise rechtfertigen? Gibt es Grenzen der Beweisbarkeit? Ist die Mathematik widerspruchsfrei? Kann man das Auffinden mathematischer Beweise Computern übertragen? Erst im 20. Jahrhundert ist es der mathematischen Logik gelungen, weitreichende Antworten auf diese Fragen zu geben: Im vorliegenden Werk werden die Ergebnisse systematisch zusammengestellt; im Mittelpunkt steht dabei die Logik erster Stufe. Die Lektüre setzt – außer einer gewissen Vertrautheit mit der mathematischen Denkweise – keine spezifischen Kenntnisse voraus. In der vorliegenden 5. Auflage finden sich erstmals Lösungsskizzen zu den Aufgaben.

Mathematical Logic

Author: H.-D. Ebbinghaus,J. Flum,Wolfgang Thomas

Publisher: Springer Science & Business Media

ISBN: 1475723555

Category: Mathematics

Page: 291

View: 9425

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.

Introduction to Model Theory

Author: Philipp Rothmaler

Publisher: CRC Press

ISBN: 9789056993139

Category: Mathematics

Page: 324

View: 8335

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.

A First Course in Mathematical Logic and Set Theory

Author: Michael L. O'Leary

Publisher: John Wiley & Sons

ISBN: 0470905883

Category: Mathematics

Page: 464

View: 1032

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.

Einführung in die Modelltheorie

Vorlesungen

Author: Philipp Rothmaler

Publisher: Spektrum Akademischer Verlag

ISBN: 9783860254615

Category: Model theory

Page: 331

View: 4387

Formal Languages, Automata and Numeration Systems 1

Introduction to Combinatorics on Words

Author: Michel Rigo

Publisher: John Wiley & Sons

ISBN: 1119008220

Category: Computers

Page: 338

View: 3433

Formal Languages, Automaton and Numeration Systems presents readers with a review of research related to formal language theory, combinatorics on words or numeration systems, such as Words, DLT (Developments in Language Theory), ICALP, MFCS (Mathematical Foundation of Computer Science), Mons Theoretical Computer Science Days, Numeration, CANT (Combinatorics, Automata and Number Theory). Combinatorics on words deals with problems that can be stated in a non-commutative monoid, such as subword complexity of finite or infinite words, construction and properties of infinite words, unavoidable regularities or patterns. When considering some numeration systems, any integer can be represented as a finite word over an alphabet of digits. This simple observation leads to the study of the relationship between the arithmetical properties of the integers and the syntactical properties of the corresponding representations. One of the most profound results in this direction is given by the celebrated theorem by Cobham. Surprisingly, a recent extension of this result to complex numbers led to the famous Four Exponentials Conjecture. This is just one example of the fruitful relationship between formal language theory (including the theory of automata) and number theory.

Mathematical Logic

Foundations for Information Science

Author: Wei Li

Publisher: Springer Science & Business Media

ISBN: 3764399775

Category: Mathematics

Page: 273

View: 5563

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of classical mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines.

A Course on Mathematical Logic

Author: Shashi Mohan Srivastava

Publisher: Springer Science & Business Media

ISBN: 1461457467

Category: Mathematics

Page: 198

View: 4565

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 Concise Introduction to Mathematical Logic

Author: Wolfgang Rautenberg

Publisher: Springer Science & Business Media

ISBN: 0387342419

Category: Mathematics

Page: 256

View: 5254

While there are already several well known textbooks on mathematical logic this book is unique in treating the material in a concise and streamlined fashion. This allows many important topics to be covered in a one semester course. Although the book is intended for use as a graduate text the first three chapters can be understood by undergraduates interested in mathematical logic. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text.

Mathematical logic

Author: Heinz-Dieter Ebbinghaus,Jörg Flum,Wolfgang Thomas

Publisher: Springer

ISBN: 9780387908953

Category: Mathematics

Page: 216

View: 9933

This careful, self-contained introduction to first-order logic includes an exposition of certain topics not usually found in introductory texts (such as Trachtenbrot's undecidability theorem, Fraisse's characterization of elementary equivalence, and Lindstr m's theorem on the maximality of first-order logic). The presentation is detailed and systematic without being long-winded or tedious. The role of first-order logic in the foundations of mathematics is worked out clearly, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. Many exercises accompany the text.

Introduction to Mathematical Logic

Set Theory Computable Functions Model Theory

Author: Jerome Malitz

Publisher: Springer Science & Business Media

ISBN: 1461394414

Category: Mathematics

Page: 198

View: 1224

This book is intended as an undergraduate senior level or beginning graduate level text for mathematical logic. There are virtually no prere quisites, although a familiarity with notions encountered in a beginning course in abstract algebra such as groups, rings, and fields will be useful in providing some motivation for the topics in Part III. An attempt has been made to develop the beginning of each part slowly and then to gradually quicken the pace and the complexity of the material. Each part ends with a brief introduction to selected topics of current interest. The text is divided into three parts: one dealing with set theory, another with computable function theory, and the last with model theory. Part III relies heavily on the notation, concepts and results discussed in Part I and to some extent on Part II. Parts I and II are independent of each other, and each provides enough material for a one semester course. The exercises cover a wide range of difficulty with an emphasis on more routine problems in the earlier sections of each part in order to familiarize the reader with the new notions and methods. The more difficult exercises are accompanied by hints. In some cases significant theorems are devel oped step by step with hints in the problems. Such theorems are not used later in the sequence.

