Mathematical Logic

  1. Model Theory
    1. Fundamental Concepts
      1. Theories and their Models
        1. Definition of a Theory
          1. Consistency and Satisfiability
            1. Complete Theories
            2. Elementary Equivalence vs. Isomorphism
              1. Definition of Isomorphism
                1. Definition of Elementary Equivalence
                  1. Examples and Distinctions
                    1. Ehrenfeucht-Fraïssé Games
                    2. Substructures and Elementary Substructures
                      1. Definition of Substructure
                        1. Elementary Substructure Criteria
                          1. Tarski-Vaught Test
                          2. Diagrams of Models
                            1. Elementary Diagram
                              1. Atomic Diagram
                            2. Key Techniques and Theorems
                              1. The Ultraproduct Construction
                                1. Ultrafilters
                                  1. Definition and Construction
                                    1. Properties of Ultraproducts
                                    2. Łoś's Theorem
                                      1. Statement and Proof Outline
                                        1. Consequences
                                        2. Types and Saturated Models
                                          1. Definition of Types
                                            1. Complete Types
                                              1. Realization of Types
                                                1. Saturated Models
                                                  1. Omitting Types Theorem
                                                  2. Quantifier Elimination
                                                    1. Definition and Examples
                                                      1. Criteria for Quantifier Elimination
                                                    2. Categoricity
                                                      1. Definition of Categoricity
                                                        1. ω-Categoricity
                                                          1. Definition and Examples
                                                            1. Ryll-Nardzewski Theorem
                                                            2. Categoricity in Uncountable Cardinals
                                                              1. Morley's Categoricity Theorem
                                                                1. Statement and Significance
                                                              2. Applications
                                                                1. Non-standard Analysis
                                                                  1. Construction of Non-standard Models
                                                                    1. Hyperreal Numbers
                                                                      1. Applications in Analysis
                                                                      2. Model Theoretic Algebra
                                                                        1. Applications to Algebraic Structures
                                                                          1. Definable Sets
                                                                          2. Stability Theory
                                                                            1. Stable Theories
                                                                              1. Forking and Independence

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                                                                          5. Computability Theory