TheInfoList

Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

. He created
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, which has become a fundamental theory in mathematics. Cantor established the importance of
one-to-one correspondence In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
between the members of two sets, defined
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American mus ...
and well-ordered sets, and proved that the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s are more numerous than the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

of infinities. He defined the
cardinal Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardin ...
and
ordinal Ordinal may refer to: * Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale * Ordinal date, a simple form of expressing a date using only the year and the day number within that year * O ...
numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Cantor's theory of
transfinite number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s was originally regarded as so counter-intuitive – even shocking – that it encountered
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''Th ...
from mathematical contemporaries such as
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...

and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the s ...
and later from
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens o ...

and L. E. J. Brouwer, while
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationali ...

raised philosophical objections. Cantor, a devout
Lutheran Christian Lutheranism is one of the largest branches of Protestantism that identifies with the teachings of Jesus Christ and was founded by Martin Luther, a 16th-century German monk and Protestant Reformers, reformer whose efforts to reform the theology ...
, believed the theory had been communicated to him by God. Dauben 2004, pp. 8, 11, 12–13. Some
Christian theologians #REDIRECT Christian theology #REDIRECT Christian theology #REDIRECT Christian theology Christian theology is the theology of Christianity, Christian belief and practice. * help them better understand Christian tenets * make comparative religion, ...
(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 Pantheism is the belief that reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary Imaginary may refer to: * Imaginary (sociology), a concept in sociology * The ...

– a proposition that Cantor vigorously rejected. The objections to Cantor's work were occasionally fierce:
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...

'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 A mental disorder, also called a mental illness or psychiatric disorder, is a behavioral or mental pattern that causes significant distress or impairment of ...

. 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 called 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 In internal medici ...
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 A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exis ...
awarded Cantor its
Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society A learned society (; also known as a learned academy, scho ...
, the highest honor it can confer for work in mathematics.
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."

# Life of Georg Cantor

## Youth and studies

Georg Cantor was born in 1845 in the western merchant colony of

, Russia, and brought up in the city until he was eleven. Cantor, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist
Joseph Böhm Joseph Böhm ( hu, Böhm József; 4 April 1795 – 28 March 1876) was a violin The violin, sometimes known as a '' fiddle'', is a wooden chordophone ( string instrument) in the violin family. Most violins have a hollow wooden body. It is th ...
'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 The Stock Exchange Saint-Petersburg (SPBEX) is located in Saint Petersburg, Russia. The stock exchange was founded on January 31, 1991, and it is now the third most active stock exchange in Russia by volume, and the largest outside of Moscow. SPBE ...
; 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 German states, state in Germany. Its ...

, then to
Frankfurt Frankfurt, officially Frankfurt am Main (; Hessian dialects, Hessian: , "Franks, Frank ford (crossing), ford on the Main (river), Main"; french: Francfort-sur-le-Main), is the most populous city in the States of Germany, German state of Hess ...

, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in
Darmstadt Darmstadt (, also , , ) is a city in the States of Germany, state of Hesse in Germany, located in the southern part of the Frankfurt Rhine Main Area, Rhine-Main-Area (Frankfurt Metropolitan Region). Darmstadt has around 160,000 inhabitants, m ...

; his exceptional skills in mathematics,
trigonometry Trigonometry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the
Technische Universität Darmstadt The Technische Universität Darmstadt (official English name Technical University of Darmstadt, sometimes also referred to as Darmstadt University of Technology), commonly known as TU Darmstadt, is a research university in the city of Darmstadt ...

. In 1862, Cantor entered the Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 1863, Cantor shifted his studies to the
University of Berlin Humboldt University of Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an ...
, attending lectures by
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...

,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

and
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see ...
. He spent the summer of 1866 at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...
, then and later a center for mathematical research. Cantor was a good student, and he received his doctorate degree in 1867.

## Teacher and researcher

Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, Cantor took up a position at the
University of Halle Martin Luther University of Halle-Wittenberg (german: Martin-Luther-Universität Halle-Wittenberg), also referred to as MLU, is a public, research Research is "creative and systematic work undertaken to increase the stock of knowledge" ...
, where he spent his entire career. He was awarded the requisite
habilitation Habilitation is the procedure to achieve the highest university degree in many European countries in which the candidate fulfills certain criteria set by the university which require excellence in research, teaching, and further education. Its qu ...
for his thesis, also on number theory, which he presented in 1869 upon his appointment at
Halle University Martin Luther University of Halle-Wittenberg (german: Martin-Luther-Universität Halle-Wittenberg), also referred to as MLU, is a public, research-oriented university in the cities of Halle, Saxony-Anhalt, Halle and Wittenberg in the State of Ge ...
. In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the
Harz mountains The Harz () is a highland area in northern Germany. It has the highest elevations for that region, and its rugged terrain extends across parts of Lower Saxony Lower Saxony (german: Niedersachsen ; nds, Neddersassen; stq, Läichsaksen) is ...

, Cantor spent much time in mathematical discussions with
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citiz ...
, whom he had met two years earlier while on Swiss 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 One of the basic pieces of furniture Furniture refers to movable objects intended to support various human activities such as seating (e.g., chairs, stools, and sofas), eating (table (furniture), tables), and sleeping (e.g., beds). Furn ...

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 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 it 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 Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and
Franz Mertens Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a Polish mathematician. He was born in Środa Wielkopolska, Schroda in the Grand Duchy of Posen, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in ...

, 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 mathematics. He also began an intense study of
Elizabethan literature Elizabethan literature refers to bodies of work produced during the reign of Queen Elizabeth I Elizabeth I (7 September 153324 March 1603) was Queen of England and Ireland Ireland ( ; ga, Éire ; Ulster-Scots: ) is an island ...
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 for England and Wales, Attorney General and as Lord Chancellor of K ...

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

(see
Shakespearean authorship question Image:ShakespeareCandidates1.jpg, alt=Portraits of Shakespeare and four proposed alternative authors, Oxford, Bacon, Derby, and Marlowe (clockwise from top left, Shakespeare centre) have each been proposed as the true author. ''(Clickable image ...
); 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 A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: *Cantor's diagonal argument (the earliest) *Cantor's theorem *Russell's paradox *Diagonal lemma **Gödel's incompleteness theorems, Gödel's first ...
and
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. 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 The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or ...
and 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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
, which was held in Zürich, Switzerland, in 1897.

## Later years and death

After Cantor's 1884 hospitalization, there is no record that he was in any
sanatorium A sanatorium (also spelled sanitarium or sanitorium) is a medical facility for long-term illness, most typically associated with the treatment of tuberculosis Tuberculosis (TB) is an infectious disease An infection is the invasion o ...

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 The Baconian theory of Shakespeare authorship question, Shakespeare authorship holds that Francis Bacon, Sir Francis Bacon, philosopher, essayist and scientist, wrote the Shakespeare's plays, plays which were publicly attributed to William Shakes ...
and
William Shakespeare William Shakespeare (bapt. 26 April 1564 – 23 April 1616) was an English playwright, poet and actor, widely regarded as the greatest writer in the English language and the world's greatest dramatist. He is often called England's national p ...

), 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 International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
. 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 Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for ...
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 in 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 In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of ...
,
Cantor's paradox In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of ...
, and
Russell's paradox In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...

) to a meeting of the ''Deutsche Mathematiker-Vereinigung'' in 1903, and attending the International Congress of Mathematicians at
Heidelberg Heidelberg () is a university town in the German state The Federal Republic of Germany, as a federal state, consists of sixteen partly sovereign federated states (german: Land (state), plural (states); commonly informally / federated s ...

in 1904. In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the
University of St. Andrews (Aien aristeuein) , motto_lang = grc , mottoeng = Ever to ExcelorEver to be the Best , established = , type = Public In public relations and communication science, publics are groups of individual people, and the public (a.k.a. the g ...
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 polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
, whose newly published ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fun ...
'' repeatedly cited Cantor's work, but this 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, living in poverty and suffering from
malnourishment Malnutrition is 'a state of nutrition in which a deficiency or excess (or imbalance) of energy, protein and other nutrients causes measurable adverse effect on tissue and body form (body shape, size and composition) and function and clinical ou ...
during
World War I World War I, often abbreviated as WWI or WW1, also known as the First World War or the Great War, was a global war A world war is "a war engaged in by all or most of the principal nations of the world". The term is usually reserved for ...

. 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 of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
. 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 A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

. 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 that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
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 (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

,
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
and
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

) 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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is "more numerous" than the set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 correspondence In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
s (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define
finite Finite is the opposite of infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infin ...
and
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch ...
s, subdividing the latter into
denumerable In mathematics, a Set (mathematics), set is countable if it has the same cardinality (the cardinal number, number of elements of the set) as some subset of the set of natural numbers N = . Equivalently, a set ''S'' is ''countable'' if there exist ...
(or countably infinite) sets and nondenumerable sets (uncountably infinite sets). Cantor developed important concepts in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and their relation to
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. For example, he showed that the
Cantor set 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. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

, discovered by
Henry John Stephen Smith Prof Henry John Stephen Smith FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * F ...

in 1875, is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the
rationals In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. Cantor introduced fundamental constructions in set theory, such as the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of a set ''A'', which is the set of all possible
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 In elementary set theory, Cantor's theorem is a fundamental result which states that, for any Set (mathematics), set A, the set of all subsets of A (the power set of A, denoted by \mathcal(A)) has a strictly greater cardinality than A itself. F ...
. Cantor developed an entire theory and arithmetic of infinite sets, called
cardinals Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardina ...
and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $\aleph$ (
aleph Aleph (or alef or alif, transliterated ʾ) is the first letter Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that indicates, sig ...
) with a natural number subscript; for the ordinals he employed the Greek letter ω (
omega Omega (; capital Capital most commonly refers to: * Capital letter Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (o ...

). This notation is still in use today. The ''
Continuum hypothesis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
'', introduced by Cantor, was presented by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
as the first of his twenty-three open problems in his address at the 1900
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, ian, mathematician and scientist who is sometimes known as "the father of ". He was known as a somewhat unusual character. Educated as a chemist an ...

praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897,
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, se ...

and
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily t ...
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 Philip Edward Bertrand Jourdain (16 October 1879 – 1 October 1919) was a British logician and follower of Bertrand Russell. He was born in Ashbourne, Derbyshire, Ashbourne in Derbyshire* one of a large family belonging to Emily Clay and his f ...

on the history of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
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 on
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, his thesis topic. At the suggestion of
Eduard Heine Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
, the Professor at Halle, Cantor turned to
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
. Heine proposed that Cantor solve an open problem that had eluded
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For c ...

,
Rudolf Lipschitz Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a Germany, German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity, Lipschitz continuity condition) and different ...
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
, and Heine himself: the uniqueness of the representation of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
by
trigonometric series In mathematics, a trigonometric series is a series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment ...
. 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''''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 point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), s ...
s 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 number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s as
convergent sequences In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real number, real or complex numbers. Equivalently, it is a function space whose elements are functions from the ...
of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 cut In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s of his contemporaries
Otto Stolz Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Something associated wi ...
and
Paul du Bois-Reymond Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For ...
, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s.

## Set theory

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