A course in model theory. An introduction to contemporary mathematical logic. Transl. from the French by Moses Klein
It is with great satisfaction that I welcome the English translation of Poizat's \textit{Cours de théorie des modèles} (1985; Zbl 0583.03001) -- after all, this is the book where I learned not only my model and stability theory, but also my mathematical French. Incidentally, the French version is still available from the author, considerably cheaper than \textit{chez Springer}, and would be my first choice --- unless you think that mathematics is already difficult enough on its own. Except for Section 6.6 on real closed fields, some additional notes, a LATeX-assembled index with page numbers (much more convenient than the original), and the preface to the English edition, the content is unaltered from the French version (the Post Scriptum of the second edition has been incorporated into the main text, though, slightly changing the theorem numbering). Even the inimitable literary style of the author shines through, although at places the translation is inaccurate (e.g. \textit{la lumière \(\ldots\) eut été plus aveuglante} becomes \textit{the light \(\ldots\) was blinding}), or bows to a political correctness rather uncharacteristic for the author (the hitherto male reader suddenly and unaccountably becomes female on various occasions). Maybe the text is even a bit too unchanged: few mathematical books discuss the problems relating to \textit{dope} (usually called the DOP), and an extension of a type is not standardly referred to as its \textit{son}. However, on the whole I found the Gallicisms entertaining rather than distracting. As far as the mathematical content is concerned, the above mentioned review of the French version is still valid. The first half of the book is a very good, if slightly nonstandard, introduction to general model theory, with a heavy emphasis on semantic aspects and a healthy neglect of syntactics. The second half deals with stability theory, taking the Parisian approach to the subject via the fundamental order; as such it is a nice and highly readable account of the basic theory. However, development in the subject has been very rapid in the last 15 years; obviously none of the more recent developments is reflected in this book. In particular, after the groundbreaking work of Kim and Pillay on simple theories [\textit{B. Kim}, J. Lond. Math. Soc., II. Ser. 57, No. 2, 257-267 (1998; Zbl 0922.03048); \textit{B. Kim} and \textit{A. Pillay}, Bull. Symb. Log. 4, No. 1, 17-36 (1998; Zbl 0897.03034), Ann. Pure Appl. Logic 88, No. 2-3, 149-164 (1997; Zbl 0897.03036)], not only were stability-theoretic methods extended to a wider class of theories, but the basic emphasis shifted away from the fundamental order and back to Shelah's original approach using indiscernible sequences, which turned out to be more fundamental after all. I should therefore recommend reading the second half of the book under review in conjunction with a more modern introduction to simplicity theory.
algebraically and differentially closed fields
scientific article; zbMATH DE number 1479599