Model theory
Chang, C. C.
- 3rd ed.
- New York Dover Publications 2012
- xvi, 650 p.
Table of Contents:
Chapter - I
1 Introduction
1.1.What is model theory?
1.2.Model theory for sentential logic
1.3.Languages, models and satisfaction
1.4.Theories and examples of theories
1.5.Elimination of quantifiers
Chapter - II
2 Models constructed from constants
2.1.Completeness and compactness
2.2.Refinements of the method. Omitting types and interpolation theorems
2.3.Countable models of complete theories
2.4.Recursively saturated models
2.5.Lindstrom's characterization of first order logic
Chapter - III
3 Further model-theoretic constructions
3.1.Elementary extensions and elementary chains
3.2.Applications of elementary chains
3.3.Skolem functions and indiscernibles
3.4.Some examples
3.5.Model completeness
Chapter - IV
4 Ultraproducts
4.1.The fundamental theorem
4.2.Measurable cardinals
4.3.Regular ultrapowers
4.4.Nonstandard universes
Chapter - V
5 Saturated and special models
5.1.Saturated and special models Contents note continued:
5.2.Preservation theorems
5.3.Applications of special models to the theory of definability
5.4.Applications to field theory
5.5.Application to Boolean algebras
Chapter - VI
6 More about ultraproducts and generalizations
6.1.Ultraproducts which are saturated
6.2.Direct products, reduced products, and Horn sentences
6.3.Direct products, reduced products, and Horn sentences (continued)
6.4.Limit ultrapowers and complete extensions
6.5.Iterated ultrapowers
Chapter - VII
7 Selected topics
7.1.Categoricity in power
7.2.An extension of Ramsey's theorem and applications; some two-cardinal theorems
7.3.Models of large cardinality
7.4.Large cardinals and the constructible universe.
Model theory deals with a branch of mathematical logic showing connections between a formal language and its interpretations or models. This is the first and most successful textbook in logical model theory. Extensively updated and corrected in 1990 to accommodate developments in model theoretic methods — including classification theory and nonstandard analysis — the third edition added entirely new sections, exercises, and references. Each chapter introduces an individual method and discusses specific applications. Basic methods of constructing models include constants, elementary chains, Skolem functions, indiscernibles, ultraproducts, and special models. The final chapters present more advanced topics that feature a combination of several methods. This classic treatment covers most aspects of first-order model theory and many of its applications to algebra and set theory.