Mathematical Logic

1. Model Theory
   1. Structures and Interpretations
      1. Definitions
         1. Models
         2. Signatures
         3. Interpretations in structures
      2. Types of Structures
         1. Algebraic Structures
            1. Groups
            2. Rings
            3. Fields
         2. Order Structures
            1. Linear Orders
            2. Partial Orders
         3. Graphs and Combinatorial Structures
      3. Homomorphisms
         1. Types of Homomorphisms
            1. Isomorphisms
            2. Endomorphisms
            3. Automorphisms
         2. Homomorphism Properties
   2. Elementary Classes
      1. Definition and Properties
         1. Characterization of Elementary Classes
         2. Examples of Elementary Classes
      2. Definability
         1. First-order Definable Classes
         2. Elementary Equivalence
   3. Ultraproducts and Ultraproperties
      1. Constructing Ultraproducts
         1. Unary Ultra Filters
         2. Application of the Construction
      2. Łoś's Theorem
         1. Statement and Proof Ideas
         2. Applications and Examples
      3. Ultraproperties
         1. Influence on Model Theory
         2. Compactness and Completeness
   4. Compactness Theorem
      1. Statement of the Theorem
      2. Proof Techniques
         1. Finite Satisfiability
         2. Applications in Model Theory
      3. Consequences and Corollaries
         1. Consistency and Inconsistency
         2. Applications in Various Logical Systems
   5. Löwenheim-Skolem Theorems
      1. Löwenheim-Skolem Downward Theorem
         1. Statement and Proof Sketch
         2. Implications for Countable Models
      2. Löwenheim-Skolem Upward Theorem
         1. Statement and Proof Sketch
         2. Existence of Uncountable Models
      3. Impact on Set Theory
         1. Skolem Paradox
         2. Connection with Axiomatic Languages
   6. Omitting Types Theorem
      1. Formal Statement
         1. Conditions for the Theorem
         2. Type Omittability
      2. Proof Outline
         1. Construction methods
         2. Applications in Elementary Extensions
      3. Relationship with Consistency
         1. Type Spaces
         2. Non-standard Models
   7. Types and Type Spaces
      1. Definition of Types in Logic
         1. Type Spaces over Structures
         2. Complete Types
      2. Stone Spaces
         1. Topological Representation
         2. Applications in Model Theory
   8. Stability Theory
      1. Stability and Instability
         1. Classification of Theories
         2. Examples of Stable and Unstable Theories
      2. Morley's Categoricity Theorem
         1. Introduction and Concepts
         2. Proof Sketch and Applications
      3. Classification Theory
         1. Categoricity
         2. Stability Spectrum Functions
   9. Applications of Model Theory
      1. Algebra
         1. Algebraically Closed Fields
         2. Differential Algebra
      2. Geometry
         1. Non-Euclidean Geometry
         2. O-minimal Structures
      3. Computer Science
         1. Logical Databases
         2. Constraint Solving