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by Lev Beklemishev, Moscow The ?eld of mathematical logic-evolving around the notions of logical validity, provability, and computation-was created in the ?rst half of the previous century by a cohort of brilliant mathematicians and philosophers such as Frege, Hilbert, Godel, Turing, Tarski, Malcev, Gentzen, and some others. The development of this discipline is arguably among the highest achievements of science in the twentieth century: it expanded mat- matics into a novel area of applications, subjected logical reasoning and computability to rigorous analysis, and eventually led to the creation of computers. The textbook by Professor Wolfgang Rautenberg is a well-written - troduction to this beautiful and coherent subject. It contains classical material such as logical calculi, beginnings of model theory, and Godel's incompleteness theorems, as well as some topics motivated by appli- tions, such as a chapter on logic programming. The author has taken great care to make the exposition readable and concise; each section is accompanied by a good selection of exercises. A special word of praise is due for the author's presentation of Godel's second incompleteness theorem, in which the author has succeeded in giving an accurate and simple proof of the derivability conditions and the provable ? -completeness, a technically di?cult point that is usually 1 omittedintextbooksofcomparablelevel. Thisworkcanberecommended to all students who want to learn the foundations of mathematical logic.

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From the reviews of the third edition: Wolfgang Rautenberg's A Concise Introduction to Mathematical Logic is a pretty ambitious undertaking, seeing that at the indicated introductory level it covers 'classical material ! and Godel's incompleteness theorems, as well as some topics motivated by applications, such as chapter on logic programming' (from the Foreword by Lev Beklemishev). ! The third edition ! is a fine piece of scholarship and will more than repay the efforts of the committed student who chooses this means as an entry into modern mathematical logic. (Michael Berg, The Mathematical Association of America, June, 2010)

From the reviews of the third edition: Wolfgang Rautenberg's A Concise Introduction to Mathematical Logic is a pretty ambitious undertaking, seeing that at the indicated introductory level it covers 'classical material ... and Godel's incompleteness theorems, as well as some topics motivated by applications, such as chapter on logic programming' (from the Foreword by Lev Beklemishev). ... The third edition ... is a fine piece of scholarship and will more than repay the efforts of the committed student who chooses this means as an entry into modern mathematical logic. (Michael Berg, The Mathematical Association of America, June, 2010) This is essentially the English translation of the third edition of the German version [Einfuhrung in die mathematische Logik. Ein Lehrbuch. Wiesbaden: Vieweg+Teubner (2008; Zb1 1152.03-002)] of this well-written textbook ... . The book remains one of the most recommendable introductions into mathematical logic for mathematicians, and well-suited for computer scientists too. (Siegfried J. Gottwald, Zentralblatt MATH, Vol. 1185, 2010)

From the reviews of the third edition: Wolfgang Rautenberg's A Concise Introduction to Mathematical Logic is a pretty ambitious undertaking, seeing that at the indicated introductory level it covers `classical material ... and Godel's incompleteness theorems, as well as some topics motivated by applications, such as chapter on logic programming' (from the Foreword by Lev Beklemishev). ... The third edition ... is a fine piece of scholarship and will more than repay the efforts of the committed student who chooses this means as an entry into modern mathematical logic. (Michael Berg, The Mathematical Association of America, June, 2010) This is essentially the English translation of the third edition of the German version [Einfuhrung in die mathematische Logik. Ein Lehrbuch. Wiesbaden: Vieweg+Teubner (2008; Zb1 1152.03-002)] of this well-written textbook ... . The book remains one of the most recommendable introductions into mathematical logic for mathematicians, and well-suited for computer scientists too. (Siegfried J. Gottwald, Zentralblatt MATH, Vol. 1185, 2010)

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