Set Theory

The foundation of mathematics, formalizing collections of objects (sets), their elements, and membership relations.

• Founder: Georg Cantor
• Axiomatization: ZFC (Zermelo-Fraenkel with Choice)
• core concepts:: set, element, subset, union, intersection, complement, power set
• cardinality measures the size of a set, distinguishing countable from uncountable infinities
• The diagonal argument proves the reals are uncountable, establishing a hierarchy of infinities
• Provides the language for logic, topology, algebra, category theory, and probability
• Russell’s paradox motivated axiomatic approaches replacing naive set theory
• Related: number theory, combinatorics, graph theory, information theory