German mathematician who created set theory and proved, against the intuitions of most of his contemporaries, that infinite sets come in different sizes: the real numbers are strictly more numerous than the natural numbers, shown by his 1891 diagonal argument. He introduced the notion of cardinality (aleph numbers) to measure the sizes of infinite sets and proposed the continuum hypothesis about the possible sizes between them. His work was fiercely opposed by influential contemporaries, notably Leopold Kronecker, and he suffered repeated episodes of severe depression later in life, dying in a sanatorium in Halle.
Facts
BirthplaceSaint Petersburg, Russian Empire 2 Death PlaceHalle, German Empire, in a sanatorium 2 Nationality / Culture Defining ContributionCreated set theory and proved that infinite sets have different sizes (the diagonal argument), overturning the assumption that all infinities are the same infinity. 3 Notable WorkUeber eine elementare Frage der Mannigfaltigkeitslehre (1891, the diagonal argument paper) 3 Connections
Associated With
Georg Cantor introduced the set in 1883 as an example within his work on point-set topology and the theory of trigonometric series.
Source Cantor Set (Wikipedia)
Source Wikipedia: Cantor's Diagonal Argument
Infinity, Concepts Founded the rigorous mathematical treatment of completed infinities and their differing sizes, from 1874.
Source The Stanford Encyclopedia of Philosophy
Weierstrass was Cantor's teacher in Berlin; the atlas has no dedicated advisor and student relation, so this is recorded as a general association.
Source Karl Weierstrass, Biography (MacTutor History of Mathematics)
Founder of set theory, 1870s.
Source Set Theory (Wikipedia)
Sets, Concepts Created set theory from 1874 onward.
Source The Stanford Encyclopedia of Philosophy
Attributed Works
Source Absolute Infinite (Wikipedia)
Source Actual and Potential Infinity (Wikipedia)
Source Wikipedia: Cardinal number
Source Wikipedia: Cantor's Diagonal Argument
Source Wikipedia: Cardinal number
Source Cardinality (Wikipedia)
Source Cardinality of the Continuum (Wikipedia)
Source Countable Set (Wikipedia)
Source Wikipedia: Ordinal number
Conjectures Posed
Source Continuum Hypothesis (Wikipedia)
In Branch
Source The Stanford Encyclopedia of Philosophy
Source The Stanford Encyclopedia of Philosophy
Namesake Of
Proofs Credited
Source Cantor's Intersection Theorem (Wikipedia)
Source Cantor's Theorem (Wikipedia)
In the Other Atlases
Nationality
Also in Geography Atlas: the nationality of this subject.
Sources
1. Wikipedia: Georg Cantor
WikipediaLead section, gender referenceQuote, Lead section, gender reference
He defined the cardinal and ordinal numbers and their arithmetic.
2. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statistics
3. The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford University
Cantor's Theorem (Wikipedia)
Wikimedia FoundationProofs Credited: Cantor's Theorem, proof and attribution sectionQuote, Proofs Credited: Cantor's Theorem, proof and attribution section
the diagonal argument for the uncountability of the reals also first appears
Continuum Hypothesis (Wikipedia)
Wikimedia FoundationConjectures Posed: Continuum Hypothesis, HistoryQuote, Conjectures Posed: Continuum Hypothesis, History
Cantor first introduced the continuum hypothesis in his 1878 paper "Ein Beitrag zur Mannigfaltigkeitslehre", in a form now called the weak continuum hypothesis which is equivalent to the standard formulation under the then-undeveloped axiom of choice.
Wikipedia: Cantor's Diagonal Argument
Wikimedia FoundationAssociated With: Cantor's Diagonal Argument, lead paragraph, publication date
Georg Cantor published this proof in 1891
Attributed Works: Cantor's Diagonal Argument, Lead section
Cantor's diagonal argument (among various similar names) is a mathematical proof
View the Source Set Theory (Wikipedia)
WikipediaAssociated With: Set Theory, Origins sectionQuote, Associated With: Set Theory, Origins section
Cantor is commonly considered the founder of set theory
Karl Weierstrass, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsAssociated With: Karl Weierstrass, Students sectionQuote, Associated With: Karl Weierstrass, Students section
In 1870 Sofia Kovalevskaya came to Berlin and Weierstrass taught her privately since she was not allowed admission to the university.
Cantor Set (Wikipedia)
Absolute Infinite (Wikipedia)
Attributed Works: Absolute Infinite, Lead paragraphQuote, Attributed Works: Absolute Infinite, Lead paragraph
proposed by mathematician Georg Cantor
Wikipedia: Ordinal number
Attributed Works: Ordinal Number, Lead sentence, attributionQuote, Attributed Works: Ordinal Number, Lead sentence, attribution
Ordinals were introduced by Georg Cantor in 1883
Cardinality of the Continuum (Wikipedia)
Attributed Works: Cardinality of the Continuum, Lead sentence, attributionQuote, Attributed Works: Cardinality of the Continuum, Lead sentence, attribution
This was proven by Georg Cantor in his uncountability proof of 1874
Cardinality (Wikipedia)
Wikimedia FoundationAttributed Works: Cardinality, LeadQuote, Attributed Works: Cardinality, Lead
It is considered to have been first introduced formally to mathematics by Georg Cantor at the turn of the 20th century.
Actual and Potential Infinity (Wikipedia)
Attributed Works: Actual and Potential Infinity, Lead paragraphQuote, Attributed Works: Actual and Potential Infinity, Lead paragraph
The concept of actual infinity was introduced into mathematics near the end of the 19th century by Georg Cantor with his theory of infinite sets, and was later formalized into Zermelo-Fraenkel set theory.
Countable Set (Wikipedia)
Wikimedia FoundationAttributed Works: Countable Set, Lead sectionQuote, Attributed Works: Countable Set, Lead section
The concept is attributed to Georg Cantor, who proved the existence of uncountable sets
Cantor's Intersection Theorem (Wikipedia)
Wikimedia FoundationProofs Credited: Cantor's Intersection Theorem, Lead sectionQuote, Proofs Credited: Cantor's Intersection Theorem, Lead section
named after Georg Cantor
Wikipedia: Cardinal number
Attributed Works: Aleph Number, Lead
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874-1884
Attributed Works: Cardinal Number, Lead
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874-1884
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