Set theory - Definition and Overview

Set theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed.

For more information on set theory in Wikipedia, see:

• Set gives a basic introduction to elementary set theory.
• Naïve set theory is the original set theory developed by mathematicians at the end of the 19th century.
• Zermelo set theory is the theory developed by the German mathematician Ernst Zermelo.
• Zermelo-Fraenkel set theory is the most commonly used system of axioms, based on Zermelo set theory and further developed by Abraham Fraenkel and Thoralf Skolem.
• Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's Paradox) in naïve set theory.
• 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.