Set theory is the mathematical theory of sets. Set theory is closely associated with the branch of mathematics known as logic.
There are a number of different versions of set theory, each with its own rules and axioms. In order of increasing consistency
strength, several versions of set theory include Peano
arithmetic (ordinary algebra), second-order arithmetic
(analysis), Zermelo-Fraenkel
set theory, Mahlo, weakly compact, hyper-Mahlo, ineffable, measurable, Ramsey,
supercompact, huge, and 
-huge
set theory.
See also
Abstract Algebra, Analysis, Axiomatic Set Theory, Consistency Strength, Continuum Hypothesis, Descriptive Set Theory, Impredicative, Kuratowski's Closure-Complement Problem, Naive Set Theory, Peano Arithmetic, Sentence, Set, Theory, Zermelo-Fraenkel Axioms, Zermelo-Fraenkel Set Theory, Zermelo Set Theory
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References
Courant, R. and Robbins, H. "The Algebra of Sets." Supplement to Ch. 2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 108-116, 1996.Devlin, K. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed. New York: Springer-Verlag, 1993.Ferreirós, J. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, 1999.Halmos, P. R. Naive Set Theory. New York: Springer-Verlag, 1974.MacTutor History of Mathematics Archive. "The Beginnings of Set Theory." http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html.MathPages. "Set Theory and Foundations." http://www.mathpages.com/home/ifoundat.htm.Stewart, I. The Problems of Mathematics, 2nd ed. Oxford: Oxford University Press, p. 96, 1987.Weisstein, E. W. "Books about Set Theory." http://www.ericweisstein.com/encyclopedias/books/SetTheory.html.
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Cite this as:
Weisstein, Eric W. "Set Theory." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SetTheory.html