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Mathematics
Mathematical Logic
1. Introduction to Mathematical Logic
2. Propositional Logic
3. First-Order Logic
4. Model Theory
5. Computability Theory
6. Incompleteness Theorems
7. Axiomatic Set Theory
8. Proof Theory
9. Modal Logic
10. Intuitionistic Logic
11. Higher-Order Logic
8.
Proof Theory
8.1.
Structural Proof Theory
8.1.1.
Analysis of Sequent Calculus
8.1.1.1.
Structure and Rules
8.1.1.2.
Invertibility of Rules
8.1.1.3.
Admissibility
8.1.2.
Gentzen's Cut-Elimination Theorem
8.1.2.1.
Statement and Proof Outline
8.1.2.2.
Consequences for Proof Structure
8.1.2.3.
Normalization
8.1.3.
Subformula Property
8.1.3.1.
Definition and Applications
8.1.3.2.
Analytic Proofs
8.1.4.
Applications of Cut-Elimination
8.1.4.1.
Consistency Proofs
8.1.4.2.
Interpolation Theorems
8.1.4.3.
Decidability Results
8.2.
Ordinal Analysis
8.2.1.
Assigning Ordinals to Proofs
8.2.1.1.
Ordinal Notations
8.2.1.2.
Proof-Theoretic Ordinals
8.2.1.3.
Ordinal Bounds
8.2.2.
Gentzen's Consistency Proof for Peano Arithmetic
8.2.2.1.
Outline of the Proof
8.2.2.2.
Transfinite Induction
8.2.2.3.
Significance for Hilbert's Program
8.3.
Proof Complexity
8.3.1.
Measuring Proof Length
8.3.2.
Lower Bounds
8.3.3.
Propositional Proof Systems
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7. Axiomatic Set Theory
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9. Modal Logic