Editing Georg Cantor

{{Short description|German mathematician (1845–1918)}}
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{{Use dmy dates|date=August 2016}}
{{Infobox scientist
| name              = Georg Cantor
| image             = Georg Cantor (Porträt).jpg
| caption           = Cantor, {{circa|1910}}
| birth_name        = Georg Ferdinand Ludwig Philipp Cantor
| birth_date        = {{Birth date|1845|3|3|df=y}}
| birth_place       = [[Saint Petersburg]], [[Russian Empire]]
<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->
| death_date        = {{Death date and age|1918|1|6|1845|3|3|df=y}}
| death_place       = [[Halle (Saale)|Halle]], Province of Saxony<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->, German Empire<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->
| nationality       = [[Germans|German]]-[[Russians|Russian]]
| alma_mater        = {{plainlist|
* [[ETH Zurich|Swiss Federal Polytechnic]]
* [[Humboldt University of Berlin|University of Berlin]]
* [[University of Göttingen]]
}}
| thesis_title      = De aequationibus secundi gradus indeterminatis
| thesis_url        = https://web.archive.org/web/20150704015954/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN237853094&DMDID=DMDLOG_0008&L=1
| thesis_year       = 1867
| doctoral_advisor  = {{plainlist|
* [[Ernst Kummer]]
* [[Karl Weierstrass]]}}
| doctoral_students = 
| known_for         = [[Set theory]]
| spouse            = {{marriage|Vally Guttmann|1874}}
| field             = [[Mathematics]]
| work_institutions = [[University of Halle]]
| prizes            = [[Sylvester Medal]] (1904)
| signature         = Georg Cantor Signature.png
}}

'''Georg Ferdinand Ludwig Philipp Cantor''' ({{IPAc-en|ˈ|k|æ|n|t|ɔr}} {{respell|KAN|tor}}; {{IPA|de|ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯|lang}}; {{OldStyleDate|3 March|1845|19 February}}&nbsp;– 6 January 1918<ref>[[#Guinness2000|Grattan-Guinness 2000]], p. 351.</ref>) was a <!--please do not add a nationality--> mathematician who played a pivotal role in the creation of [[set theory]], which has become a [[foundations of mathematics|fundamental theory]] in mathematics. Cantor established the importance of [[one-to-one correspondence]] between the members of two sets, defined [[infinite set|infinite]] and [[well-order|well-ordered sets]], and proved that the [[real number]]s are more numerous than the [[natural number]]s. Cantor's method of proof of this theorem implies the existence of an [[infinity]] of infinities. He defined the [[cardinal number|cardinal]] and [[ordinal number|ordinal]] numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.<ref>The biographical material in this article is mostly drawn from [[#Dauben1979|Dauben 1979]]. [[#Guinness1971|Grattan-Guinness 1971]], and [[#Purkert|Purkert and Ilgauds 1985]] are useful additional sources.</ref>

Originally, Cantor's theory of [[transfinite number]]s was regarded as counter-intuitive&nbsp;– even shocking. This caused it to encounter resistance from mathematical contemporaries such as [[Leopold Kronecker]] and [[Henri Poincaré]]<ref>[[#Dauben2004|Dauben 2004]], p. 1.</ref> and later from [[Hermann Weyl]] and [[L.&nbsp;E.&nbsp;J. Brouwer]], while [[Ludwig Wittgenstein]] raised [[philosophical objections to Cantor's theory|philosophical objections]]; see [[Controversy over Cantor's theory]]. Cantor, a devout [[Lutheran Christian]],<ref>{{cite book|last1=Dauben|first1=Joseph Warren|title=Georg Cantor His Mathematics and Philosophy of the Infinite|url=https://archive.org/details/georgcantorhisma0000daub|url-access=registration|date=1979|publisher=[[Princeton University Press]]|isbn=9780691024479|pages=introduction}}</ref> believed the theory had been communicated to him by God.<ref name = "xdpfir">[[#Dauben2004|Dauben 2004]], pp.&nbsp;8, 11, 12–13.</ref> Some [[Christian theology|Christian theologians]] (particularly [[Neo-Scholasticism|neo-Scholastics]]) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God<ref name = "nuozkv">[[#Dauben1977|Dauben 1977]], p. 86; [[#Dauben1979|Dauben 1979]], pp.&nbsp;120, 143.</ref>&nbsp;– on one occasion equating the theory of transfinite numbers with [[pantheism]]<ref name = "daub77102"/>&nbsp;– a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher {{ill|Konstantin Gutberlet|de|Konstantin Gutberlet|it}} was in favor of it and Cardinal [[Johann Baptist Franzelin]] accepted it as a valid theory (after Cantor made some important clarifications).{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}}

