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Cantor's diagonal argument | A Wisdom Archive on Cantor's diagonal argument | | Cantor's diagonal argument A selection of articles related to Cantor's diagonal argument | |
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Cantor's diagonal argument
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ARTICLES RELATED TO Cantor's diagonal argument | |
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| | |  Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theoryThe important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory |
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| | | | Cantor's diagonal argument: Encyclopedia - Cardinal number In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). See names of numbers in English.
In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. While for finite sets the size is given by a natural number, the number of elements, cardinal numbers (cardinality ...
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Read more here: » Cardinal number: Encyclopedia - Cardinal number |
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| | | | Cantor's diagonal argument: Encyclopedia - Cardinality In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers.
Cardinality - Comparing sets.
We say that two sets A and B have the same cardinality if there exists a bijection, i.e. a injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positi ...
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Read more here: » Cardinality: Encyclopedia - Cardinality |
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| | | | Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - 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. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory |
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| | | | Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness:
a set, x0, is well founded iff it has no infinite descending membership sequence:
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In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets |
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| | | | Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics From these initial axioms for sets one can construct all other mathematical concepts and objects: number - discrete and continuous, order, relation, function , etc.
For example, whilst the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair ( a, b ) which represents the pairing of two objects in this order. The defining property of an ordered pair is that ( a, b ) = ( c, d ) if and only if a = c and b = d. The approach is basically to specify th ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Set theory ZFC foundations for mathematics |
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| | | | Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - Axioms for set theory The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.
An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory |
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| | | | Cantor's diagonal argument: Encyclopedia II - Axiomatic set theory - Independence in ZFC Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Independence in ZFC |
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| | | | Cantor's diagonal argument: Encyclopedia - Mathematics Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.
Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...
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Read more here: » Mathematics: Encyclopedia - Mathematics |
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| | | | Cantor's diagonal argument: Encyclopedia II - Crank person - Topics typically associated with the crank label
Crank person - Physics computer science and mathematics.
Claims to have produced solutions to problems which have been proven to be unsolvable, such as the geometric construction problems of squaring the circle, doubling the cube and trisecting the angle. (It should be noted that all of these problems have solutions if one is permitted tools beyond a straightedge and compass).
producing unified Theories of Everything, and particularly doing so with high school or undergraduate level physics knowled ...
See also:Crank person, Crank person - Crank tactics, Crank person - Cranks on the Internet, Crank person - Related terminology, Crank person - Topics typically associated with the crank label, Crank person - Physics computer science and mathematics, Crank person - Medicine, Crank person - Politics economics and law, Crank person - Paranormal and spiritual Read more here: » Crank person: Encyclopedia II - Crank person - Topics typically associated with the crank label |
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