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Cantor established these results using two constructions.

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Cantor and established

__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

**.**

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**.**
The show

__established____Cantor__as a leading comedian, and his scriptwriter, David Freedman, as “ the Captain of Comedy .”__Cantor__soon became the world's highest-paid radio star**.**
The Hollywood Stock Exchange, a virtual market game

__established__in 1996 and now a division of__Cantor__Fitzgerald, LP, in which players buy and sell prediction shares of movies, actors, directors, and film-related options, correctly predicted 32 of 2006's 39 big-category Oscar nominees and 7 out of 8 top category winners**.**
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**.**

Cantor and these

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**.**
Each of

__these__strings p < sub > i </ sub > determines a subset S < sub > i </ sub > of__Cantor__space ; the set S < sub > i </ sub > contains all sequences in cantor space that begin with p < sub > i </ sub >.
Note that while both the elements of the

__Cantor__space ( regarded as sets of integers ) and subsets of the__Cantor__space are classified in arithmetic hierarchies but__these__are not the same hierarchy**.**
The term transfinite was coined by Georg

__Cantor__, who wished to avoid some of the implications of the word infinite in connection with__these__objects, which were nevertheless not finite**.**
It is constant on intervals of the form ( 0

**.**x < sub > 1 </ sub > x < sub > 2 </ sub > x < sub > 3 </ sub >... x < sub > n </ sub > 022222 ..., 0**.**x < sub > 1 </ sub > x < sub > 2 </ sub > x < sub > 3 </ sub >... x < sub > n </ sub > 200000 ...), and every point not in the__Cantor__set is in one of__these__intervals, so its derivative is 0 outside of the__Cantor__set**.**
This gives additional justification to the practice of restricting attention to Baire space and

__Cantor__space, since__these__and any other Polish spaces are all isomorphic at the level of Borel sets**.**
Included among

__these__are Nora Bayes, Fanny Brice, Ruth Etting, W**.**C**.**Fields, Eddie__Cantor__, Marilyn Miller, Will Rogers, Bert Williams and Ann Pennington**.**
By convention

__these__features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to__Cantor__**.**
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**.**
Of

__these__, only George White's Scandals ( 1939 ) and Banjo Eyes with Eddie__Cantor__( 1941 ) had long runs**.**
Among

__these__is the current Mayor's office ( former Ballet Hall ), a water reservoir, and the__Cantor__church, where a Russian pilot is buried**.**
However, some of

__these__sets contain subclasses that are not sets, which makes them different from__Cantor__( ZF ) finite sets and they are called infinite in AST**.**

Cantor and results

That the algebra of the real numbers can be employed to yield

__results__about the linear continuum of geometry relies on the__Cantor__– Dedekind axiom**.**
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**.**
By considering only the points of the tori that are not removed at some stage, an embedding

__results__of the sphere with a__Cantor__set removed**.**
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 >, ...
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**.**

Cantor and using

* Fractals derived from standard geometry by

__using__iterative transformations on an initial common figure like a straight line ( the__Cantor__dust or the von Koch curve ), a triangle ( the Sierpinski triangle ), or a cube ( the Menger sponge ).
In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass,

__Cantor__, Dedekind, and others__using__the ( ε, δ )- definition of limit and set theory**.**
* 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**.**
A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of Baire space or

__Cantor__space,__using__formulas with several free variables**.**
In order to do this, Deutsch invents the notion of a CantGoTu environment ( named after

__Cantor__, Gödel, and Turing ),__using__Cantor's diagonal argument to construct an ' impossible ' Virtual Reality which a physical VR generator would not be able to generate**.**
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 ".
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**.**
In the nineteenth century,

__using__the same methods,__Cantor__showed that this restriction is not necessary**.**
This can be done either by

__using__the Tietze extension theorem on each of the components of, or by simply extending " linearly " ( that is, on each of the deleted open interval in the construction of the__Cantor__set, we define the extension part of on to be the line segment within the unit square joining the values and ).
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**.**
If is a set of subsets of the natural numbers, then the set (

__using__the__Cantor__pairing function ) is required to be recursively enumerable**.**

Cantor and two

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**.**
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 ).__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

**.**

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 ).0.200 seconds.