Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive — even shocking — that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections.

A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result.

Prompted by Eddie Cantor's insistence, at that meeting, that any response to that producer's agreement help all actors, not just the already established ones, it took only three weeks for SAG membership to go from around 80 members to more than 4, 000.

Cantor's popularity led to merchandising of such products as Eddie Cantor's Tell It to the Judge game from Parker Brothers.

When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.

Skolem ( 1922 ) pointed out the seeming contradiction between the Löwenheim – Skolem theorem on the one hand, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which states that uncountable sets exist, and which is provable from Zermelo's axioms.

In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the collection of " infinite sizes " is itself infinite.

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.

Cantor's work between 1874 and 1884 is the origin of set theory.

George Burns, in his memoir All My Best Friends, claimed that Warner Bros. created a miracle producing the movie in that " it made Eddie Cantor's life boring ".

And yet Cantor's diagonal argument shows that real numbers have higher cardinality.

Symeon eventually became precentor of the priory, and examples of his handwriting appear to survive in several Durham books, including the Liber Vitae, the so-called Cantor's Book ( whose text he would have had to keep up to date as part of his duties as precentor ), and in copies of his own historical works.

An opponent of ethnomathematics may claim that pointing out that many cultures have arrived at different ways of counting on their fingers is not as insightful, on objective terms, as Cantor's work on infinity, for example.

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.

Cantor's father had been a member of the Saint Petersburg stock exchange ; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg.

The suggestion has therefore been made that Hume's Principle ought better be called " Cantor's Principle ".

Brooks was born in Beverly Hills, California, the son of Thelma Leeds ( née Goodman ), a singer and actress, and Harry Einstein, a radio comedian who performed on Eddie Cantor's radio program and was known as Parkyakarkus.

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.

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.

He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ( Burali-Forti paradox, Cantor's paradox, and Russell's paradox ) to a meeting of the Deutsche Mathematiker – Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

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

Cantor's first ten papers were on number theory, his thesis topic.

Given a guaranteed halting language, the computable function which is produced by Cantor's diagonal argument on all computable functions in that language is not computable in that language.

The original statement of the paradox, due to Richard ( 1905 ), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers.

On September 10, 1933, Durante appeared on Eddie Cantor's The Chase and Sanborn Hour, continuing until November 12 of that year.

After a solo appearance by Gracie on Eddie Cantor's radio show, they were heard together on Rudy Vallee's The Fleischmann's Yeast Hour and on February 15, 1932 they became regulars on The Guy Lombardo Show on CBS.

Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's ' new ' use of axiomatic method as a research tool.

Cantor's eyes became his trademark, often exaggerated in illustrations, and leading to his appearance on Broadway in the musical Banjo Eyes ( 1941 ).

They ( famously ) had five daughters, Marjorie, Natalie, Edna, Marilyn and Janet, who provided comic fodder for Cantor's longtime running gag, especially on radio, about his five unmarriageable daughters.

Cantor's appearance with Rudy Vallee on Vallee's The Fleischmann's Yeast Hour on February 5, 1931 led to a four-week tryout with NBC's The Chase and Sanborn Hour.

He criticized Cantor's work on set theory, and was quoted by as having said, " God made natural numbers ; all else is the work of man ".

These cases demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true: the situations " there is a guest to every room " and " no more guests can be accommodated " are not equivalent when there are infinitely many rooms ( an analogous situation is presented in Cantor's diagonal proof ).

By his own admission in his 1919 Introduction to Mathematical Philosophy, he " attempted to discover some flaw in Cantor's proof that there is no greatest cardinal ".

The lemma is called " diagonal " because it bears some resemblance to Cantor's diagonal argument.

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.

This is in accordance with Cantor's original vision of a cardinals: to take a set and abstract its elements into canonical " units " and collect these units into another set, such that the only thing special about this set is its size.

The Absolute Infinite is mathematician Georg Cantor's concept of an " infinity " that transcended the transfinite numbers.

However, as in Cantor's argument ( above ), this idea leads to difficulties.

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's construction of the real numbers is similar to the above construction ; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances.

Norman F. Cantor's summary of the effects of late medieval deforestation applies equally well to Early Modern Europe:

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers.

For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

Ellis Island as a port of entry is described in detail in Mottel the Cantor's Son by Sholom Aleichem.

Cantor's work is of great philosophical interest, a fact of which he was well aware.

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.

" Kronecker even objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum.

Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work.

Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair.