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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.

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## Some Related Sentences

Cantor and Dedekind

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**.**__Cantor__also mentioned the idea in his letters to Richard

__Dedekind__( text in square brackets not present in original ):

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**.**
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**.**
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**.**__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

**.**

Owing to the gigantic simultaneous efforts

**of**Frege**,**__Dedekind__**and**__Cantor__**,**the infinite was set**on****a**throne**and**revelled in its total triumph**.**
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**.**__Dedekind__

**,**Riemann

**,**Moritz

__Cantor__

**and**Alfred Enneper

**,**although they had all already earned their PhDs

**,**attended Dirichlet's classes to study with him

**.**

* the various ( but equivalent ) constructions

**of**the real numbers by__Dedekind__**and**__Cantor__resulting in the modern axiomatic definition**of**the real number field ;
The first three

**,**due to Georg__Cantor__/ Charles Méray**,**Richard__Dedekind__**and**Karl Weierstrass / Otto Stolz all occurred within**a**few years**of**each other**.**
Bolzano's posthumously published work Paradoxien des Unendlichen ( The Paradoxes

**of**the Infinite ) was greatly admired by many**of**the eminent logicians who came after him**,**including Charles Sanders Peirce**,**Georg__Cantor__**,****and**Richard__Dedekind__**.**
Given the

__Cantor__–__Dedekind__axiom**,**this algorithm can be regarded as an algorithm to decide the truth**of**any statement in Euclidean geometry**.**
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**.**
Of

**a**totally different orientation < nowiki > the " Old Formalist School "**of**Richard__Dedekind__|__Dedekind__**,**Georg__Cantor__|__Cantor__**,**Giuseppe Peano | Peano**,**Ernst Zermelo | Zermelo**,****and**Louis Couturat | Couturat**,**etc .< nowiki ></ nowiki > was the Pre-Intuitionist School**,**mainly led by Poincaré**,**Borel**and**Lebesgue**.**
In mathematical logic

**,**the phrase__Cantor__–__Dedekind__axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear**continuum****of**geometry**.**

Cantor and were

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 ).__Cantor__Fitzgerald's corporate headquarters

**and**New York City office

**,**

**on**the 101st-105th floors

**of**One World Trade Center in Lower Manhattan ( 2-6 floors above the impact zone

**of**

**a**hijacked airliner ),

__were__destroyed during the September 11

**,**2001 attacks

**.**

The objections to his work

__were__occasionally fierce: Poincaré referred to Cantor's ideas as**a**" grave disease " infecting the discipline**of**mathematics**,****and**Kronecker's public opposition**and**personal attacks included describing__Cantor__as**a**" scientific charlatan ",**a**" renegade "**and****a**" corrupter**of**youth**.**
In 1860

**,**__Cantor__graduated with distinction from the Realschule in Darmstadt ; his exceptional skills in mathematics**,**trigonometry in particular**,**__were__noted**.**
Before

__Cantor__**,**there__were__only finite sets ( which are easy to understand )**and**" the infinite " ( which was considered**a**topic for philosophical**,**rather than mathematical**,**discussion ).
Several hit songs

__were__also recorded during the mahjong fad**,**most notably " Since Ma is Playing Mah Jong " by Eddie__Cantor__**.**
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**.**
In 1951 the duo became one

**of**the rotating hosts**of**The Colgate Comedy Hour ( Eddie__Cantor__**and**Bob Hope__were__among the others )**and**the following year they inaugurated their own situation comedy**,**The Abbott**and**Costello Show**.**
Eddie

__Cantor__**,**Fred Allen**,**Ethel Merman**and**Smith**and**Dale__were__among the top acts seen in Paramount shorts**.**
The implications

**of**the infinite void__were__revolutionary ; to have pursued them would have threatened the singular relationship**of**man**and**this natural world to God (__Cantor__2001 ); in it he treated theology mathematically**.**
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**.**
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**.**
Among popular artists

**of**the time who recorded for " Hit**of**the Week Records "__were__Gene Austin**,**Duke Ellington ( as Harlem Hot Chocolates ), Eddie__Cantor__**,**Morton Downey**,****and**Rudy Vallee**.**
In 1933

**,****a**set**of**12 Eddie__Cantor__caricatures by Frederick J**.**Garner__were__published by Brown & Bigelow**.**
Eddie

**and**Ida__Cantor____were__seen in**a**brief prologue**and**epilogue set in**a**projection room**,**where they are watching Brasselle in action ; at the end**of**the film Eddie tells Ida**,**" I never looked better in my life "...**and**gives the audience**a**knowing**,**incredulous look**.**
In 1749 he was involved in

**a**conflict with the__Cantor__**of**the Liebfrauenkirche**,**Gottfried Mittag**,**who had misappropriated funds that__were__due to Friedemann**.**
Both

__were__especially hard-hit by 9 / 11**,**as Joanne's brother**and**Rick's closest friend**,**Billy Minardi**,**was working as**a**bond trader for__Cantor__Fitzgerald**on**the 105th floor**of**the north tower**of**the World Trade Center when it was struck by American Airlines Flight 11**.**
His earliest movie jobs

__were__**on**Samuel Goldwyn's Eddie__Cantor__musicals**,**where he began**developing**such techniques as**a**“ parade**of**faces ” ( individualizing each chorus girl with**a**loving close-up ),**and**moving his dancers all over the stage (**and**often beyond ) in as many kaleidoscopic patterns as possible**.****As**radio

**and**television matured

**,**many

**of**its most famous comedians

**,**including Jack Benny

**,**Sid Caesar

**,**George Burns

**,**Eddie

__Cantor__

**,**Jack Carter

**,**Henny Youngman

**and**Milton Berle

**,**

__were__Jewish

**.**

Some

**of**them__were__Jack Benny**,**Burns**and**Allen**,**Fred Allen**,**Eddie__Cantor__**,****and**Rudy Vallee**,**among others**.**0.150 seconds.