Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of

_{''n''} of a set ''S'' of zeros of a trigonometric series. 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''_{1} as its set of zeros, where ''S''_{1} is the set of limit points of ''S''. If ''S''_{''k+1''} is the set of limit points of ''S''_{''k''}, then he could construct a trigonometric series whose zeros are ''S''_{''k+1''}. Because the sets ''S''_{''k''} were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets ''S'', ''S''_{1}, ''S''_{2}, ''S''_{3},... formed a limit set, which we would now call ''S''_{''ω''}, and then he noticed that ''S''_{ω} would also have to have a set of limit points ''S''_{ω+1}, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ''ω'', ''ω'' + 1, ''ω'' + 2, ...
Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of

set theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...

, which has become a fundamental theory in mathematics. 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 number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s are more numerous than the natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

s. In fact, Cantor's method of proof of this theorem implies the existence of an infinity
Infinity is that which is boundless, endless, or larger than any natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples ar ...

of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...

raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Dauben 2004, pp. 8, 11, 12–13. 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 Dauben 1977, p. 86; Dauben 1979, pp. 120, 143. – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. It is important to note that not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).
The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, Dauben 1979, p. 280: "... the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression. though some have explained these episodes as probable manifestations of a bipolar disorder
Bipolar disorder, previously known as manic depression, is a mental disorder characterized by periods of depression and periods of abnormally elevated mood that last from days to weeks each. If the elevated mood is severe or associated wit ...

. Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness
A mental disorder, also referred to as a mental illness or psychiatric disorder, is a behavioral or mental pattern that causes significant distress or impairment of personal functioning. Such features may be persistent, relapsing and remitt ...

as "cyclic manic-depression".
The harsh criticism has been matched by later accolades. In 1904, the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, ...

awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."
Biography

Youth and studies

Georg Cantor, born in 1845 in the western merchant colony ofSaint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...

, Russia, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden
Wiesbaden () is a city in central western Germany and the capital of the state of Hesse
Hesse (, , ) or Hessia (, ; german: Hessen ), officially the State of Hessen (german: links=no, Land Hessen), is a state in Germany
Germany ...

, then to Frankfurt
Frankfurt, officially Frankfurt am Main (; Hessian: , "Frank
Frank or Franks may refer to:
People
* Frank (given name)
* Frank (surname)
* Franks (surname)
* Franks
The Franks ( la, Franci or ) were a group of Germanic peoples w ...

, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry
Trigonometry () is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. T ...

in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867.
Teacher and researcher

Cantor submitted his dissertation onnumber theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number
In mathematics, the natura ...

at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including ...

for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle University.
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met at Interlaken in Switzerland
two years earlier while on holiday.
Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair
A chair is a type of seat, typically designed for one person and consisting of one or more legs, a flat or slightly angled seat and a back-rest. They may be made of wood, metal, or synthetic materials, and may be padded or upholstered in var ...

at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Dauben 1979, p. 163. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, Dauben 1979, p. 34. perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.
In 1881, Cantor's Halle colleague Eduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
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''. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to ''Acta''. Dauben 1979, p. 138. He asked Cantor to withdraw the paper from ''Acta'' while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about ''Acta Mathematica''." Dauben 1979, p. 139.
Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of Elizabethan literature, thinking there might be evidence that Francis Bacon
Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England
England is a country that ...

wrote the plays attributed to William Shakespeare
William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...

(see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.
Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
In 1889, Cantor was instrumental in founding the German Mathematical Society, and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which took place in Zürich, Switzerland, in 1897.
Later years and death

After Cantor's 1884 hospitalization there is no record that he was in any sanatorium again until 1899. Dauben 1979, p. 282. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views on Baconian theory andWilliam Shakespeare
William Shakespeare ( 26 April 1564 – 23 April 1616) was an English playwright, poet and actor. He is widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist. He is often called England's natio ...

), and this tragedy drained Cantor of much of his passion for mathematics. Dauben 1979, p. 283. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Dauben 1979, p. 248. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ( Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the ''Deutsche Mathematiker-Vereinigung'' in 1903, and attending the International Congress of Mathematicians at Heidelberg
Heidelberg (; Palatine German: ''Heidlberg'') is a city in the German state of Baden-Württemberg, situated on the river Neckar in south-west Germany
Germany,, officially the Federal Republic of Germany, is a country in Centra ...

in 1904.
In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is t ...

, whose newly-published '' Principia Mathematica'' repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.
Cantor retired in 1913, and lived in poverty and suffering from malnourishment during World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe
Europe is a large peninsula conventionally considered a continent in ...

. Dauben 1979, p. 284. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.
Mathematical work

Cantor's work between 1874 and 1884 is the origin ofset theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...

. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

. No one had realized that set theory had any nontrivial content. 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). 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. Set theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...

has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra
Algebra () is one of the broad areas of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their c ...

, analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics
Mathematics is an area of kno ...

, and topology
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.
In one of his earliest papers, Cantor proved that the set of real number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s is "more numerous" than the set of natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

s; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).
Cantor developed important concepts in topology
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

and their relation to cardinality. For example, he showed that the Cantor set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

, discovered by Henry John Stephen Smith in 1875, is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

.
Cantor introduced fundamental constructions in set theory, such as the power set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

of a set ''A'', which is the set of all possible subset
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

s of ''A''. He later proved that the size of the power set of ''A'' is strictly larger than the size of ''A'', even when ''A'' is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $\backslash aleph$ (aleph
Aleph (or alef or alif, transliterated ʾ) is the first letter of the Semitic abjads, including Phoenician , Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the s ...

) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega
Omega (; capital: Ω, lowercase: ω; Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world
Ancient history is a time period from the beginning of writing and ...

). This notation is still in use today.
The '' Continuum hypothesis'', introduced by Cantor, was presented by David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory
Set theory is the branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...

and on Cantor's religious ideas. This was later published, as were several of his expository works.
Number theory, trigonometric series and ordinals

Cantor's first ten papers were onnumber theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number
In mathematics, the natura ...

, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics
Mathematics is an area of kno ...

. Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices ''n'' in the ''n''th derived set ''S''rational number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

s. 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. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.
Set theory

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Ueber