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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 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)