The purpose of the book is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with significant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research.
Beyond the core techniques and results of model theory, this book places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the field of real numbers (real closed fields) and show how this can be used to obtain the characterization of positive semi-definite rational functions that gives a solution to Hilbert's 17th Problem.
This open book is licensed under a Creative Commons License (CC BY). You can download Model Theory ebook for free in PDF format (0.7 MB).
Table of Contents
Ultraproducts and the Compactness Theorem
Theories and Types
Algebraically Closed Fields
Model Theoretic Algebraic Closure
Algebraic Closure in Minimal Structures
Real Closed Ordered Fields
Morley rank and ω-stability
Morley's uncountable categoricity theorem