Beyond First Order Model Theory, Volume I and II
Model theory is the meta-mathematical study of the concept of mathematical truth. After Afred Tarski coined the term Theory of Models in the early 1950’s, it rapidly became one of the central most active branches of mathematical logic. In the last few decades, ideas that originated within model theory have provided powerful tools to solve problems in a variety of areas of classical mathematics, including algebra, combinatorics, geometry, number theory, and Banach space theory and operator theory.
The two volumes of Beyond First Order Model Theory present the reader with a fairly comprehensive vista, rich in width and depth, of some of the most active areas of contemporary research in model theory beyond the realm of the classical first-order viewpoint. Each chapter is intended to serve both as an introduction to a current direction in model theory and as a presentation of results that are not available elsewhere. All the articles are written so that they can be studied independently of one another.
The first volume is an introduction to current trends in model theory, and contains a collection of articles authored by top researchers in the field. It is intended as a reference for students as well as senior researchers.
This second volume contains introductions to real-valued logic and applications, abstract elementary classes and applications, interconnections between model theory and function spaces, nonstucture theory, and model theory of second-order logic.
- A coherent introduction to current trends in model theory.
- Contains articles by some of the most influential logicians of the last hundred years. No other publication brings these distinguished authors together.
- Suitable as a reference for advanced undergraduate, postgraduates, and researchers.
- Material presented in the book (e.g, abstract elementary classes, first-order logics with dependent sorts, and applications of infinitary logics in set theory) is not easily accessible in the current literature.
- The various chapters in the book can be studied independently.
I. Model Theory of Strong Logics. 1. Expressive Power of Infinitary [0, 1]-logics. 2. Scott Processes. 3. Failure of 0-1 Law for Sparse Random Graph in Strong Logics. II. Model Theory of Special Classes of Structures. 4. Maximality of Continuous Logic. 5. Model Theory and Metric Convergence I: Metastability and dominated convergence. 6. Randomizations of Scattered Sentences. 7 Existentially Closed Locally Finite Groups. 8. Analytic Zariski Structures and Non-Elementary Categoricity. III. Abstract Elementary Classes. 9. Hanf Numbers and Presentation Theorems in AECs. 10. A Survey on Tame Abstract Elementary Classes.
Table of Contents of Volume II
I. Real-Valued Structures and Applications. 1. Metastable Convergence and Logical Compactness. 2. Model Theory for Real-Valued Structures. 3. Spectral Gap and Definability. II. Abstract Elementary Classes and Applications. 4. Lf groups, AEC amalgamation, few automorphisms. III. Model Theory and Topology of Spaces of Functions. 5. Cp-Theory for Model Theorists. IV. Constructing Many Models. 6. General Non-Structure Theory. V. Model Theory of Second Order Logic. 7. Model Theory of Second Order Logic.