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**Mathematical logic and model theory. A brief introduction.**
*(English)*
Zbl 1241.03001

Universitext. Berlin: Springer (ISBN 978-1-4471-2175-6/pbk; 978-1-4471-2176-3/ebook). x, 193 p. (2011).

The first edition of this book, published in 1986, was in German [A. Prestel, Einführung in die mathematische Logik und Modelltheorie. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn (1986; Zbl 0616.03001)], and intended to give “a thorough and self-contained presentation of the model-theoretic aspects” of algebraic theories. This new edition has the same intention and “is essentially just a translation to address” “a larger audience”. (The authors state: “German is sometimes considered to be a difficult language.” Ganz richtig!) There are two major changes: the expansion of exercises and the addition of a second appendix. This review is addressed to these, leaving that of the main text, which is a pleasure to read, to J. Flum’s review in Zbl 0616.03001.

Exercises for Chapter 1, ‘First-order logic’, are routine: formalize notions, check if so and so are tautologies, deduce…. Those for Chapters 2 and 3, ‘Model constructions’ and ‘Properties of model classes’, include some important topics which are not covered in the main text. They include the back-and-forth argument, Beth’s theorem about definability, and model companions. Exercises of the last chapter, ‘Model theory of several algebraic theories’, are about \(p\)-adically closed fields. The new appendix is titled ‘Remarks on second-order logic’, and divided into three parts. First, set variables are interpreted naively, and all important first-order properties are lost. Then many-sorted formulation is introduced which is appropriate for valued fields, because valuation rings and value groups are associated. Finally, set variables are regarded as a new sort. Then, their range is not the full power-set.

The authors explicitly state that subsequent developments are not included: they are delegated to “the many good books”. And yet, it’s a pity that one cannot see the authors’ succinct and lucid overview of recent work. Another appendix, perhaps?

Exercises for Chapter 1, ‘First-order logic’, are routine: formalize notions, check if so and so are tautologies, deduce…. Those for Chapters 2 and 3, ‘Model constructions’ and ‘Properties of model classes’, include some important topics which are not covered in the main text. They include the back-and-forth argument, Beth’s theorem about definability, and model companions. Exercises of the last chapter, ‘Model theory of several algebraic theories’, are about \(p\)-adically closed fields. The new appendix is titled ‘Remarks on second-order logic’, and divided into three parts. First, set variables are interpreted naively, and all important first-order properties are lost. Then many-sorted formulation is introduced which is appropriate for valued fields, because valuation rings and value groups are associated. Finally, set variables are regarded as a new sort. Then, their range is not the full power-set.

The authors explicitly state that subsequent developments are not included: they are delegated to “the many good books”. And yet, it’s a pity that one cannot see the authors’ succinct and lucid overview of recent work. Another appendix, perhaps?

Reviewer: M. Yasuhara (Princeton)

### MSC:

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03C07 | Basic properties of first-order languages and structures |

03C52 | Properties of classes of models |

03C60 | Model-theoretic algebra |

03C98 | Applications of model theory |

11U09 | Model theory (number-theoretic aspects) |