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In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.
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mathematics and Cantor
In mathematics, the continuum hypothesis ( abbreviated CH ) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets.
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – January 6, 1918 ) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics.
The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a " grave disease " infecting the discipline of mathematics, and Kronecker's public opposition and personal attacks included describing Cantor as a " scientific charlatan ", a " renegade " and a " corrupter of youth.
In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.
In 1860, Cantor graduated with distinction from the Realschule in Darmstadt ; his exceptional skills in mathematics, trigonometry in particular, were noted.
Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a " corrupter of youth " for teaching his ideas to a younger generation of mathematicians.
Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly ( while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare ), and this tragedy drained Cantor of much of his passion for mathematics.
While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both " an abomination " and " a cholera bacillus of mathematics ".
* April 10 – Moritz Cantor, German historian of mathematics ( b. 1829 )
Cantor distinguished three realms of infinity: ( 1 ) the infinity of God ( which he called the " absolutum "), ( 2 ) the infinity of reality ( which he called " nature ") and ( 3 ) the transfinite numbers and sets of mathematics.
One of the most vigorous and fruitful branches of mathematics [...] a paradise created by Cantor from which nobody shall ever expel us [...] the most admirable blossom of the mathematical mind and altogether one of the outstanding achievements of man's purely intellectual activity.
Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing.
In the foundations of mathematics, Russell's paradox ( also known as Russell's antinomy ), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction.
The usage of the word abscissa is first recorded in 1659 by Stefano degli Angeli, a mathematics professor in Rome, according to Moritz Cantor.
Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century.
* Moritz Cantor ( 1829 – 1920 ), German historian of mathematics
Many fields of mathematics bear the imprint of their creators for notation: the differential operator is due to Leibniz, the cardinal infinities to Georg Cantor ( in addition to the lemniscate (∞) of John Wallis ), the congruence symbol (≡) to Gauss, and so forth.
Moritz Benedikt Cantor ( 23 August 1829 – 10 April 1920 ) was a German historian of mathematics.
* Florian Cajori, Moritz Cantor, The historian of mathematics, Bull.
In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous.
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set.
mathematics and set
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X
In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.
The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).
This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.
In mathematics, a binary operation on a set is a calculation involving two elements of the set ( called operands ) and producing another element of the set ( more formally, an operation whose arity is two ).
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics.
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 set theory concepts in most contemporary mathematics.
The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.
Certain categories called topoi ( singular topos ) can even serve as an alternative to axiomatic set theory as a foundation of mathematics.
In mathematics, a countable set is a set with the same cardinality ( number of elements ) as some subset of the set of natural numbers.
Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.
In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.
mathematics and is
This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.
Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.
But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.
Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.
In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.
So, too, is the mathematical competence of a college graduate who has majored in mathematics.
The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.
The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.
The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.
In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.
There is no prize awarded for mathematics, but see Abel Prize.
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