Mathematical Logic

Author: George Tourlakis

Publisher: John Wiley & Sons

ISBN: 1118030699

Category: Mathematics

Page: 294

View: 4450

A comprehensive and user-friendly guide to the use of logic inmathematical reasoning Mathematical Logic presents a comprehensive introductionto formal methods of logic and their use as a reliable tool fordeductive reasoning. With its user-friendly approach, this booksuccessfully equips readers with the key concepts and methods forformulating valid mathematical arguments that can be used touncover truths across diverse areas of study such as mathematics,computer science, and philosophy. The book develops the logical tools for writing proofs byguiding readers through both the established "Hilbert" style ofproof writing, as well as the "equational" style that is emergingin computer science and engineering applications. Chapters havebeen organized into the two topical areas of Boolean logic andpredicate logic. Techniques situated outside formal logic areapplied to illustrate and demonstrate significant facts regardingthe power and limitations of logic, such as: Logic can certify truths and only truths. Logic can certify all absolute truths (completeness theorems ofPost and Gödel). Logic cannot certify all "conditional" truths, such as thosethat are specific to the Peano arithmetic. Therefore, logic hassome serious limitations, as shown through Gödel'sincompleteness theorem. Numerous examples and problem sets are provided throughout thetext, further facilitating readers' understanding of thecapabilities of logic to discover mathematical truths. In addition,an extensive appendix introduces Tarski semantics and proceeds withdetailed proofs of completeness and first incompleteness theorems,while also providing a self-contained introduction to the theory ofcomputability. With its thorough scope of coverage and accessible style,Mathematical Logic is an ideal book for courses inmathematics, computer science, and philosophy at theupper-undergraduate and graduate levels. It is also a valuablereference for researchers and practitioners who wish to learn howto use logic in their everyday work.

Philosophical and Mathematical Logic

Author: Harrie de Swart

Publisher: Springer

ISBN: 3030032558

Category: Philosophy

Page: 539

View: 2666

This book was written to serve as an introduction to logic, with in each chapter – if applicable – special emphasis on the interplay between logic and philosophy, mathematics, language and (theoretical) computer science. The reader will not only be provided with an introduction to classical logic, but to philosophical (modal, epistemic, deontic, temporal) and intuitionistic logic as well. The first chapter is an easy to read non-technical Introduction to the topics in the book. The next chapters are consecutively about Propositional Logic, Sets (finite and infinite), Predicate Logic, Arithmetic and Gödel’s Incompleteness Theorems, Modal Logic, Philosophy of Language, Intuitionism and Intuitionistic Logic, Applications (Prolog; Relational Databases and SQL; Social Choice Theory, in particular Majority Judgment) and finally, Fallacies and Unfair Discussion Methods. Throughout the text, the author provides some impressions of the historical development of logic: Stoic and Aristotelian logic, logic in the Middle Ages and Frege's Begriffsschrift, together with the works of George Boole (1815-1864) and August De Morgan (1806-1871), the origin of modern logic. Since "if ..., then ..." can be considered to be the heart of logic, throughout this book much attention is paid to conditionals: material, strict and relevant implication, entailment, counterfactuals and conversational implicature are treated and many references for further reading are given. Each chapter is concluded with answers to the exercises.

Mathematical Logic

Author: Ian Chiswell,Wilfrid Hodges

Publisher: OUP Oxford

ISBN: 9780198571001

Category: Mathematics

Page: 258

View: 3959

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.

Lectures in Logic and Set Theory: Volume 1, Mathematical Logic

Author: George Tourlakis

Publisher: Cambridge University Press

ISBN: 9781139439428

Category: Mathematics

Page: N.A

View: 2677

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

Fundamentals of Mathematical Logic

Author: Peter G. Hinman

Publisher: A K Peters/CRC Press

ISBN: 9781568812625

Category: Mathematics

Page: 896

View: 9840

This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.

Logic for Mathematicians

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521368650

Category: Mathematics

Page: 228

View: 3220

This is an introductory textbook which is designed to be useful not only to intending logicians but also to mathematicians in general.

Real Mathematical Analysis

Author: Charles C. Pugh

Publisher: Springer Science & Business Media

ISBN: 9780387952970

Category: Mathematics

Page: 440

View: 8288

Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises.

Mathematical Logic

Author: Stephen Cole Kleene

Publisher: Courier Corporation

ISBN: 0486317072

Category: Mathematics

Page: 416

View: 6289

Contents include an elementary but thorough overview of mathematical logic of 1st order; formal number theory; surveys of the work by Church, Turing, and others, including Gödel's completeness theorem, Gentzen's theorem, more.

Logic for Mathematics and Computer Science

Author: Stanley Burris

Publisher: Pearson College Division

ISBN: N.A

Category: Computers

Page: 420

View: 3445

This book provides an elementary "hands-on" presentation of important mathematical logic topics.Explores topics that are at the cutting edge of developments in computer science, while preserving the integrity of traditional logic. Stresses several self-contained proof systems of interest to mathematical logic, some more suitable than others for particular kinds of questions. For anyone interested in Computer Science or Mathematics.