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".<ref>[[#Dauben2004|Dauben 2004]], p. 1; [[#Dauben1977|Dauben 1977]], p. 89 ''15n''.</ref> 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".{{sfn|Rodych|2007}} 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,<ref name="daub280">[[#Dauben1979|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.</ref> though some have explained these episodes as probable manifestations of a [[bipolar disorder]].<ref name="bipolar">[[#Dauben2004|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]] as "cyclic manic-depression".</ref>

The harsh criticism has been matched by later accolades. In 1904, the [[Royal Society]] awarded Cantor its [[Sylvester Medal]], the highest honor it can confer for work in mathematics.<ref name = "daub248"/> [[David Hilbert]] defended it from its critics by declaring, "No one shall expel us from the [[Cantor's paradise|paradise that Cantor has created]]."<ref>{{harvtxt|Hilbert|1926|p=170}}: "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.")</ref><ref name="encomium"/>

== Biography ==

===Youth and studies===
[[File:Georg Cantor3.jpg|thumb|left|Cantor, around 1870]]
Georg Cantor, born in 1845 in [[Saint Petersburg]], Russian Empire<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->, 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.<ref>[http://dic.academic.ru/dic.nsf/enc_music/924/Бём ru: The musical encyclopedia (Музыкальная энциклопедия)].</ref> Cantor's father had been a member of the [[Saint Petersburg Bourse|Saint Petersburg stock exchange]]; when he became ill, the family moved to Germany in 1856, first to [[Wiesbaden]], then to [[Frankfurt]], 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]] in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the [[Technische Universität Darmstadt]].<ref>{{Cite web|url=http://www-groups.dcs.st-and.ac.uk/history/Biographies/Cantor.html|title=Georg Cantor (1845-1918)|website= www-groups.dcs.st-and.ac.uk|access-date= 2019-09-14}}</ref><ref>{{Cite book|title= Georg Cantor 1845-1918|publisher= Birkhauser|year= 1985|isbn= 978-3764317706}}</ref> In 1862 Cantor entered the [[ETH Zurich|Swiss Federal Polytechnic]] in Zurich. After receiving a substantial inheritance upon his father's death in June 1863,<ref name=":0">{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html|title=Cantor biography|website=www-history.mcs.st-andrews.ac.uk|access-date=2017-10-06}}</ref> Cantor transferred to the [[Humboldt University of Berlin|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.<ref name=":0" /><ref name=":1">{{Cite book|title=Math and mathematicians: the history of math discoveries around the world|last1= Bruno|first1= Leonard C.|year= 1999|publisher= U X L|last2= Baker|first2= Lawrence W.|isbn= 978-0787638139 |location=Detroit, Mich.|page=[https://archive.org/details/mathmathematicia00brun/page/54 54]|oclc= 41497065|url=https://archive.org/details/mathmathematicia00brun/page/54}}</ref>

===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, he took up a position at the [[Martin Luther University of Halle-Wittenberg|University of Halle]], where he spent his entire career. He was awarded the requisite [[habilitation]] for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.<ref name=":1" /><ref>{{cite web |author1=O'Connor, John J |author2=Robertson, Edmund F. |year= 1998 |url= http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html |title= Georg Ferdinand Ludwig Philipp Cantor |publisher=MacTutor History of Mathematics}}</ref>

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| 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.<ref>{{cite web |last1=O'Connor |first1=JJ |last2=Robertson |first2=E F |title=Georg Ferdinand Ludwig Philipp Cantor |url=https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |website=Maths History |publisher=University of St Andrews |access-date=9 February 2025 |archive-url=https://archive.today/20250209194516/https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |archive-date=9 February 2025 |language=English |date=October 1998 |quote=They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind. |url-status=live}}</ref>

