Bertrand Russell's Paradox and Set Theory Lecture

Introduction

• Bertrand Russell discovered a paradox in 1901 affecting mathematics and science.
• The paradox involves set theory, a foundational branch of mathematics.
• Aim of lecture: Teach set theory in 8 minutes and demonstrate the paradox.

Understanding Numbers and Sets

• Number 4:
  • Not tangible; has properties like divisibility by 2, square root of 16.
• Set Theory:
  • Mathematics branch studying collections of objects (sets).
  • Invented by Georg Cantor in the 1870s.
  • Examples:
    • Set of three markers.
    • Set of all people watching a video.

Logicism and Mathematical Truth

• Immanuel Kant:
  • Mathematics is a human construct; subjective truths.
• Gottlob Frege and Bertrand Russell:
  • Believed mathematics must be objective.
  • Developed logicism: mathematics as a branch of logic and set theory.

Basics of Set Theory

• Naive Set Theory:
  • Uses ordinary language to define sets.
  • Formal Set Theory: Uses logical language.
• Set: A collection of objects.
  • Examples: LeBron James and the top half of the Eiffel Tower.
  • Objects in a set need not be related.
• Notation:
  • Squiggly brackets for set (e.g., {LeBron, 4}).
  • Set builder notation for large sets (e.g., the set of all x such that x is a cat).

Key Rules of Set Theory

1. Unrestricted Composition:
   • Any conceivable set can be formed.
2. Set Identity and Membership:
   • Defined by members of the set.
3. Order of Elements:
   • Does not matter in a set.
4. Repetition:
   • Does not change set identity.
5. Description:
   • Labeling items doesn’t change set identity.
6. Union of Sets:
   • Combination of two sets is a set.
7. Subsets:
   • A set containing some items of another set is also a set.
8. Singleton Set:
   • Set with one member.
9. Empty Set:
   • Set with no members.
10. Sets of Sets:
    • Sets can contain other sets.
11. Self-Contemplation:
    • Sets can contain themselves (leads to paradox).

Russell's Paradox

• Consider sets that do or do not contain themselves.
• Contradiction: The set of all sets that do not contain themselves.
  • If it contains itself, it doesn’t.
  • If it doesn’t contain itself, it does.
• Conclusion: Set theory rules lead to a paradox.

Responses to the Paradox

• Mathematicians tried to change the rules (e.g., no sets containing themselves).
• But this approach may not fully resolve the issue.

Linguistics and Predication

• Paradox exists in language too:
  • Subject (e.g., Garfield) and predicate (e.g., is a cat).
  • Rule 1 for Predication: Any characteristic can be a predicate.

Regenerating the Paradox in Language

• Predicates true of themselves:
  • Example: