Introduction to Model Theory

Model Theory investigates mathematical structures by means of formal languages. This is an introductory text for graduates and undergraduates who wish to learn the basics. Model Theory is attracting more and more interest in the field of mathematics in view of the growing applications to classical parts of mathematics. This book is suitable for readers who have not studied mathematical logic previously and unlike most introductory texts this book avoids syntactical issues which are not too relevant to model theory. Selected exercises have hints and solutions and some reflect very recent research in the field. It also contains a detailed bibliography, an epilogue and hints for further reading including some of the most recent literature of active research. The final chapter contains, for the very first time in an introductory level textboook, two previously unpublished applictions of Model Theory.

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect.This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory.Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.

Basics, Structures, Languages, Semantics. Beginnings of Model Theory, The Finiteness Theorem, First Consequences of Finiteness Theorem, Malcev's Applications to Group Theory, Some Theory of Ordering. Basic Properties of Theories, Elementary Maps, Elimination, Chains. Theories and Types, Types, Thick and Thin Models, Countable Complete Theories. Two Applications, Strong Minimal Theories, Hints to Selected Exercise, Solutions for Selected Exercises.