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879.<ref name=":1" /><ref name=":0" /> To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a [[professor|chair]] 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.<ref name="daub163">[[#Dauben1979|Dauben 1979]], p. 163.</ref> Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,<ref name="daub34">[[#Dauben1979|Dauben 1979]], p. 34.</ref> perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.<ref>[[#Dauben1977|Dauben 1977]], p. 89 ''15n.''</ref> 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.<ref name=":1" /> Kronecker, now seen as one of the founders of the [[Constructivism (mathematics)|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,<ref name=":1" /> so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.<ref>{{cite book |last1=Dauben |first1=Joseph Warren |title=GEORG CANTOR His Mathematics and Philosophy of the Infinite |date=20 September 1990 |publisher=Princeton University Press |isbn=978-0691024479 |url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Cantor/Cantor.html |access-date=9 February 2025 |archive-url=https://archive.today/20250209195322/https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Cantor/Cantor.html |archive-date=9 February 2025 |chapter=Chapter 6}}</ref>

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.<ref>{{cite web |last1=O'Connor |first1=JJ |last2=Robertson |first2=E F |title=Georg Ferdinand Ludwig Philipp Cantor |url=https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |website=Maths History |publisher=University of St Andrews |access-date=9 February 2025 |archive-url=https://archive.today/20250209194516/https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |archive-date=9 February 2025 |language=English |date=October 1998 |quote=They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind. |url-status=live}}</ref>

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.<ref>[[#Dauben1979|Dauben 1979]], pp. 2–3; [[#Guinness1971|Grattan-Guinness 1971]], pp. 354–355.</ref> 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''.<ref name="daub138">[[#Dauben1979|Dauben 1979]], p. 138.</ref> He asked Cantor to withdraw the paper from ''Acta'' while it was in proof, writing that it was "...&nbsp;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''."<ref name="daub139">[[#Dauben1979|Dauben 1979]], p. 139.</ref>

Cantor suffered his first known bout of depression in May 1884.<ref name=":0" /><ref name = "daub282"/> 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:

{{Blockquote|... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.<ref>[[#Dauben1979|Dauben 1979]], p. 136; [[#Guinness1971|Grattan-Guinness 1971]], pp. 376–377. Letter dated June 21, 1884.</ref>}}

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]] wrote the plays attributed to [[William Shakespeare]] (see [[Shakespearean authorship question]]); this ultimately resulted in two pamphlets, published in 1896 and 1897.<ref>[[#Dauben1979|Dauben 1979]], pp. 281–283.</ref>

Cantor recovered soon thereafter, and subsequently made further important contributions, including his [[Cantor's diagonal argument|diagonal argument]] and [[Cantor's theorem|theorem]]. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891.<ref name=":1" /> 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]],<ref name=":1" /> 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.<ref name=":1" />

===Later years and death===
After Cantor's 1884 hospitalization there is no record that he was in any [[sanatorium]] again until 1899.<ref name="daub282">[[#Dauben1979|Dauben 1979]], p. 282.</ref> Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (Cantor was delivering a lecture on his views on [[Baconian theory]] and [[William Shakespeare]]), and this tragedy drained Cantor of much of his passion for mathematics.<ref name="daub283">[[#Dauben1979|Dauben 1979]], p. 283.</ref> 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.<ref>For a discussion of König's paper see [[#Dauben1979|Dauben 1979]], pp. 248–250. For Cantor's reaction, see [[#Dauben1979|Dauben 1979]], pp. 248, 283.</ref> Although [[Ernst Zermelo]] demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God.<ref name="daub248">[[#Dauben1979|Dauben 1979]], p. 248.</ref> 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.<ref>[[#Dauben1979|Dauben 1979]], pp. 283–284.</ref> 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 [[German Mathematical Society|''Deutsche Mathematiker-Vereinigung'']] in 1903, and attending the International Congress of Mathematicians at [[Heidelberg]] 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]], 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 suffered from [[Malnutrition|malnourishment]] during [[World War I]].<ref name="daub284">[[#Dauben1979|Dauben 1979]], p. 284.</ref> 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 6 January 1918, in the sanatorium where he had spent the last year of his life.<ref name=":0" />

