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A Course in Mathematical Logic for Mathematicians de Yu I. Manin

A Course in Mathematical Logic for Mathematicians: Graduate Texts in Mathematics, cartea 53

Autor Yu I. Manin Traducere de Neal Koblitz
en Limba Engleză Paperback – 3 mar 2012

Recomandăm A Course in Mathematical Logic for Mathematicians ca o resursă fundamentală pentru programele de masterat și doctorat în matematică pură, fiind concepută special pentru a răspunde nevoilor de rigoare ale cercetătorilor. Considerăm că relevanța acestui curs pentru examenele de grad și pentru fundamentarea cercetării în bazele matematicii este dublată de perspectiva unică a lui Yu I. Manin, un matematician care abordează logica nu ca pe o ramură izolată, ci ca pe un instrument viu, integrat în algebra modernă și geometria algebrică.

Această a doua ediție aduce actualizări esențiale într-un domeniu care a evoluat spectaculos din 1977. În timp ce primele opt capitole păstrează nucleul solid despre limbaje formale, deductibilitate și teoremele de incompletitudine ale lui Gödel, noutatea absolută constă în Partea a IV-a, dedicată Teoriei Modelelor. Această secțiune, semnată de Boris I. Zilber, demonstrează cum perspectivele din logica formală pot soluționa probleme clasice de matematică convențională. Structura este riguros organizată pe patru direcții: Demonstrabilitate, Calculabilitate, sinteza acestora și Teoria Modelelor, culminând cu discuții despre problemele P/NP și aspectele cuantice ale computației.

În contextul operei sale, A Course in Mathematical Logic for Mathematicians reflectă aceeași rigoare structurală pe care o regăsim în Homological Algebra, unde Yu I. Manin explorează categoriile derivate. Spre deosebire de manualele introductive, această lucrare face tranziția de la limbajul set-teoretic spre intuiția categoriilor superioare. Considerăm volumul o alternativă sofisticată la A Course on Mathematical Logic de Shashi Mohan Srivastava pentru cursurile de fundamentele matematicii, având avantajul unei profunzimi teoretice superioare și al unei conexiuni directe cu cercetarea actuală din model theory și calculabilitate cuantică.

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Specificații

ISBN-13: 9781461424796
ISBN-10: 1461424798
Pagini: 404
Ilustrații: XVIII, 384 p. 12 illus.
Dimensiuni: 155 x 235 x 22 mm
Greutate: 0.61 kg
Ediția:2nd edition 2010
Editura: Springer
Colecția Graduate Texts in Mathematics
Seria Graduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Graduate

De ce să citești această carte

Recomandăm acest volum studenților avansați și cercetătorilor care doresc o înțelegere a logicii matematice din perspectiva unui practician al algebrei. Cititorul câștigă acces la o sinteză modernă ce include teoria modelelor și computație cuantică, elemente adesea omise în textele standard. Este motivul ideal pentru a aprofunda legătura dintre structurile formale și matematica de vârf, trecând dincolo de simpla manipulare a simbolurilor logice.

Despre autor

Yu. I. Manin este un matematician de renume mondial, cunoscut pentru contribuțiile sale fundamentale în geometria algebrică, algebra omologică și fizica matematică. Profesor la Institutul Max Planck pentru Matematică, opera sa include lucrări de referință precum Homological Algebra și Introduction to Modern Number Theory. Abordarea sa în domeniul logicii matematice este marcată de o viziune unificatoare, căutând mereu punți de legătură între fundamentele teoretice și aplicațiile lor în structuri matematice complexe, fiind unul dintre primii care au propus ideea computației cuantice.

Descriere scurtă

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.

Cuprins

PROVABILITY.- to Formal Languages.- Truth and Deducibility.- The Continuum Problem and Forcing.- The Continuum Problem and Constructible Sets.- COMPUTABILITY.- Recursive Functions and Church#x2019;s Thesis.- Diophantine Sets and Algorithmic Undecidability.- PROVABILITY AND COMPUTABILITY.- G#x00F6;del#x2019;s Incompleteness Theorem.- Recursive Groups.- Constructive Universe and Computation.- MODEL THEORY.- Model Theory.

