This book is an introductory text in model theory. It is a translation into English of the author’s earlier Spanish original. The translation takes into account the author’s additions to and corrections of the Spanish text.
This book is about the first era of model theory (1919-1960+), starting with Löwenheim and Skolem and reaching its apogee with the Tarski school of the 1950’s and with Abraham Robinson. It has seven chapters. In keeping with its use as a teaching text, problems and exercises are generously scattered throughout, although it is left to the user to distinguish the difference between “problem” and “exercise”. There is an extensive glossary of symbols and abbreviations, an unannotated bibliography, and a concise index. There is a very brief appendix on transfinite induction, cardinals and ordinals. The reader is assumed to have some previous knowledge of naive set theory. Just what background is expected of the reader is never spelled out. The author reports using the book in courses for third year undergraduates and for fourth and fifth year undergraduates.
Here is a list of the chapters along with a brief description of their contents. “Basic notions: universal algebra”: relational structures, substructures, homomorphisms; “First order languages: semantics”: the language \(L(\mathfrak A)\) for the structure \(\mathfrak A\), the concepts of consequence and validity, the theorems of coincidence and substitution, definability in a structure; “Completeness of first order logic”: completeness and soundness relative to a sequent calculus – proved in the style of Henkin-Hasenjaeger; “Basic notions: model theory”: model theory as a combination of algebra and logic, elementary equivalence, elementary embedding, theories of (classes of) structures, models of (sets of) sentences, diagrams; “The compactness theorem and its mathematical applications”: proof of compactness by diagrams, questions of axiomatizability, ultraproducts and Łos’s theorem, a second proof of compactness; “Löwenheim-Skolem theorems and their consequences”: L-S theorems, nonstandard models of Peano arithmetic and of the reals, Skolem’s paradox; “Complete and categorical theories”: Vaught’s test for completeness, quantifier elimination and a test for q.e., model completeness and Robinson’s test.