David Marker - Böcker

This book offers an introductory course in model theory emphasizing connections to algebra. It will be an appropriate introduction both for graduate students interested in advanced work in model theory and for students and researchers in logic or algebra who want to learn the basic results and themes of model theory. In the end, the reader will have a firm background in model theory and be well motivated and well prepared for more advanced treatments like Pillay's "Geometric Model Theory" or Buechler's "Essential Stability Theory. "While some familiarity at the undergraduate level with mathematical logic would be helpful, it is not assumed. The author assumes familiarity with algebra at the first-year graduate level.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the fifth publication in the Lecture Notes in Logic series, the authors give an insightful introduction to the fascinating subject of the model theory of fields, concentrating on its connections to stability theory. In the first two chapters David Marker gives an overview of the model theory of algebraically closed, real closed and differential fields. In the third chapter Anand Pillay gives a proof that there are 2א non-isomorphic countable differential closed fields. Finally, Margit Messmer gives a survey of the model theory of separably closed fields of characteristic p > 0.

Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in forty years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, model existence techniques, indiscernibles and end extensions are covered before more modern topics are surveyed. Zilber's categoricity theorem for quasiminimal excellent classes is proved and an application is given to covers of multiplicative groups. Infinitary methods are also used to study uncountable models of counterexamples to Vaught's conjecture, and effective aspects of infinitary model theory are reviewed, including an introduction to Montalbán's recent work on spectra of Vaught counterexamples. Self-contained introductions to effective descriptive set theory and hyperarithmetic theory are provided, as is an appendix on admissible model theory.

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

The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of the Mordell-Lang conjecture for function fields. This volume provides an insightful introduction to this active area, concentrating on connections to stability theory.

The material should be accessible to first year graduate students or advanced undergraduates in mathematics, graduate students in philosophy with a solid math background, or students in computer science who want a mathematical introduction to logic.