Item description for Introduction to Model Theory (Algebra, Logic and Applications Volume 15) by Philipp Rothmaler...
Model theory investigates mathematical structures by means of formal languages. These so-called first-order languages have proved particularly useful. The text introduces the reader to the model theory of first-order logic, avoiding syntactical issues that are not too relevant to model-theory. In this spirit, the compactness theorem is proved via the algebraically useful ultraproduct 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, also 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 sets. The final chapter is on the models of the first-order theory of the integers as an abelian group. This material appears here for the first time in a textbook of introductory level, and is used to give hints
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Studio: CRC Press
Est. Packaging Dimensions: Length: 8.9" Width: 6" Height: 0.9" Weight: 1.05 lbs.
Release Date Oct 31, 2000
ISBN 9056993135 ISBN13 9789056993139
Reviews - What do customers think about Introduction to Model Theory (Algebra, Logic and Applications Volume 15)?
great introduction to model theory Dec 5, 2007
I just finished a year long graduate course with the author and I highly recommend this book to anyone learning the basics of mathematical logic or model theory. The book is organized very well and contains LOTS of references to the original literature and historical points as well as LOTS of exercises throughout. We used the first half of the book in a first-year graduate introduction to logic and the second half in a second-year introduction to model theory. So, a knowledge of mathematical logic (though helpful) is not assumed.
The author only assumes a basic knowledge of advanced algebra (groups etc.). The book starts by introducing basic concepts such as signature, structures, satisfaction, truth etc. My favorite thing about the book is it's treatment of the compactness theorem. The compactness theorem is proven using ultraproducts. Something missing from most books (not this one) is a discussion of why the theorem is called the "compactness" theorem; the author discusses, stone space, which is the topological space asserted to be compact by the theorem. Other more advanced topics include the Lowenhiem-Skolem theorems, the omitting types theorem, and the structure theorem for strongly minimal theories.
The book would be suitable for a first course in model theory, or as an introduction to mathematical logic. I highly recommend this book.