Set Theory

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 $n$-huge set theory.

Given a set of Real Numbers, there are 14 versions of set theory which can be obtained using only closure and complement (Beeler et al. 1972, Item 105).

See also Axiomatic Set Theory, Consistency Strength, Continuum Hypothesis, Descriptive Set Theory, Impredicative, Naive Set Theory, Peano Arithmetic, Set, Zermelo-Fraenkel Set Theory

References

Set Theory

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 36-44, Feb. 1972.

Brown, K. S. ``Set Theory and Foundations.'' http://www.seanet.com/~ksbrown/ifoundat.htm.

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.

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.

Stewart, I. The Problems of Mathematics, 2nd ed. Oxford: Oxford University Press, p. 96, 1987.