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Encyclopedia > Set theory

Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, 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 mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... As used in philosophy, in general, an object is something that can have properties and relations. ... Primary or elementary education is the first years of formal, structured education that occurs during childhood. ... Logic, from Classical Greek λόγος logos (meaning word, account, reason or principle), is the study of the principles and criteria of valid inference and demonstration. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...


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 abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...


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. This article or section is in need of attention from an expert on the subject. ... This article does not adequately cite its references or sources. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...


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 e.g. by Paul Bernays, and closely investigated by Crispin Wright, among others. In this article, Cantors Theory refers to the pre-formal ideas about set theory introduced by Georg Cantor in the latter part of the nineteenth century. ... Errett Albert Bishop (1928-1983) was an American mathematician known for is work on analysis. ... Wittgenstein and Hitler in school photograph taken at the Linz Realschule in 1903. ... Paul Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who played a crucial role in the development of mathematical logic in the 20th century. ... Crispin Wright (born 1942) is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgensteins later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity. ...


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. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...


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. For discussion of topoi in literary theory, see literary topos. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... An artistic representation of a Turing Machine . ...


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".

Academic Games is a U.S. competition in which players win by out-thinking each other in mathematics, language arts, and social studies. ...

See also

Wikibooks
Wikibooks has a book on the topic of
Topology/Set Theory
Wikibooks
Wikibooks has a book on the topic of
Discrete mathematics/Set theory

  Results from FactBites:
 
Set Theory (Stanford Encyclopedia of Philosophy) (0 words)
The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.
There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe.
Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
PlanetMath: set theory (937 words)
Set theory is special among mathematical theories, in two ways: It plays a central role in putting mathematics on a reliable axiomatic foundation, and it provides the basic language and apparatus in which most of mathematics is expressed.
A category is not a set, and a functor is not a mapping, despite similarities in both cases.
This is version 8 of set theory, born on 2003-01-01, modified 2003-02-07.
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