Information about 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, 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.
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.
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.
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.
See also
- The article on Sets gives a basic introduction to elementary set theory.
- 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).
Major fields of mathematics |
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Logic
Set theory
Algebra (Abstract algebra – Linear algebra)
Discrete mathematics
Number theory
Analysis
Geometry
Topology
Applied mathematics
Probability
Statistics
Mathematical physics
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SET may stand for:
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- Sanlih Entertainment Television, a television channel in Taiwan
- Secure electronic transaction, a protocol used for credit card processing,
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abstract or concrete. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing. concretum).
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primary school is an institution where children receive the first stage of compulsory education known as primary or elementary education. Primary school is the preferred term in the United Kingdom and many Commonwealth Nations, and in most publications of the United Nations
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Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show all of the possible mathematical or logical relationships between sets (groups of things).
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Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified.
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Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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naive set theory[1] is one. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of
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In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. The basis of set theory was created principally by the German mathematician Georg Cantor at the end of the 19th century.
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axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.
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A spatial point is a concept used to define an exact location in space. It has no volume, area or length, making it a zero dimensional object. Points are used in the basic language of geometry, physics, vector graphics (both 2D and 3D), and many other fields.
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line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
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Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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Some statements may be disputed, incorrect, , biased or otherwise objectionable. Please read the discussion on the before making substantial changes.
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Errett Albert Bishop (1928–1983) was an American mathematician known for his work on analysis. He is the father of constructivist analysis, by virtue of his 1967 Foundations of Constructive Analysis
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Ludwig Josef Johann Wittgenstein (IPA: ['luːtvɪç 'joːzɛf 'joːhan 'vɪtgənʃtaɪn]
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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. He was an assistant and close collaborator of David Hilbert.
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Crispin Wright (born 1942) is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity.
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constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its
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naive set theory[1] is one. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of
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topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory.
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constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its
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Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent
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Turing machines are extremely basic abstract symbol-manipulating devices which, despite their simplicity, can be adapted to simulate the logic of any computer that could possibly be constructed. They were described in 1936 by Alan Turing.
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SET may stand for:
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- Sanlih Entertainment Television, a television channel in Taiwan
- Secure electronic transaction, a protocol used for credit card processing,
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Reasoning: Deduction Induction Abduction Informal Logic: Proposition Inference Argument Validity Cogency Term logic Critical Thinking Fallacies Syllogism Mathematical logic:
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naive set theory[1] is one. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of
..... Click the link for more information.
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In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. The basis of set theory was created principally by the German mathematician Georg Cantor at the end of the 19th century.
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Part of the foundation of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.
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