Recenzii

From the reviews of the second edition:
"As one might expect from a graduate text on logic by a very distinguished algebraic geometer, this book assumes no previous acquaintance with logic, but proceeds at a high level of mathematical sophistication. Chapters I and II form a short course. Chapter I is a very informal introduction to formal languages, e.g., those of first order Peano arithmetic and of ZFC set theory. Chapter II contains Tarski's definition of truth, Gödel's completeness theorem, and the Löwenheim-Skolem theorem. The emphasis is on semantics rather than syntax. Some rarely-covered side topics are included (unique readability for languages with parentheses, Mostowski's transitive collapse lemma, formalities of introducing definable constants and function symbols). Some standard topics are neglected. (The compactness theorem is not mentioned!) The latter part of Chapter II contains Smullyan's quick proof of Tarski's theorem on the undefinability of truth in formal arithmetic, and an account of the Kochen-Specker "no hidden variables" theorem in quantum logic. There are digressions on philosophical issues (formal logic vs. ordinary language, computer proofs). A wealth of material is introduced in these first 100 pages of the book..."--MATHEMATICAL REVIEWS
“Manin’s book on mathematical logic is addressed to a working-mathematician with some knowledge of naive set theory … . incorporate some of the exciting developments in mathematical logic of the last four decades into this edition. … The exquisite taste and the elegant style of the author have produced an outstanding treatment of mathematical logic that allows one to understand some of the pillars of this area of mathematical research … and Manin’s original treatment of the subject provides an extraordinary introduction to mathematical logic.” (F. Luef, Internationale Mathematische Nachrichten, Issue 217, August, 2011)
“The new extended title of thisbook corresponds more to its concept, contents, spirit and style. The book is really addressed to mathematicians and introduces the reader to the glorious discoveries in logic during the last century through the difficult and subtle results, problems, proofs and comments. … due to the author’s brilliant style, each part of the book provokes new opinions and pleasure of a different understanding of basic results and ideas of contemporary mathematical logic.” (Branislav Boričić, Zentralblatt MATH, Vol. 1180, 2010)

Textul de pe ultima copertă

A Course in Mathematical Logic for Mathematicians, Second Edition offers a straightforward introduction to modern mathematical logic that will appeal to the intuition of working mathematicians. The book begins with an elementary introduction to formal languages and proceeds to a discussion of proof theory. It then presents several highlights of 20th century mathematical logic, including theorems of Gödel and Tarski, and Cohen's theorem on the independence of the continuum hypothesis. A unique feature of the text is a discussion of quantum logic.
The exposition then moves to a discussion of computability theory that is based on the notion of recursive functions and stresses number-theoretic connections. The text present a complete proof of the theorem of Davis–Putnam–Robinson–Matiyasevich as well as a proof of Higman's theorem on recursive groups. Kolmogorov complexity is also treated.
Part III establishes the essential equivalence of proof theory and computation theory and gives applications such as Gödel's theorem on the length of proofs. A new Chapter IX, written by Yuri Manin, treats, among other things, a categorical approach to the theory of computation, quantum computation, and the P/NP problem. A new Chapter X, written by Boris Zilber, contains basic results of model theory and its applications to mainstream mathematics. This theory has found deep applications in algebraic and diophantine geometry.
Yuri Ivanovich Manin is Professor Emeritus at Max-Planck-Institute for Mathematics in Bonn, Germany, Board of Trustees Professor at the Northwestern University, Evanston, IL, USA, and Principal Researcher at the Steklov Institute of Mathematics, Moscow, Russia. Boris Zilber, Professor of Mathematical Logic at the University of Oxford, has contributed the Model Theory Chapter for the second edition.

Caracteristici

Contains a new chapter on categorical approach to theory of computations, quantum computations, and P/NP problem New chapter containing basic results of Model Theory and its applications to mainstream mathematics Presents several highlights of mathematical logic of the 20th century including Gödel's and Tarski's Theorems, Cohen's Theorem on the independence of Continuum Hypothesis Complete proof of Davis-Putnam-Robinson-Matiyasevich theorem Discusses Kolmogorov complexity Includes supplementary material: sn.pub/extras