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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.

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## Some Related Sentences

Cantor and points

In mathematics

**,**the__Cantor__set is**a**set**of**__points__lying on**a**single line segment**that**has**a**number**of**remarkable and deep properties**.**
Given

**a**trigonometric series f ( x ) with S as its set**of**zeros**,**__Cantor__had discovered**a**procedure**that**produced another trigonometric series**that**had S ' as its set**of**zeros**,**where S ' is the set**of**limit__points__**of**S**.**If p ( 1 ) is the set**of**limit__points__**of**S**,**then he could construct**a**trigonometric series whose zeros are p ( 1 ).
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**.**
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**.**
In mathematics

**,**the Smith – Volterra –__Cantor__set ( SVC ), fat__Cantor__set**,**or ε-Cantor set is an example**of****a**set**of**__points__on the real line R**that**is nowhere dense ( in particular it**contains**no intervals ), yet has positive measure**.**
Continuing indefinitely with this removal

**,**the Smith – Volterra –__Cantor__set is then the set**of**__points__**that**are never removed**.**
Thus there are

__points__where V ′ takes values 1 and − 1 in every neighborhood**of**each**of**the endpoints**of**intervals removed in the construction**of**the Smith – Volterra –__Cantor__set S**.**In fact**,**V ′ is discontinuous at every point**of**S**,**even though V itself is differentiable at every point**of**S**,**with derivative 0**.**
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**.**
More abstractly

**,****a**natural class**of**objects to study in topology are objects**that**are homogeneous ( all__points__are topologically the same**:**the group**of**self-homeomorphisms acts transitively ) and " finite type " or " tame " ( to rule**out**spaces such as the__Cantor__set**,**where each open set**contains**uncountably**many**connected components );**more**generally**,****a**space**of**" finite type " where the self-homeomorphism group has finitely**many**orbits**,**forming the strata**.**

Cantor and out

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 ).
( Konig is now remembered as having only pointed

__out__**that**some sets cannot be well-ordered**,**in disagreement with__Cantor__**.**
Dedekind

**,**whom__Cantor__befriended in 1872**,**cited this paper later**that**year**,**in the paper where he first set__out__**his**celebrated definition**of**real**numbers**by Dedekind cuts**.**
In Waguespack and

__Cantor__( 1996 ), the authors point__out__**that**JIT would require**a**significant change in the supplier / refiner relationship**,**but the changes in inventories in the oil industry exhibit none**of**those tendencies**.**
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**.**
It is not

**a**true ordering because the trichotomy law need not hold**:**if both and**,**it is true by the__Cantor__– Bernstein – Schroeder**theorem****that**i**.**e**.**A and B are equinumerous**,**but**they**do not have to be literally equal ;**that**at least one case holds turns__out__to be equivalent to the Axiom**of**choice**.**
After Marge bails them

__out__**,**Bart and Homer can speak fluent Japanese**,**and have fully absorbed**,**as__Cantor__writes**,**the " exclusionary " character**of**the Japanese culture**,**as Homer asks Bart ( in Japanese**,**with English subtitles ): " Should we tell them and Lisa the secret to inner peace?__Cantor__comes to the jail to bail them

__out__

**,**but finds there is not

**a**Yussel Rabinovitch there

**,**only

**a**Jess Robin

**.**

Along with him are

**a**few other__out__**of**work actors**that**appear as caricatures**of**Al Jolson**,**Jack Benny**,**Eddie__Cantor__and Bing Crosby**.**

Cantor and constructions

__Cantor__introduced fundamental

__constructions__in set theory

**,**such as the power set

**of**

**a**set A

**,**which is the set

**of**all possible subsets

**of**A

**.**

* the various ( but equivalent )

__constructions__**of**the real**numbers**by Dedekind and__Cantor__resulting in the modern axiomatic definition**of**the real number field ;

