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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.

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## Some Related Sentences

Cantor and established

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 importance

Whole chapters

**are**devoted to**the**emergence**of**algebraic logic in**the**19th century UK**,**__Cantor__**and****the**emergence**of**set theory**,****the**emergence**of**mathematical logic in Germany told in a way**that**downplays Frege's__importance__**,****and**to Peano**and**his followers**.**

Cantor and one-to-one

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****.**
Cantor's diagonal argument

**,**also called**the**diagonalisation argument**,****the**diagonal slash argument or**the**diagonal method**,**was published in 1891 by Georg__Cantor__as a mathematical proof**that**there**are****infinite****sets**which cannot be put into__one-to-one__**correspondence**with**the****infinite**set**of****natural****numbers****.**
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**.**
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**.**

Cantor and correspondence

'' My impassioned plea for civil rights created a landslide

**of**__correspondence__**and**one sponsor even asked me to consider replacing**the**Eddie__Cantor__comedy hour on a permanent basis**.**
In 1882

**,****the**mathematical__correspondence__**between**__Cantor__**and**Dedekind came to an end**,**apparently as a result**of**Dedekind's declining**the**chair at Halle**.**__Cantor__also began another important

__correspondence__

**,**with Gösta Mittag-Leffler in Sweden

**,**

**and**soon began to publish in Mittag-Leffler's journal Acta Mathematica

**.**

**"**

__Cantor__complied

**,**but then curtailed his relationship

**and**

__correspondence__with Mittag-Leffler

**,**writing to a third party:

Cantor and between

Norman F

**.**__Cantor__notes**the**three roles Boniface played**that**made him**"**one**of****the**truly outstanding creators**of****the**first Europe**,**as**the**apostle**of**Germany**,****the**reformer**of****the**Frankish church**,****and****the**chief fomentor**of****the**alliance__between__**the**papacy**and****the**Carolingian family**.**
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**.**
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**.**
The

__Cantor__function challenges naive intuitions about continuity**and**measure ; though it is continuous everywhere**and**has zero derivative almost everywhere**,**c goes from 0 to 1 as x goes from 0 to 1**,****and**takes on every value in__between__**.**
In addition

**,**he edited one**of****the**most influential compilations**of**American historical documents**,**Documents**of**American History**,**which went through ten editions__between__1938**and**1988 (**the**tenth**,****and**last**,**coedited with Commager's former student Milton__Cantor__**.**
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**.**
A similar relationship

__between__**the**projective hierarchy**and****the**relativized analytical hierarchy holds for subsets**of**__Cantor__space**and****,****more**generally**,**subsets**of**any effective Polish space**.**__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

**.**

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****.**
According to

__Cantor__**,**this is where**the**family find a difference__between__Japanese**and**American culture**,**as Wink**,****the**game-show host**,**explains to them:**"**Our game shows**are**a little different from yours**.**
This paradox is named for Georg

__Cantor__**,**who is often credited with first identifying it in 1899 ( or__between__1895**and**1897 ).

Cantor and members

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**.**
On July 12

**,**1938**,**with**the**support**of**radio stars Eddie__Cantor__**,**Edgar Bergen**,**Jack Benny**,**Bing Crosby**,****and**others**,**AFRA__members__negotiate**the**first collectively bargained agreement on a national scale – with NBC**and**CBS – resulting in a wage increase**of**125 %.
Presently

**,**seven**of****the**nine top Republican leaders — Majority Leader Eric__Cantor__**,**Chief Deputy Whip Peter Roskam**,**Conference Chairman Jeb Hensarling**,**Conference Vice-Chairwoman Cathy McMorris Rodgers**,**Conference Secretary John Carter**,**Policy Committee chairman Tom Price**,****and**National Republican Congressional Committee chairman Pete Sessions --**are**__members__**of****the**RSC**.**

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**.**
* 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 ).
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****.**
She co-starred with Eddie

__Cantor__in__two__features**,**Show Business ( 1944 )**and**If You Knew Susie ( 1948 ).0.312 seconds.