Marco Benini

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 and intended interpretation
  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. Coding Peano arithmetic into itself
  6. Fixed point lemma
  7. Gödel’s first incompleteness theorem
  8. Gödel’s second incompleteness theorem
  9. Mathematical meaning of incompleteness
  10. Natural incompleteness
  11. Incompleteness in set theory