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 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 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.

* An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic ; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space.

The arithmetical hierarchy can be defined on any effective Polish space ; the definition is particularly simple for Cantor space and Baire space because they fit with the language of ordinary second-order arithmetic.

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

Formally, the Cantor function c: → is defined as follows:

* Given a Gödel numbering of the computable functions, the set ( where is the Cantor pairing function and indicates is defined ) is recursively enumerable.

The analytical hierarchy can be defined on any effective Polish space ; the definition is particularly simple for Cantor and Baire space because they fit with the language of ordinary second-order arithmetic.

An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic ; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space.

The function is defined by making use of the Smith – Volterra – Cantor set and " copies " of the function defined by f ( x )

For ordinal numbers α, the α-th Cantor – Bendixson derivative of a topological space is defined by transfinite induction as follows:

While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901.

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.

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.

) As Cantor realized, this collection could not itself have a cardinality, as this would lead to a paradox of the Burali-Forti type.

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

For one example ( a variant of the Cantor set ), remove from all dyadic fractions, i. e. fractions of the form a / 2 < sup > n </ sup > in lowest terms for positive integers a and n, and the intervals around them: ( a / 2 < sup > n </ sup > − 1 / 2 < sup > 2n + 1 </ sup >, a / 2 < sup > n </ sup > + 1 / 2 < sup > 2n + 1 </ sup >).

This reduction of real numbers and continuous functions in terms of rational numbers and thus of natural numbers, was later integrated by Cantor in his set theory, and axiomatized in terms of second order arithmetic by Hilbert and Bernays.

The principle that cardinal number was to be characterized in terms of one-to-one correspondence had previously been used to great effect by Georg Cantor, whose writings Frege knew.

But Frege criticized Cantor on the ground that Cantor defines cardinal numbers in terms of ordinal numbers, whereas Frege wanted to give a characterization of cardinals that was independent of the ordinals.

Cantor describes Southern in Arthurian terms, with a group of devotees who surrounded their master following the publication of The Making of the Middle Ages.

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.

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.

A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase ( a term also used for singular functions in general ).

As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.

Note that elements of the Cantor space can be identified with sets of integers and elements of the Baire space with functions from integers to integers.

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.

* The set of elements of Cantor space which are the characteristic functions of well orderings of is a set which is not.

Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith – Volterra – Cantor set ; in other words, the function V is the limit of the sequence of functions f < sub > 1 </ sub >, f < sub > 2 </ sub >, ...

During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

Cantor established these results using two constructions.

The carpet is a generalization of the Cantor set to two dimensions ( another is Cantor dust ).

Eddie Cantor is broadcast from NBC's Manhattan station WNBT to Philco's Philadelphia station WPTZ, via an automatic relay tower halfway between the two cities.

* A Few Moments with Eddie Cantor, Star of " Kid Boots " ( 1923 ) A six-minute film made in Phonofilm by Lee De Forest featuring Cantor telling monologues and singing two songs.

This embedding extends to the whole sphere, since points approaching two different points of the Cantor set will be at least a fixed distance apart in the construction.

This consisted of the two producers ( Cantor and Harvey ), a scratch DJ Jason Tunbridge ( Mad Doctor X ), a guitarist ( Tony Ayiotou ), drummer ( Clive Jenner ), bass guitarist ( Joe Henson ), two MCs ( MC Navigator and Tenor Fly ) and three breakdancers ( Coza, Marat, Tim ).

* 1940: Eddie Cantor, one of only two balloons based on a living person or people, The Tin Man

Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers.

In his book Gilligan Unbound, American literary critic Paul Cantor described how " Thirty Minutes Over Tokyo " references and mocks several aspects of Japanese and American culture, as well as differences between the two.

She co-starred with Eddie Cantor in two features, Show Business ( 1944 ) and If You Knew Susie ( 1948 ).

With that, Cantor Rabinovitch forgives his son and the two embrace.