Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment.

Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

In this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable.

Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes ( see Frápolli 1991 ); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

For example, Georg Cantor ( who introduced this concept ) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers ( non-negative integers ), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

* The Cantor set is compact.

In fact, every compact metric space is a continuous image of the Cantor set.

* Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

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.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

`` It is as though '', I said on the historic three-hour, coast-to-coast radio broadcast which I bought ( following Father Coughlin and pre-empting the Eddie Cantor, Manhattan Merry-go-round and Major Bowes shows ) `` That Man in the White House, like some despot of yore, insisted on reading my diary, raiding my larder and ransacking my lingerie!!

Cantor is quoted as saying:

It is undisputed that, by 1900, Cantor was aware of some of the paradoxes and did not believe that they discredited his theory.

In mathematics, the continuum hypothesis ( abbreviated CH ) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets.

The Cantor space is the collection of all infinite sequences of 0s and 1s.

The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string x the set of sequences that begin with x has measure 2 < sup >-| x |</ sup >.

Cantor Fitzgerald L. P. is a global financial services firm specializing in bond trading.

The company's effort to regain its footing is the subject of Tom Barbash's 2003 book On Top of the World: Cantor Fitzgerald, Howard Lutnick, and 9 / 11: A Story of Loss and Renewal.

Cantor Gaming is a Cantor Fitzgerald affiliate that operates race and sports books.

( Konig is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.

'' My impassioned plea for civil rights created a landslide of correspondence and one sponsor even asked me to consider replacing the Eddie Cantor comedy hour on a permanent basis.

It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets.

David Hilbert defended it from its critics by famously declaring: " No one shall expel us from the Paradise that Cantor has created.

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.

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.

Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.

Everything and More: A Compact History of Infinity is a book by American novelist and essayist David Foster Wallace that examines the history of infinity, focusing primarily on the work of Georg Cantor, the 19th-century German mathematician who created set theory.

The Bickersons was created by Philip Rapp, the one-time Eddie Cantor writer who had also created the Fanny Brice skits ( for The Ziegfeld Follies of the Air and Maxwell House Coffee Time ) that grew into radio's Baby Snooks.

The term was originated by Georg Cantor.

It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874 – 1884.

Headquartered in Midtown Manhattan, New York City, Cantor Fitzgerald was formerly based in the World Trade Center and was the company hardest hit by the September 11, 2001 attacks, which killed all 658 of its employees who were in the office at the time ( out of 960 who were based there ).

Seconds after Cantor's building was struck by the plane, a Goldman Sachs server issued an alert saying that its trading system had gone offline because it wasn't able to connect with a Cantor server.

Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.

In his professorial doctoral dissertation, On the Concept of Number ( 1886 ) and in his Philosophy of Arithmetic ( 1891 ), Husserl sought, by employing Brentano's descriptive psychology, to define the natural numbers in a way that advanced the methods and techniques of Karl Weierstrass, Richard Dedekind, Georg Cantor, Gottlob Frege, and other contemporary mathematicians.

Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass, published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.

It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.

After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris.

The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration.

Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series.