Set theory
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. In the last two decades, transformational theory in music has taken the concepts of mathematical set theory more rigorously.
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