Model Theory : An Introduction
This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry.
David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.
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A great introduction for anyone interested in studying applied model theory. Applications to algebra, in particular model theory of fields, are especially well- presented. The exercises are interesting and plentiful.
The book is perfect preparation for studying more advanced topics in applied model theory. The book also is good preparation for studying stability theory in more general cases (in this book, the totally transcendental or omega stable (the languages here are countable here) cases are carefully presented). I think studying this book leads naturally to studying stability theory, more advanced model theory of fields, and other applied model theory (like say o-minimality). One of the best aspects of the text is the historical section at the end of each chapter. These sections often mention current avenues of research in addition to describing the history of the work in the chapter.
For other model theory topics, like model theory of modules or groups, there may be more suitable texts like Poizat's book (though one should be warned that the language of model theory has evolved, and the modern working language is used in this text, but not in older ones). For those not interested in becoming model theorists, but interested in picking up some interesting model theory and applications to their own branch of mathematics, this is the ideal book. Marker's text is much better suited to this sort of study than, say, Chang and Keisler's text or the more modern book by Hodges.
Additionally, it should be said that Dave Marker's writing style is very nice, keeping the reader excited and organized. Professor Marker won the 2007 Inaugural Shoenfield Prize from the ASL for this textbook.
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A First Course in Logic: An Introduction to Model Theory, Proof ..., Issue 9
No preview available - 2004