Mathematical Logic – Video

Part 1: Introduction

1. Bureaucracy
2. A brief history
3. Syntax, semantics, and intended interpretation
4. An infinite variety of logics
5. Foundational issues
6. Soundness and completeness

Part 2: Propositional classical logic

1. Induction
2. Formulae: syntax and intended interpretation
3. Natural deduction
4. Examples of derivation
5. Truth tables
6. Interdependence of connectives
7. Soundness with respect to truth tables
8. Boolean algebras
9. Semantics and Boolean algebras
10. Soundness with respect to Boolean algebras
11. Completeness with respect to Boolean algebras

Part 3: First-order classical logic

1. Signatures, terms, and formulae
2. Substitution
3. Formal definitions
4. Natural deduction
5. Examples of derivations
6. Tarski’s semantics
7. Examples of interpretations
8. Soundness
9. Completeness
10. Compactness
11. Examples about compactness
12. Löwenheim–Skolem theorem
13. Topological interpretation of compactness
14. An alternative notion of completeness

Part 4: Formal set theory

1. Language
2. Paradoxes
3. Comparing sets
4. Examples of set comparison
5. Axioms
6. Ordinals
7. Transfinite induction
8. Ordinal arithmetic
9. More on well orders
10. Cardinals
11. Cardinal arithmetic
12. Hierarchy of cardinals
13. Axiom of choice
14. Continuum hypothesis
15. What is a set?

Part 5: Computability theory

1. Computable functions
2. Primitive recursive functions
3. Examples of primitive recursive functions
4. Partial recursive functions
5. Enumeration
6. Universal function and fixed points
7. Pure λ-calculus
8. Representable functions in the λ-calculus
9. Simple theory of types
10. Strong normalisation of the simple theory of types

Part 6: Intuitionistic logic

1. Motivation
2. Syntax and natural deduction
3. Expressive power
4. Heyting algebras
5. Semantics
6. Soundness
7. Completeness
8. Semantics of first-order intuitionistic logic
9. Propositions as types
10. Variations on the theme
11. Normalisation

Part 7: Limiting results

1. Peano arithmetic
2. Standard and non-standard models
3. A discussion about limiting results
4. Representable entities
5. Representable entities (extended version)
6. Coding Peano arithmetic into itself
7. Fixed point lemma
8. Gödel’s first incompleteness theorem
9. Gödel’s second incompleteness theorem
10. Mathematical meaning of incompleteness
11. Natural incompleteness
12. Incompleteness in set theory