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