==Mathematical work==
Cantor's work between 1874 and 1884 is the origin of [[set theory]].<ref name="Johnson p. 55">{{Cite journal |last=Johnson |first= Phillip E.|year=1972|title=The Genesis and Development of Set Theory|journal=The Two-Year College Mathematics Journal|jstor=3026799|volume=3|issue=1|pages= 55–62|doi=10.2307/3026799}}</ref> 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]]. 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. [[Axiomatic set theory|Set theory]] has come to play the role of a [[foundations of mathematics|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]], [[Mathematical analysis|analysis]], and [[topology]]) 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.<ref>{{Cite book |title=Axiomatic Set Theory|first=Patrick|last=Suppes|author-link=Patrick Suppes|year=1972|publisher=Dover|isbn=9780486616308|page=1|url=https://books.google.com/books?id=sxr4LrgJGeAC&pg=PA1|quote=With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.... As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.}}</ref>

In one of his earliest papers,<ref>{{Harvnb|Cantor|1874}}</ref> Cantor proved that the set of [[real number]]s is "more numerous" than the set of [[natural number]]s; this showed, for the first time, that there exist infinite sets of different [[Cardinality|sizes]]. He was also the first to appreciate the importance of [[one-to-one correspondence]]s (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define [[finite set|finite]] and [[infinite set]]s, subdividing the latter into [[countable set|denumerable]] (or countably infinite) sets and [[uncountable set|nondenumerable set]]s (uncountably infinite sets).<ref>A [[countable set]] is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".</ref>

Cantor developed important concepts in [[topology]] and their relation to [[cardinality]]. For example, he showed that the [[Cantor set]], discovered by [[Henry John Stephen Smith]] in 1875,<ref>[https://www.maa.org/press/periodicals/convergence/the-cantor-set-before-cantor-a-mini-primary-source-project-for-analysis-and-topology-students The Cantor Set Before Cantor] {{Webarchive|url=https://web.archive.org/web/20220829183355/https://www.maa.org/press/periodicals/convergence/the-cantor-set-before-cantor-a-mini-primary-source-project-for-analysis-and-topology-students |date=29 August 2022 }} Mathematical Association of America</ref> is [[Nowhere dense set|nowhere dense]], but has the same cardinality as the set of all real numbers, whereas the [[Rational number|rationals]] are everywhere dense, but countable. He also showed that all countable dense [[total order|linear orders]] without end points are order-isomorphic to the [[rational numbers]].

Cantor introduced fundamental constructions in set theory, such as the [[power set]] of a set ''A'', which is the set of all possible [[subset]]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 [[Ordinal arithmetic|arithmetic of infinite sets]], called [[Cardinal number|cardinals]] and [[Ordinal number|ordinals]], which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter <math>\aleph</math> ([[ℵ]], [[aleph]]) with a natural number subscript; for the ordinals he employed the Greek letter <math>\omega</math> ({{script|Grek|ω}}, [[omega]]). This notation is still in use today.

The ''[[Continuum hypothesis]]'', introduced by Cantor, was presented by [[David Hilbert]] as the first of his [[Hilbert's problems|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.<ref name="encomium">{{Cite book |last=Reid |first=Constance|year=1996|title=Hilbert|place=New York|publisher=Springer-Verlag|isbn=978-0-387-04999-1|page=[https://archive.org/details/hilbert0000reid_y1l6/page/177 177]|url=https://archive.org/details/hilbert0000reid_y1l6/page/177}}</ref> 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]] 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]], his thesis topic. At the suggestion of [[Eduard Heine]], the Professor at Halle, Cantor turned to [[Mathematical analysis|analysis]]. Heine proposed that Cantor solve [[Open problem|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 (mathematics)|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 (mathematics)|derived set]] ''S''<sub>''n''</sub> 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''<sub>1</sub> as its set of zeros, where ''S''<sub>1</sub> is the set of [[limit point]]s of ''S''. If ''S''<sub>''k+1''</sub> is the set of limit points of ''S''<sub>''k''</sub>, then he could construct a trigonometric series whose zeros are ''S''<sub>''k+1''</sub>. Because the sets ''S''<sub>''k''</sub> were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets ''S'', ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>,... formed a limit set, which we would now call ''S''<sub>''ω''</sub>, and then he noticed that ''S''<sub>ω</sub> would also have to have a set of limit points ''S''<sub>ω+1</sub>, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ''ω'', ''ω''&nbsp;+&nbsp;1, ''ω''&nbsp;+&nbsp;2, ...<ref>{{Cite journal |last1=Cooke|first1=Roger|title=Uniqueness of trigonometric series and descriptive set theory, 1870–1985|journal=Archive for History of Exact Sciences| volume=45|page=281|year=1993|doi=10.1007/BF01886630|issue=4|s2cid=122744778}}</ref>

