Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are " more numerous " than the natural numbers.

In one of his earliest papers, Cantor proved that the set of real numbers is " more numerous " than the set of natural numbers ; this showed, for the first time, that there exist infinite sets of different sizes.

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

In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable.

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.

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.

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.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem.

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.

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

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

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.

* Cantor – Bernstein – Schroeder theorem, after Felix Bernstein

In set theory, the Cantor – Bernstein – Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function.

The theorem is also known as the Schroeder – Bernstein theorem, the Cantor – Bernstein theorem, or the Cantor – Schroeder – Bernstein theorem.

Knaster – Tarski theorem can be used for a simple proof of Cantor – Bernstein – Schroeder theorem.

* The Cantor set is meagre as a subset of the reals, but not as a space, since it is a complete metric space – it is thus a Baire space, by the Baire category theorem.

A constructive version of " the famous theorem of Cantor, that the real numbers are uncountable " is: " Let

*" Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable.

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

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

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.

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

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.

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

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.

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.

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

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.

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.

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

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.

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 ".

Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities.

" 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.

It may happen that a continuous function f is differentiable almost everywhere on, its derivative f ′ is Lebesgue integrable, and nevertheless the integral of f ′ differs from the increment of f. For example, this happens for the Cantor function, which means that this function is not absolutely continuous.

Since the Smith – Volterra – Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure.

* Overture to Glory ( Der Vilner Shtot Khazn ; the Yiddish title literally means " The Vilnius City Cantor ") 1940, USA, B & W, 85 min, Yiddish with English subtitles.

One of the greatest achievements of Georg Cantor was the construction of a one-to-one correspondence between the points of a square and the points of one of its edges by means of continued fractions.