Model Theory: An Introduction (Graduate Texts in Mathematics, Vol. 217)

This is a graduate level text--you will need mathematical maturity as well as a decent background in both logic and abstract algebra (the deeper your background the more you can gain). When I first purchased this book I had a difficult time appreciating the subtleties of the model theoretic approach to logic. Having had some time to ponder them, I have developed a deep appreciation of its power. Model theory is to predicate logic what analysis is to engineering calculus, it is enlightening, it is logic for grown-ups. Marker's presentation is terse, for the most part he gives his definitions and theorems with very little comment. This is unfortunate because the essence of these definitions and theorems can usually be explained intuitively with just a sentence or two of plain English, much to the benefit of the learner. Also, there are a fair amount of typos, some of them damaging. For these two reasons, this book is not friendly to the beginner, and I myself did not like it at all when I first purchased it. With that said, I have since grown very fond of this text. Marker knows his subject well and this is reflected in the logical development. The theorems, their applications, and the many examples he gives are actually quite interesting, once you are with the program. I suspect that someone who has already had some model theory will find this book especially enjoyable. I also think this text can be put to very profitable use in the classroom--there is a great deal of power lying dormant here that can be unlocked by a professor with a good intuitive grasp of the subject. Briefly, Marker's text is difficult for the beginner but well worth the reward if you perservere. Remove the typos and this is a five star book in my opinion. 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. Part A: Structures And Theories. Basics. Algebraic Examples -- Part B: Realizing And Omitting Types. Indiscernibles -- Part C: Categoricity. Omega-stable Groups. Geometry Of Strongly Minimal Sets -- Appendices. David Marker. Includes Bibliographical References (p. [329]-335) And Index. "This book is a modern introduction to model theory that 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."--BOOK JACKET. Introduction....Pages 1-6 Structures and Theories....Pages 7-32 Basic Techniques....Pages 33-69 Algebraic Examples....Pages 71-113 Realizing and Omitting Types....Pages 115-173 Indiscernibles....Pages 175-205 ω-Stable Theories....Pages 207-249 ω-Stable Groups....Pages 251-288 Geometry of Strongly Minimal Sets....Pages 289-313 Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures