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05177922 Kaye, Richard The mathematics of logic. A guide to completeness theorems and their applications. Cambridge: Cambridge University Press (ISBN 978-0-521-88219-4/hbk; 978-0-521-70877-7/pbk). xii, 204~p. \sterling~19.99, \$~39.99/pbk; \sterling~55.00, \$~99.00/hbk (2007). 2007 Cambridge: Cambridge University Press EN first-order logic completeness theorem Boolean logic compactness theorem nonstandard analysis lattice theory textbook This textbook presents several basic topics for a course in formal logic for undergraduates in mathematics or computer science or first-year graduate students in philosophy or engineering. Special attention is given to the usual completeness theorem of first-order logic. The author proceeds from K\"onig's Lemma (graph theory), Zorn's Lemma, elementary Boolean algebra (but no minimization theory for the economic design of logical systems, say, for computers) to the completeness and compactness of first-order logic. Since no second-order logic is considered, there is no discussion of the incompleteness of axiomatic arithmetic; that is, no G\"odel's Incompleteness Theorem. On the other hand, recall that G\"odel gave a (non-effective) proof of the completeness of first-order logic. A novel and useful feature of the book is discussions of elementary model theory and nonstandard analysis. A limitation of the book is the lack of presentations on Turing machines for an understanding of the notion of computation and recursively enumerable sets. Surprisingly, there is no specific discussion of mathematical induction, (or its connection to ``well ordering''), which is a basic mathematical tool for proving, say, the deduction theorem of logic. This reviewer looks forward to future editions of the book, which might include new topics on elements of (1) modal logics, (2) many-valued logics and (3) tense/temporal logics. Albert A. Mullin (Madison)