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.

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

In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument ; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society.

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.

Bernard Bolzano who introduced the notion of set ( in German: Menge ) and Georg Cantor who introduced set theory opposed the general attitude.

In the late 1950s, friends introduced Layton to Aviva Cantor, who had emigrated to Montreal from her native Australia in 1955.

The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.

Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets.

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.

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.

Georg Cantor considered the well-ordering theorem to be a " fundamental principle of thought.

Cantor established these results using two constructions.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

* the various ( but equivalent ) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field ;

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.

* 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.

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.

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments.

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

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

Some Christian theologians ( particularly neo-Scholastics ) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God — on one occasion equating the theory of transfinite numbers with pantheism — a proposition which Cantor vigorously rejected.

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

In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin.

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.

By proving that there are ( infinitely ) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.

Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers.

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.

From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas.

They both postulated that if the stars in the universe were distributed in a hierarchical fractal cosmology ( e. g., similar to Cantor dust )— the average density of any region diminishes as the region considered increases — it would not be necessary to rely on the Big Bang theory to explain Olbers ' paradox.

Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:

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.

It was suggested to him as an alternate form of Russell's paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions.

In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem.

Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

This implies that for each natural number n, the set of sequences f in Cantor space such that f ( n )

President Calvin Coolidge, opera singer Abbie Mitchell, and vaudeville stars such as Phil Baker, Ben Bernie, Eddie Cantor, and Oscar Levant appeared in the firm's pictures.

In 2007, Stanford's Cantor Arts Center opened an exhibition, " Art of Being Tuareg: Sahara Nomads in a Modern World ", the first such exhibit in the United States.

" Shakespeare scholar Paul A. Cantor argues that this association is appropriate — the warlike Klingons find their literary matches in the characters Othello, Mark Antony, and Macbeth — but that it also reinforces a claim that the end of the Cold War means the end of heroic literature such as Shakespeare's.

They are typically hosted by a senior faculty member or noted scholar ( such as historian Charles Adams and literary critic Paul Cantor ).

Young " Tony " Benedetto grew up listening to Al Jolson, Eddie Cantor, Judy Garland and Bing Crosby as well as jazz artists such as Louis Armstrong, Jack Teagarden and Joe Venuti.

Based on Merovingian ad hoc arrangements, using the form missus regis ( the " king's envoy ") and sending a layman and an ecclesiastic in pairs, the use of missi dominici was fully exploited by Charlemagne ( ruling 768 — 814 ), who made them a regular part of his administration, " a highly intelligent and plausible innovation in Carolingian government ", Norman F. Cantor observes, " and a tribute to the administrative skill of the ecclesiastics, such as Alcuin and Einhard ".

In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.

( The restriction of the Cantor function to the Cantor set is an example of such a function.

During its heyday, the Cotton Club served as a hip meeting spot featuring regular " Celebrity Nights " on Sundays which featured celebrity guests such as Jimmy Durante, George Gershwin, Sophie Tucker, Paul Robeson, Al Jolson, Mae West, Richard Rodgers, Irving Berlin, Eddie Cantor, Fanny Brice, Langston Hughes, Judy Garland, Moss Hart, and New York City Mayor Jimmy Walker, among others.

His earliest movie jobs were on Samuel Goldwyn's Eddie Cantor musicals, where he began developing such techniques as a “ parade of faces ” ( individualizing each chorus girl with a loving close-up ), and moving his dancers all over the stage ( and often beyond ) in as many kaleidoscopic patterns as possible.

Examples of such spaces include the Cantor space and Baire space.

He published stories by such writers as Booth Tarkington and F. Scott Fitzgerald, nonfiction pieces by women such as Shirley Temple's mother and Eleanor Roosevelt, and articles on the Wall Street Crash of 1929 by men like Cornelius Vanderbilt and Eddie Cantor, as well as a complete novel in each issue.

Performers such as Eddie Cantor, Anna Held, W. C. Fields, Bert Williams, The Marx Brothers and the Fairbanks Twins found great success on the revue stage.

Over the next couple of decades many of his songs were used in the famous Ziegfeld Follies, and were performed by the premier singers of the day, such as Eddie Cantor, Anna Held and Edith Day.

Past and present residents of the building include such famous personalities as Stephen Sondheim, Tiger Woods, Steven Spielberg, Donna Karan, Tony Randall, Demi Moore, Glenn Close, Dustin Hoffman, U2 frontman Bono, Steve Martin, Bruce Willis, Trey Parker, Eddie Cantor, Robert Stigwood, Marshall Brickman, Jackie Leo, Don Hewitt, Billy Squier, and Texas natural-gas heiress Adelaide de Menil but also some businessmen: Dodi Fayed, Andrew Tobias, Aaron Spelling, Jack Sherwall, Hedy Lamarr.

Another example is the theory of dense linear orders with no endpoints ; Cantor proved that any such countable linear order is isomorphic to the rational numbers.

One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the Cantor set.

In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant.

Other guests have included actors such as Ethel Barrymore, Charles Boyer, Eddie Cantor, Mary Pickford, Ginger Rogers, Bette Davis ( who was married at the Inn in 1945 ), W. C. Fields, Clark Gable, Cary Grant, Spencer Tracy, Fess Parker, James Brolin and Barbra Streisand, Raquel Welch and Drew Barrymore.

" According to Reed University Professor of Music David Schiff, " With the appearance of black musicals like Shuffle Along and the emergence of black stars such as Paul Robeson and Ethel Waters, the minstrel convention of blackface, which survived in the vastly popular performances of Al Jolson and Eddie Cantor, had become an embarrassment-at least to some critics.