A guide to classical and modern model theory
As the title suggests, this is a guide rather than an introduction to model theory, directed primarily at non-specialists (graduate students, non-model theorists, or model theorists working in a different domain) who want to get an idea about what model theory is, and what its principal lines of research are. It thus presents a reasonably streamlined exposition from the basic definitions (structures, formulas) and theorems (compactness) to recent and current results (Shelah's classification theory, \(\omega\)-stability and Morley rank, the relation with algebraic geometry (including a sketch of Hrushovski's proof of the Mordell-Lang conjecture), and o-minimality, emphasising quantifier elimination, model completeness and elimination of imaginaries on the way. (This basically lists the chapters of the book.) So although some classical ideas and results (e.g.\ on model companions) are mentioned insofar as they have become fashionable again, the stress is definitely on the \textit{modern} rather than on the \textit{classical}. There, however, the book does fulfill its purpose: To give an overview to the non-expert of the principal ideas in contemporary model theory. The book is easy to read, even if the occasional leftover Italian word and slightly rough translation (amusing as they are) indicate a certain lack of proof-reading by the editor. Quite a number of statements are given without proof, which is understandable given the scope and the aims of the book. Some quibbles: The appeal to compactness in the proof of Lemma 4.3.3 seems incorrect, as we are working in a not necessarily saturated structure. I would prefer a remark that orthogonality as defined in 7.5.13 is \textit{over the given model}, and possibly remark already there that is is invariant under non-forking extension. In Theorem 7.5.20 two types are RK-equivalent iff they are \textit{non}orthogonal. The notation GM\((p)\) for Morley degree is non-standard, at least in English. And finally, the groups of finite \textit{type} of page 305 become groups of finite \textit{rank} on page 306. On the whole, the book does what its title promises, and the name of the series suggests -- and it does it quite well.
scientific article; zbMATH DE number 1970863