A Course in Mathematical Logic for Mathematicians
Authors: Manin, Yu. I.
 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 DavisPutnamRobinsonMatiyasevich theorem
 Discusses Kolmogorov complexity
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 About this Textbook

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 numbertheoretic 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 MaxPlanckInstitute 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.
 Reviews

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 LöwenheimSkolem theorem. The emphasis is on semantics rather than syntax. Some rarelycovered 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 KochenSpecker "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 workingmathematician 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 this book 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)
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Bibliographic Information
 Bibliographic Information

 Book Title
 A Course in Mathematical Logic for Mathematicians
 Authors

 Yu. I. Manin
 Translated by
 Koblitz, N.
 Series Title
 Graduate Texts in Mathematics
 Series Volume
 53
 Copyright
 2010
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9781441906151
 DOI
 10.1007/9781441906151
 Hardcover ISBN
 9781441906144
 Softcover ISBN
 9781461424796
 Series ISSN
 00725285
 Edition Number
 2
 Number of Pages
 XVIII, 384
 Number of Illustrations and Tables
 12 b/w illustrations
 Topics