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining [[irrational number]]s as [[Sequence space|convergent sequences]] of [[rational number]]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]]s.  While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of [[infinitesimal]]s of his contemporaries [[Otto Stolz]] and [[Paul du Bois-Reymond]], describing them as both "an abomination" and "a [[cholera]] [[bacillus]] of mathematics".<ref name="Infinitesimal">{{Cite journal |author=Katz, Karin Usadi |author2=Katz, Mikhail G. |author2-link=Mikhail Katz |year=2012|title= A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography|journal= [[Foundations of Science]]|doi=10.1007/s10699-011-9223-1|volume =17|number=1|pages=51–89|arxiv=1104.0375|s2cid=119250310 }}</ref> Cantor also published an erroneous "proof" of the inconsistency of [[infinitesimal]]s.<ref>{{Cite journal |author=Ehrlich, P.| year=2006| title=The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes|journal=Arch. Hist. Exact Sci.| volume=60|number=1|pages=1–121| url=http://www.ohio.edu/people/ehrlich/AHES.pdf|doi=10.1007/s00407-005-0102-4| s2cid=123157068|url-status=dead| archive-url=https://web.archive.org/web/20130215061415/http://www.ohio.edu/people/ehrlich/AHES.pdf|archive-date=15 February 2013}}</ref>

===Set theory===
[[File:Diagonal argument 2.svg|right|thumb|250px|An illustration of [[Cantor's diagonal argument]] for the existence of [[uncountable set]]s.<ref>This follows closely the first part of Cantor's 1891 paper.</ref> The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.]]
The beginning of set theory as a branch of mathematics is often marked by the publication of [[Georg Cantor's first set theory article|Cantor's 1874 paper]],<ref name="Johnson p. 55"/> "Ueber<!--[sic; see 'Talk:Gottlob Frege#Ueber']--> eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").<ref>{{Harvnb|Cantor|1874}}. English translation: [[#Ewald|Ewald 1996]], pp. 840–843.</ref> This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be [[equinumerosity|equinumerous]] (that is, of "the same size" or having the same number of elements).<ref>For example, geometric problems posed by [[Galileo Galilei|Galileo]] and [[John Duns Scotus]] suggested that all infinite sets were equinumerous&nbsp;– see {{Cite journal |last=Moore |first= A. W. |date=April 1995|title=A brief history of infinity|journal=Scientific American|volume=272|issue=4|pages=112–116 (114)|doi=10.1038/scientificamerican0495-112|bibcode=1995SciAm.272d.112M}}</ref> Cantor proved that the collection of real numbers and the collection of positive [[integers]] are not equinumerous. In other words, the real numbers are not [[countable]]. His proof differs from the [[Cantor's diagonal argument|diagonal argument]] that he gave in 1891.<ref>For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., [[#Suppes|Suppes 1972]].</ref> Cantor's article also contains a new method of constructing [[transcendental number]]s. Transcendental numbers were first constructed by [[Joseph Liouville]] in 1844.<ref>Liouville, Joseph (13 May 1844). [http://bibnum.education.fr/mathematiques/theorie-des-nombres/propos-de-lexistence-des-nombres-transcendants A propos de l'existence des nombres transcendants].</ref>

