Mathematical Logic

  1. First-Order Logic
    1. Limitations of Propositional Logic
      1. Expressive Limitations
        1. Need for Quantification and Relations
          1. Examples Requiring First-Order Logic
          2. Syntax of First-Order Logic
            1. Language Components
              1. Variables
                1. Constant Symbols
                  1. Function Symbols
                    1. Predicate Symbols
                      1. Logical Symbols
                      2. Terms and Formulas
                        1. Construction of Terms
                          1. Atomic Formulas
                            1. Formation Rules for Formulas
                            2. Quantifiers
                              1. Universal Quantifier
                                1. Existential Quantifier
                                  1. Nested Quantifiers
                                    1. Quantifier Scope
                                    2. Free and Bound Variables
                                      1. Identifying Free and Bound Variables
                                        1. Substitution Rules
                                          1. Variable Capture
                                            1. Alpha-Conversion
                                            2. Sentences
                                              1. Definition and Importance
                                                1. Open vs. Closed Formulas
                                              2. Semantics of First-Order Logic
                                                1. Structures and Interpretations
                                                  1. Domain of Discourse
                                                    1. Interpretation of Constant Symbols
                                                      1. Interpretation of Function Symbols
                                                        1. Interpretation of Predicate Symbols
                                                        2. Satisfaction and Truth in a Model
                                                          1. Assignment Functions
                                                            1. Satisfaction Relation
                                                              1. Tarski's Definition of Truth
                                                              2. Models of a Set of Sentences
                                                                1. Definition of a Model
                                                                  1. Finite vs. Infinite Models
                                                                  2. Logical Validity and Consequence
                                                                    1. Validity in First-Order Logic
                                                                      1. Semantic Consequence
                                                                        1. Logical Equivalence
                                                                      2. Proof Systems for First-Order Logic
                                                                        1. Extending Axiomatic Systems
                                                                          1. Additional Axioms for Quantifiers
                                                                            1. Universal Instantiation
                                                                              1. Existential Generalization
                                                                                1. Rules for Quantifier Manipulation
                                                                                2. Extending Natural Deduction
                                                                                  1. Rules for Universal Quantifier
                                                                                    1. Rules for Existential Quantifier
                                                                                      1. Proof Strategies
                                                                                        1. Working with Assumptions
                                                                                        2. Extending Sequent Calculus
                                                                                          1. Quantifier Rules in Sequents
                                                                                            1. Eigenvariable Conditions
                                                                                              1. Cut-Elimination in First-Order Logic
                                                                                            2. Metatheory of First-Order Logic
                                                                                              1. Gödel's Completeness Theorem
                                                                                                1. Statement and Significance
                                                                                                  1. Outline of Proof
                                                                                                    1. Henkin's Proof Method
                                                                                                    2. Compactness Theorem for First-Order Logic
                                                                                                      1. Statement and Applications
                                                                                                        1. Proof via Completeness
                                                                                                        2. Löwenheim-Skolem Theorems
                                                                                                          1. Downward Löwenheim-Skolem
                                                                                                            1. Upward Löwenheim-Skolem
                                                                                                              1. Implications for Model Theory
                                                                                                                1. Skolem Paradox
                                                                                                                2. Undecidability of First-Order Logic
                                                                                                                  1. Church's Theorem
                                                                                                                    1. Statement and Consequences
                                                                                                                      1. Reduction from Halting Problem

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                                                                                                                  2. Propositional Logic

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                                                                                                                  4. Model Theory