Overview

Set Theory is the foundation of modern mathematics. It studies sets—collections of objects—and their properties. Almost all mathematical concepts (numbers, functions, spaces) can be defined in terms of sets.

Core Idea

The core idea is that collection is a primitive concept. By defining rules for how collections can be formed and manipulated, we can build the entire universe of mathematics.

Formal Definition

In Zermelo-Fraenkel Set Theory (ZFC), everything is a set. The theory is defined by a list of axioms (Extensionality, Pairing, Union, Power Set, Infinity, etc.) that describe how sets behave.

Intuition

Imagine a bag. You can put things in the bag. You can have a bag of apples, or a bag containing other bags. Set theory is the rigorous study of these “bags” (sets) and what happens when you combine them, count them, or compare them.

Examples

  • Empty Set ($\emptyset$): The set with no elements.
  • Subset ($\subseteq$): Set A is a subset of B if every element of A is also in B.
  • Union ($\cup$): The set of elements in A or B.
  • Intersection ($\cap$): The set of elements in A and B.

Common Misconceptions

  • Misconception: A set is just a list.
    • Correction: Order doesn’t matter in a set, and duplicates are ignored. ${1, 2, 3}$ is the same set as ${3, 1, 2, 2}$.
  • Misconception: You can form a set of “everything.”
    • Correction: Russell’s Paradox showed that naive set theory leads to contradictions (Does the set of all sets that don’t contain themselves contain itself?). Modern set theory (ZFC) restricts set formation to avoid this.
  • Russell’s Paradox: A contradiction discovered by Bertrand Russell in naive set theory.
  • Cardinality: A measure of the “number of elements” of the set.
  • Infinity: Set theory distinguishes between different sizes of infinity (countable vs. uncountable).

Applications

  • Foundations of Mathematics: Provides the common language for all math.
  • Computer Science: Relational databases are based on set theory. Data structures like Sets and Maps are direct implementations.
  • Logic: Used to define models for logical systems.

Criticism and Limitations

  • Abstractness: It treats infinite collections as completed objects, which some constructivist mathematicians (like Brouwer) reject.
  • Independence Results: Some statements (like the Continuum Hypothesis) are independent of ZFC—they can neither be proved nor disproved.

Further Reading

  • Naive Set Theory by Paul Halmos
  • Set Theory: An Introduction to Independence Proofs by Kenneth Kunen