Introduction to Model Theory

Model theory is used to investigate mathematical structures by means of formal languages, and first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic. Avoiding syntactical issues, author proves the compactness theorem via the ultraproduct technique, which leads fairly quickly to algebraic applications like Malcev's local theorems and eventually to Hilbert's Nullstellensatz. The author obtains Steinitz dimension theory for field extensions as a special case of a more general model-theoretic treatment of strongly minimal theories. The final chapter explores the models of the first-order theory of the integers as an abelian group.