Set theory

Set theory studies collections of objects, called sets, and relations between them. Its axioms provide a foundational language for defining mathematical structures and reasoning about infinity.

Set: A set is a collection of distinct objects, called its elements. Set theory takes these collections as its central objects.
Zermelo–Fraenkel set theory: Zermelo–Fraenkel set theory is an axiomatic system governing sets and their construction. Its axioms form the standard framework for much contemporary foundational mathematics.
Type theory: Type theory is a formal system that organizes mathematical objects into types and governs how they can be used. It offers a foundational alternative to treating all mathematical objects as sets.
Georg Cantor: Georg Cantor was a German mathematician who developed set theory and the modern mathematics of infinity. His work on infinite sets and cardinality founded the subject.
Foundations of mathematics: Foundations of mathematics studies the logical, axiomatic, and philosophical bases of mathematical knowledge. Set theory is a major framework for defining mathematical objects and formalizing proofs.
Membership (mathematics): Membership is the relation stating that an object is an element of a set. The membership relation is the basic relation in the language of set theory.
Axiom of choice: The axiom of choice states that a choice function exists for every family of nonempty sets. It enables powerful existence proofs and has consequences that cannot be derived from the other usual axioms.
New Foundations: New Foundations is an axiomatic set theory that permits unrestricted comprehension for stratified formulas. Its distinct comprehension rule avoids some paradoxes in a way unlike standard Zermelo–Fraenkel axioms.
Richard Dedekind: Richard Dedekind was a German mathematician whose work shaped number theory and the foundations of mathematics. His treatments of infinite sets and number systems helped prepare the ground for set theory.
Natural number: A natural number is a number used for counting or ordering, conventionally beginning with zero or one. In set-theoretic foundations, natural numbers can be constructed from sets.

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