Cantor established these results using two constructions. His first construction shows how to write the real [[algebraic number]]s<ref>The real algebraic numbers are the real [[root]]s of [[polynomial]] equations with [[integer]] [[coefficients]].</ref> as a [[sequence]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>,&nbsp;.... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs [[nested intervals]] whose [[intersection (set theory)|intersection]] contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence&nbsp;– that is, the real numbers are not countable.  By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more&nbsp;– namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.<ref>For more details on Cantor's article, see [[Georg Cantor's first set theory article]] and {{Cite journal|last=Gray|first=Robert|year=1994|url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf|title=Georg Cantor and Transcendental Numbers|journal=[[American Mathematical Monthly]]|volume=101|issue=9|pages=819–832|doi=10.2307/2975129|jstor=2975129|access-date=6 December 2013|archive-date=21 January 2022|archive-url=https://web.archive.org/web/20220121155859/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf|url-status=dead}}. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.</ref> Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.<ref>Cantor's construction starts with the set of transcendentals ''T'' and removes a countable [[subset]] {''t<sub>n</sub>''} (for example, ''t<sub>n</sub>''&nbsp;=&nbsp;''[[e (mathematical constant)|e]]&nbsp;/&nbsp;n''). Call this set ''T''<sub>0</sub>. Then ''T''&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t<sub>n</sub>''}&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t''<sub>2''n''-1</sub>}&nbsp;∪&nbsp;{''t''<sub>2''n''</sub>}. The set of reals '''R'''&nbsp;= ''T''&nbsp;∪&nbsp;{''a<sub>n</sub>''}&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t<sub>n</sub>''}&nbsp;∪&nbsp;{''a<sub>n</sub>''} where ''a<sub>n</sub>'' is the sequence of real algebraic numbers.  So both ''T'' and '''R''' are the union of three [[pairwise disjoint]] sets: ''T''<sub>0</sub> and two countable sets. A one-to-one correspondence between ''T'' and '''R''' is given by the function: ''f''(''t'')&nbsp;=&nbsp;''t'' if ''t''&nbsp;∈&nbsp;''T''<sub>0</sub>, ''f''(''t''<sub>2''n''-1</sub>)&nbsp;=&nbsp;''t<sub>n</sub>'', and ''f''(''t''<sub>2''n''</sub>)&nbsp;=&nbsp;''a<sub>n</sub>''. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals ({{harvnb|Cantor|1879|p=4}}).</ref>

Between 1879 and 1884, Cantor published a series of six articles in ''[[Mathematische Annalen]]'' that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a [[finitism|finite]] number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of [[actual infinity]] would open the door to paradoxes which would challenge the validity of mathematics as a whole.<ref name="popeleo">[[#Dauben1977|Dauben 1977]], p. 89.</ref> Cantor also introduced the [[Cantor set]] during this period.

The fifth paper in this series, "'''''Grundlagen einer allgemeinen Mannigfaltigkeitslehre"''''' ("''Foundations of a General Theory of Aggregates"''), published in 1883,<ref>{{harvnb|Cantor|1883}}.</ref> was the most important of the six and was also published as a separate [[monograph]]. It contained Cantor's reply to his critics and showed how the [[transfinite number]]s were a systematic extension of the natural numbers. It begins by defining [[well-order]]ed sets. [[Ordinal number]]s are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the [[cardinal number|cardinal]] and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove [[Cantor's theorem]]: the [[cardinality]] of the power set of a set ''A'' is strictly larger than the cardinality of ''A''. This established the richness of the hierarchy of infinite sets, and of the [[cardinal arithmetic|cardinal]] and [[ordinal arithmetic]] that Cantor had defined. His argument is fundamental in the solution of the [[Halting problem]] and the proof of [[Gödel's first incompleteness theorem]]. Cantor wrote on the [[Goldbach conjecture]] in 1894.

