Set theory

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Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, often as Venn diagrams, 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 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.

[edit] Objections to set theory

Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Errett Bishop dismissed set theory as "God's mathematics, which we should leave for God to do." Also Ludwig Wittgenstein questioned especially the handling of infinities, which concerns also ZF. Wittgenstein's views about foundations of mathematics have been criticised by Paul Bernays, and closely investigated by Crispin Wright, among others.

The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.

Topos theory has been proposed as an alternative to traditional axiomatic set theory. Topos theory can be used to interpret various alternatives to set theory such as constructivism, fuzzy set theory, finite set theory, and computable set theory.

[edit] Cultural references

• Set theory appears in some academic games.
• On-Sets is a cube game involving set theory. The game uses cards with dots on them to represent the "universe".

[edit] See also

• The article on Sets gives a basic introduction to elementary set theory.

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• List of set theory topics
• 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.
• Rough set theory provides a means of representing crisp sets by using lower and upper approximations
• 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.
• Von Neumann–Bernays–Gödel set theory is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemata.
• 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 1987).

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