Mathematical logic and model theory. A brief introduction
The first edition of this book, published in 1986, was in German [\textit{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\dots. 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?
scientific article; zbMATH DE number 5943674