Mathematical Logic

1. Proof Theory
   1. Axiomatic Systems
      1. Hilbert-style Systems
         1. Basic Structure
            1. Axioms
            2. Inference Rules
         2. Characteristics
            1. Formalization
            2. Minimalistic Approach
         3. Advantages and Limitations
            1. Flexibility in Formalization
            2. Dense and Compact Proof Structures
      2. Natural Deduction
         1. Structure and Syntax
            1. Introduction and Elimination Rules
            2. Assumptions and Discharge
         2. Intuition and Motivation
            1. Reflects Natural Reasoning
            2. Simplifies the Understanding of Logical Inferences
         3. Applications
            1. Education in Logic
            2. Computational Implementations
         4. Proof Strategies
            1. Direct Proof
            2. Proof by Contradiction
      3. Sequent Calculus
         1. Fundamentals
            1. Sequents as Judgements
            2. Gentzen's Contributions
         2. Structural Rules
            1. Identity
            2. Weakening, Contraction, and Cut
            3. Exchange
         3. Logical Rules
            1. Introduction and Elimination for Logical Connectives
            2. Role in Normalization and Cut-elimination
         4. Importance in Consistency Proofs
            1. Consistency of Arithmetic
            2. Cut-elimination Theorem
   2. Theorems
      1. Completeness Theorems
         1. Historical Context and Importance
            1. Gödel's Completeness Theorem
            2. Impact on Mathematical Logic
         2. Proof Techniques
            1. Henkin's Construction
            2. Saturation Method
         3. Significance in Meta-logical Investigations
      2. Incompleteness Theorems
         1. Gödel's First Incompleteness Theorem
            1. Statement and Proof Sketch
            2. Implications for Formal Systems
         2. Gödel's Second Incompleteness Theorem
            1. Expansion Beyond First Theorem
            2. Consistency and Formal Theories
         3. Broader Implications
            1. Limits of Formalization in Mathematics
   3. Proof Transformations
      1. Normalization
         1. Systematic Reduction
            1. Conversion of Derivations
            2. Role in Simplifying Proofs
         2. Applications
            1. Lambda Calculus
            2. Optimization in Logic Programming
         3. Normal Forms
            1. Process of Achieving Normal Form
            2. Advantages of Normal Forms in Proof Analysis
      2. Cut-elimination
         1. Concept and Importance
            1. Reducing Intermediate Assertions
            2. Simplification of Proofs
         2. Techniques and Algorithms
            1. Gentzen's Cut-elimination Procedure
            2. Complexity Considerations
         3. Impact on Logical Consistency
            1. Proof Theoretical Consistency
            2. Strengthening Completeness Results
   4. Connections and Impacts
      1. Interaction with Other Areas of Logic
         1. Link to Model Theory
            1. Transferability of Results
         2. Set Theory
            1. Proofs and Constructive Elements
      2. Influence on Computer Science
         1. Proof Checking and Automated Theorem Proving
         2. Type Theories and Proof Assistants