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Model theory is a branch of mathematical logic that deals with the relationship between [[Formal language|formal languages]] (such as those used in logic and mathematics) and their interpretations, or models. It's a field that straddles the line between mathematics and logic, and has various sub-disciplines and specialized topics. Here's a comprehensive list:

1. Basic Concepts and Techniques¶

First-Order Model Theory
Structures and Languages
Elementary Embeddings and Elementary Equivalence
Quantifier Elimination
Completeness and Compactness Theorems
Ultraproducts and Los's Theorem
Back-and-Forth Method

2. Classification Theory¶

Stability Theory
Simple Theories
Shelah's Classification Program
Morley Rank and O-Minimality

3. Model-Theoretic Algebra¶

Algebraically Closed Fields
Real Closed Fields
Valuation Theory
Differential Fields
Modules and Groups in Model Theory

4. Definability and Interpretability¶

Definable Sets and Definable Functions
Interpretable Structures
Definability Theorems

5. Finite Model Theory¶

Expressive Power of Logics on Finite Structures
Descriptive Complexity Theory
Zero-One Laws

6. Higher-Order and Infinitary Logics¶

Second-Order Model Theory
Infinitary Logic (Lω₁ω)
Higher-Order Logics and Their Models

7. Non-Standard Models and Analysis¶

Non-Standard Analysis
Hyperreal Numbers
Transfer Principle

8. Model Theory of Particular Theories¶

Peano Arithmetic and Non-Standard Models of Arithmetic
Set Theory and Models of Set Theory
Geometric Model Theory

9. Categorical Model Theory¶

Topoi as Models of Higher-Order Logic
Functorial Semantics
Categorical Structures in Model Theory

10. Model Theory and Philosophy¶

Philosophical Implications of Model Theory
Model-Theoretic Realism
Ontological Status of Mathematical Objects

11. Model-Theoretic Semantics¶

Formal Semantics in Linguistics
Kripke Models in Modal Logic
Game-Theoretic Semantics

12. Model Theory in Computer Science¶

Automated Theorem Proving
Database Theory
Formal Verification

13. Constructive and Intuitionistic Model Theory¶

Models of Constructive and Intuitionistic Mathematics
Kripke Models and Sheaf Models

14. O-Minimality and Model Theory of Ordered Structures¶

O-Minimal Structures
Model Theory of Real Analytic and Exponential Fields

15. Universal Model Theory¶

Universal Classes
Homogeneous and Saturated Models

Model theory's diverse applications range from pure mathematics (like algebra and geometry) to practical applications in computer science and linguistics, providing deep insights into the structures and behaviors of mathematical and logical systems.

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