set theory

Table of Contents

Introduction
Introduction to naive set theory
- Fundamental set concepts
  - Operations on sets
  - Relations in set theory
- Essential features of Cantorian set theory
  - Equivalent sets
  - Cardinality and transfinite numbers
Axiomatic set theory
- The Zermelo-Fraenkel axioms
  - Schemas for generating well-formed formulas
  - Axioms for compounding sets
  - Axioms for infinite and ordered sets
  - Schema for transfinite induction and ordinal arithmetic
  - Axiom for eliminating infinite descending species
- The Neumann-Bernays-Gödel axioms
- Limitations of axiomatic set theory
- Present status of axiomatic set theory

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axiom of restriction axiom of separation axiom schema of replacement principle of abstraction axiomatic set theory generalized continuum hypothesis measure theory union axiom of elementary sets axiom of extensionality analysis probability theory cryptology computer science combinatorics algebra statistics geometry automata theory information theory

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Key People

Georg Cantor

Georg Cantor

German mathematician

Paul Erdős

Paul Erdős

Hungarian mathematician

John von Neumann

John von Neumann

American mathematician

Saul Kripke

Saul Kripke

American logician and philosopher

Stanislaw Ulam

American scientist

Paul Joseph Cohen

American mathematician

Equations written on blackboard

Numbers and Mathematics