Set theory

Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, of collections of objects, and the elements of, and membership in, such collections. In most modern mathematical formalisms, set theory provides the language in which mathematical objects are described. It is (along with logic and the predicate calculus) one of the axiomatic foundations for mathematics, allowing mathematical objects to be constructed formally from the undefined terms of "set", and "set membership". It is in its own right a branch of mathematics and an active field of ongoing mathematical research.

In naive set theory, sets are introduced and understood using what is taken to be the self-evident concept of sets as collections of objects considered as a whole.

In axiomatic set theory, the concepts of sets and set membership are defined indirectly by first postulating certain axioms which specify their properties. In this conception, sets and set membership are fundamental concepts like point and line in Euclidean geometry, and are not themselves directly defined.

See also

• List of set theory topics
• Set gives a basic introduction to elementary set theory.
• Naive set theory is the original set theory developed by mathematicians at the end of the 19th century.
• Axiomatic set theory is a rigorous axiomatic branch of mathematics developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
• Zermelo set theory is an axiomatic system developed by the German mathematician Ernst Zermelo.
• Zermelo-Fraenkel set theory is the most commonly used system of set-theoretic axioms, based on Zermelo set theory and further developed by Abraham Fraenkel and Thoralf Skolem.
• New Foundations and positive set theory are among the alternative set theories which have been proposed.
• Internal set theory is an extension of axiomatic set theory that admits infinitesimal and illimited non-standard numbers.
• Various versions of logic have associated sorts of sets (such as fuzzy sets in fuzzy logic).
• Musical set theory concerns the application of combinatorics and group theory to music; beyond the fact that it uses finite sets it has nothing to do with mathematical set theory of any kind. In the last two decades, transformational theory in music has taken the concepts of mathematical set theory more rigorously (see Lewin 1989).