The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

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

Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous.

His collection of American art was strong in Abstract expressionism, Pop Art, Minimalism, Neo-Dada, Color Field, Lyrical Abstraction, and Neo-Expressionism and he often donated important works from his collection to institutions like MoMA, and other important private museums and University collections like the Norton Simon Museum, the Sheldon Museum of Art and the Iris & B. Gerald Cantor Center for Visual Arts at Stanford University among many others.

His anthology, Documents of American History ( 1938 ), reaching its tenth edition ( coedited with his former student Milton Cantor ) in 1988, half a century after its first appearance, remains a standard collection work of primary sources.

The oldest definition of the cardinality of a set X ( implicit in Cantor and explicit in Frege and Principia Mathematica ) is as the set of all sets which are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems.

In his work on set theory, Georg Cantor denoted the collection of all cardinal numbers by the last letter of the Hebrew alphabet, ( transliterated as Taf, Tav, or Taw.

Cantor instead said that it was an " inconsistent " collection which was absolutely infinite.

* Complete collection of Cantor Mikhail Girgis El Batanouny's recordings of Coptic chant from coptichymns. net

* Five O ' Clock Tea ( 1884 ), Iris & B. Gerald Cantor collection, New York City

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

With the full set, companies could mail a single Cantor card each month for a year to their selected special customers as an ongoing promotion.

Cantor has said the fact that the gnomes themselves do not understand their own business plan or why they steal the underwear could represent the idea that businessmen themselves often lack the economic knowledge needed to explain their activity and profits to the public.

( Alternatively, we could use the theorem that every compact metric space is a continuous image of the Cantor set to get the function.

Based on his experience at Cantor Fitzgerald, Pulver developed a market where IP Telephony minutes and capacity could be traded in both spot and futures contracts.

The choice of the Polish space in the third clause above is not very important ; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

Cantor, who knew Kantorowicz at Princeton, suggested that, but for his Jewish heritage, Kantorowicz ( at least as a young scholar in the 1920s and 1930s ) could be considered a Nazi in terms of his intellectual temperament and cultural values.

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i. e., the set of all subsets of S ( here written as P ( S )), is larger than S itself.

* The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval 1 with the usual topology.

He came from a family that had emigrated to the Netherlands from Portugal, another branch of which had established itself in Russia, where Georg Cantor was born.

The Cantor set itself is of course a Cantor space.

The support of the Cantor distribution is the Cantor set, itself the ( countably infinite ) intersection of the sets

Because Cantor space is homeomorphic to any finite Cartesian power of itself, and Baire space is homeomorphic to any finite Cartesian power of itself, the analytical hierarchy applies equally well to finite Cartesian power of one of these spaces.

Thus there are points where V ′ takes values 1 and − 1 in every neighborhood of each of the endpoints of intervals removed in the construction of the Smith – Volterra – Cantor set S. In fact, V ′ is discontinuous at every point of S, even though V itself is differentiable at every point of S, with derivative 0.

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.

On September 19, Cantor Fitzgerald made a pledge to distribute 25 percent of the firm's profits for the next five years, and committed to paying for ten years of health care, for the benefit of the families of its 658 former Cantor Fitzgerald, eSpeed, and TradeSpark employees ( profits which would otherwise have been distributed to the Cantor Fitzgerald partners ).

Historian Norman Cantor who specialized in the medieval period, teaching and writing at Columbia and New York University, says in 1993: " It may be true that the Arabs had now fully extended their resources and they would not have conquered France, but their defeat ( at Tours ) in 732 put a stop to their advance to the north.

The implications of the infinite void were revolutionary ; to have pursued them would have threatened the singular relationship of man and this natural world to God ( Cantor 2001 ); in it he treated theology mathematically.

We want Can-tor !," a phrase said to have originated in vaudeville, when the audience chanted to chase off an act on the bill before Cantor.

Cantor said " no episode of South Park I have taught has raised as much raw passion, indignation, and hostility among students as ' Gnomes ' has.

It is not a true ordering because the trichotomy law need not hold: if both and, it is true by the Cantor – Bernstein – Schroeder theorem that i. e. A and B are equinumerous, but they do not have to be literally equal ; that at least one case holds turns out to be equivalent to the Axiom of choice.

Two important members of the Notre Dame administration have been suggested as possible identities for Perotinus: the theologian Petrus Cantor ( who died in 1197 ) and the Petrus who was Succentor of Notre Dame from at least 1207 until about 1238.

Ed Dyson and David Cantor have written episodes for the seventh, eighth, ninth and eleventh series.

Apart from Brenman-Gibson's work, six critical biographies have appeared by the following authors: R. Baird Shuman ( 1962 ); Edward Murray ( 1968 ); Michael Mendelsohn ( 1969 ); Gerald Weales ( 1971 ); Harold Cantor ( 1978 ); and Christopher J. Herr ( 2003 ).

They have contributed to the restoration of the Lincoln Theatre in Yountville, California, and have supported the Cantor Arts Center at Stanford University in Palo Alto, California.

If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set C in place of the " fat " ( positive-measure ) Cantor set S, one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set C instead of the positive-measure set S, and so the resulting function would have an integrable derivative.

Like Arthurian legend, however, Cantor's story does not have an entirely happy ending, and Cantor describes his sense of disappointment when Southern fails to live up to expectations.

Bill Viola, Adel Abdessemed and Mircea Cantor have been invited to screen videos in the park for the « Nuit des musées ».

He may have been a " Walter Cantor " at Ely Cathedral between 1443 and 1466, and he may have been the Walter Frye who joined the London Parish Clerks in 1456 ; he also may have been the Walter Frye who left a will at Canterbury in 1474.

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

Cantor first established cardinality as an instrument to compare finite sets ; e. g. the sets

Cantor developed important concepts in topology and their relation to cardinality.

For example, he showed that the Cantor set is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable.

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 ".

In fact the cardinality of sets fails to be totally ordered ( see Cantor – Bernstein – Schroeder theorem ).

Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis.

Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if and only if there exists a bijective function between them.

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