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.
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.
Features
- 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.
Author(s): Jose Iovino
Publisher: CRC Press/Chapman & Hall
Year: 2023
Language: English
Pages: 326
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Foreword by H. Jerome Keisler
Preface
Contributors
I. Real-Valued Structures and Applications
1. Metastable convergence and logical compactness
1.1. Preliminaries
I. Connectives and classical quantifiers
II. Metric structures and logics for metric structures
III. Examples of logics for metric structures
IV. Relativizations
V. [k, λ]-compactness and (k, λ)-compactness
1.2. [k, k]-compactness and cofinality
1.3. Metastability and uniform metastability
I. Metastability: Basic definitions and examples
II. Uniform metastability from a topological viewpoint
1.4. The Main Theorem: Uniform metastability and logical compactness
1.5. Compactness and RPCΔ-characterizability of general structures
Bibliography
2. Model theory for real-valued structures
2.1. Introduction
2.2. Basic model theory for general structures
I. General structures
II. Ultraproducts
III. Definability and types
IV. Saturated and special structures
V. Pre-metric structures
VI. Some variants of continuous model theory
2.3. Turning general structures into metric structures
I. Definitional expansions
II. Pre-metric expansions
III. The Expansion Theorem
IV. Absoluteness
2.4. Properties of general structures
I. Types in pre-metric expansions
II. Definable predicates
III. Topological and uniform properties
IV. Infinitary continuous logic
V. Many-sorted metric structures
VI. Bounded and unbounded metric structures
VII. Imaginaries
VIII. Definable sets
IX. Stable theories
X. Building stable theories
XI. Simple and rosy theories
2.5. Conclusion
Bibliography
3. Spectral gap and definability
3.1. Introduction
I. A crash course in continuous logic
3.2. Definability in continuous logic
I. Generalities on formulae
II. Definability relative to a theory
III. Definability in a structure
3.3. Spectral gap for unitary group representations
I. Generalities on unitary group representations
II. Introducing spectral gap
III. Spectral gap and definability
IV. Spectral gap and ergodic theory
3.4. Basic von Neumann algebra theory
I. Preliminaries
II. Tracial von Neumann algebras as metric structures
III. Property Gamma and the McDuff property
3.5. Spectral gap subalgebras
I. Introducing spectral gap for subalgebras
II. Spectral gap and definability
III. Relative bicommutants and e.c. II1 factors
3.6. Continuum many theories of II1 factors
I. The history and the main theorem
II. The base case
III. A digression on good unitaries and generalized McDuff factors
IV. The inductive step
Bibliography
II. Abstract Elementary Classes and Applications
4. Lf groups, aec amalgamation, few automorphisms
4.0. Introduction
I. Review
II. Amalgamation spectrum
III. Preliminaries on groups
4.1. Amalgamation bases
4.2. Definability
4.3. Density of being complete in Klfλ
Bibliography
III. Model Theory and Topology of Function Spaces
5. Cp-Theory for model theorists
5.1. Introduction
5.2. Preliminaries in Cp-theory
I. Some basic results
II. Lindelöf Σ-spaces are Grothendieck
III. Grothendieck spaces and double limit conditions
5.3. Stability, definability, and double (ultra)limit conditions
5.4. Applications and examples concerning the undefinability of pathological Banach spaces
5.5. The NIP and the Bourgain-Fremlin-Talagrand dichotomy
5.6. Appendix: Proof that all Lindelöf Σ-spaces are Grothendieck
5.7. Acknowledgement
Bibliography
IV. Constructing Many Models
6. General non-structure theory
6.0. Introduction
I. Preliminaries
6.1. Models from indiscernibles
I. Background
II. GEM models
III. Finding templates
IV. How forcing helps
6.2. Models represented in free algebras and applications
I. Representation, non-embeddability and bigness
II. Example: Unsuperstability
III. Example: Separable reduced Abelian p-groups
IV. An example: rigid Boolean algebras
V. Closure sums
VI. Back to linear orders
6.3. Order implies many nonisomorphic models
I. Skeleton like sequence and invariants
II. Representing invariants
III. Harder results
IV. Using Infinitary sequences
Bibliography
V. Model Theory of Second Order Logic
7. Model theory of second order logic
7.1. Introduction
7.2. Second order characterizable structures
7.3. Weakly second order characterizable structures
7.4. Non second order characterizable structures
7.5. Categoricity of second order theories
7.6. What is left out?
Bibliography