Cantor and prove

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 and more

__Cantor__himself only mentioned the ternary construction in passing

**,**as an example

**of**

**a**

__more__general idea

**,**

**that**

**of**

**a**perfect set

**that**is nowhere dense

**.**

__Cantor__Fitzgerald lost over two-thirds

**of**its workforce

**,**considerably

__more__than any

**of**the other World Trade Center tenants or the New York City Police Department and New York City Fire Department

**.**

__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**

**.**

To attain the latter rank at the age

**of**34 was**a**notable accomplishment**,**but__Cantor__desired**a**chair at**a**__more__prestigious university**,**in particular at Berlin**,**at**that**time the leading German university**.**
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**.**
Between 1870 and 1872

**,**__Cantor__published__more__papers on trigonometric series**,**and also**a**paper defining irrational**numbers**as convergent sequences**of**rational**numbers****.**
He withdrew from public life in 747 to take up the monastic habit

**,**" the first**of****a****new**type**of**saintly king ," according to Norman__Cantor__**,**"__more__interested in religious devotion than royal power**,**who frequently appeared in the following three centuries and who was an indication**of**the growing impact**of**Christian piety on Germanic society ".
In this case

**they**often use the style The Reverend ;__more__usually**,**however**,****a**cantor is called__Cantor__or__Cantor__Surname**.**
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**.**
At the same time

**,**__more__popular " big name " artists were hired to record for the label including Wilbur Sweatman**,**Eddie__Cantor__**,**the Six Brown Brothers**,**and the Louisiana Five**.**
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**.**
Note

**that****,**commonly**,**2 < sup > ω </ sup > is referred to simply as the__Cantor__set**,**while the term__Cantor__space is reserved for the__more__general construction**of**D < sup > S </ sup > for**a**finite set D and**a**set S which might be finite**,**countable or possibly uncountable**.**
Mistakes and philosophical weaknesses in reasoning about infinitesimal

**numbers**in the work**of**Gottfried Leibniz**,**Johann Bernoulli**,**Leonhard Euler**,**Augustin-Louis Cauchy**,**and others were the reason**that****they**were originally abandoned for the__more__cumbersome real number-based arguments developed by Georg__Cantor__**,**Richard Dedekind**,**and Karl Weierstrass**,**which were perceived as being__more__rigorous by Weierstrass's followers**.**
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**.**
This was proven by Georg

__Cantor__in**his**1874 uncountability**proof****,**part**of****his**groundbreaking study**of**different infinities**,**and later__more__simply in**his**diagonal argument**.**
As with

**many**other episodes in the series**,**" Thirty Minutes Over Tokyo " suggests**that****,**in the end**,**the Simpsons are__more__attached to the local than to the global**,**and**,**as__Cantor__writes**:**" indeed the global is ultimately important in the series only insofar as it can be made local**,****that**is**,**part**of**Springfield**.**

Cantor and —

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**.**
They both postulated

**that**if the stars in the universe were distributed in**a**hierarchical fractal cosmology ( e**.**g., similar to__Cantor__dust )— the average density**of**any region diminishes as the region considered increases__—__it would not be necessary to rely on the Big Bang theory to explain Olbers ' paradox**.**__—__The paradox told as

**a**humorous narrative

**,**featuring

**a**hotel owner and

**a**building contractor based on the feuding 19th-century mathematicians Georg

__Cantor__and Leopold Kronecker

" 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**.**
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 ".
After marrying long-time girlfriend

__—__law student Rianna Jennifer__Cantor____—__in June 2002**,**Ponting credited her as the reason for**his**increased maturity**.**
* Rodin

**:**In His Own Words__—__Selections from the Iris and B**.**Gerald__Cantor__Foundation ( June 13 – August 16**,**2009 )
Ethel Shutta ( pronounced " shoo-tay ") ( December 1

**,**1896__—__February 5**,**1976 ) was an American actress and singer**,**who came to prominence through her performances on Jack Benny's radio show**,**her role in the early Eddie__Cantor__musical Whoopee !, and her Broadway comeback in Follies at the age**of**74**.**
Born and raised in Buffalo

**,**NY**,**Nile came from**a**musical family__—__**his**grandfather was**a**vaudeville pianist who played with Bill " Bojangles " Robinson and Eddie__Cantor__;**his**uncles played boogie-woogie**.**
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**.**0.277 seconds.