[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage of Georg Cantor's article with his set definition]]
In 1895 and 1897, Cantor published a two-part paper in ''[[Mathematische Annalen]]'' under [[Felix Klein]]'s editorship; these were his last significant papers on set theory.<ref>{{harvtxt|Cantor|1895}}, {{harvtxt|Cantor|1897}}. The English translation is [[#Cantor1955|Cantor 1955]].</ref> The first paper begins by defining set, [[subset]], etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of [[well-ordered set]]s and ordinal numbers. Cantor attempts to prove that if ''A'' and ''B'' are sets with ''A'' [[equinumerous|equivalent]] to a subset of ''B'' and ''B'' equivalent to a subset of ''A'', then ''A'' and ''B'' are equivalent. [[Ernst Schröder (mathematician)|Ernst Schröder]] had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. [[Felix Bernstein (mathematician)|Felix Bernstein]] supplied a correct proof in his 1898 PhD thesis; hence the name [[Cantor–Bernstein–Schröder theorem]].

====One-to-one correspondence====
{{Main|Bijection}}
[[File:Bijection.svg|thumb|A bijective function]]
Cantor's 1874 [[Crelle's Journal|Crelle]] paper was the first to invoke the notion of a [[Bijection|1-to-1 correspondence]], though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the [[unit square]] and the points of a unit [[line segment]]. In an 1877 letter to Richard Dedekind, Cantor proved a far [[Mathematical jargon#stronger|stronger]] result: for any positive integer ''n'', there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an [[n-dimensional space|''n''-dimensional space]]. About this discovery Cantor wrote to Dedekind: "{{lang|fr|Je le vois, mais je ne le crois pas!}}" ("I see it, but I don't believe it!")<ref>{{Cite book |last=Wallace |first=David Foster |year=2003|title=Everything and More: A Compact History of Infinity|place=New York|publisher=W.&nbsp;W. Norton and Company|isbn=978-0-393-00338-3|page=[https://archive.org/details/everythingmore00davi/page/259 259]|url=https://archive.org/details/everythingmore00davi/page/259}}</ref> The result that he found so astonishing has implications for geometry and the notion of [[dimension]].

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "[[cardinality|power]]" (a term he took from [[Jakob Steiner]]) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined [[countable set]]s (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the [[natural number]]s, and proved that the rational numbers are denumerable. He also proved that ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> has the same power as the [[real number]]s '''R''', as does a countably infinite [[Cartesian product|product]] of copies of '''R'''. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about [[dimension]], stressing that his [[Map (mathematics)|mapping]] between the [[unit interval]] and the unit square was not a [[continuous function|continuous]] one.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and [[Karl Weierstrass]] supported its publication.<ref>[[#Dauben1979|Dauben 1979]], pp. 69, 324 ''63n''. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.</ref> Nevertheless, Cantor never again submitted anything to Crelle.

====Continuum hypothesis====
{{Main|Continuum hypothesis}}
Cantor was the first to formulate what later came to be known as the [[continuum hypothesis]] or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is ''exactly'' aleph-one, rather than just ''at least'' aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to [[mathematical proof|prove]] it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.<ref name="daub280" />

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by [[Kurt Gödel]] and a 1963 one by [[Paul Cohen (mathematician)|Paul Cohen]] together imply that the continuum hypothesis can be neither proved nor disproved using standard [[Zermelo–Fraenkel set theory]] plus the [[axiom of choice]] (the combination referred to as "[[ZFC]]").<ref>Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is [[W. Hugh Woodin]]. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.</ref>

====Absolute infinite, well-ordering theorem, and paradoxes====
In 1883, Cantor divided the infinite into the transfinite and the [[Absolute infinite|absolute]].<ref>{{harvnb|Cantor|1883|pp=587–588}}; English translation: [[#Ewald|Ewald 1996]], pp. 916&ndash;917.</ref>

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α&nbsp;+&nbsp;1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it.<ref>[[#Hallett|Hallett 1986]], pp. 41–42.</ref> In 1883, Cantor also introduced the [[Well-ordering theorem|well-ordering principle]] "every set can be well-ordered" and stated that it is a "law of thought".<ref>[[#Moore1982|Moore 1982]], p. 42.</ref>

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an [[aleph number|aleph]].<ref>[[#Moore1982|Moore 1982]], p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.</ref> First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem.<ref>[[#Hallett|Hallett 1986]], pp. 166–169.</ref> In 1932, Zermelo criticized the construction in Cantor's proof.<ref>Cantor's proof, which is a [[proof by contradiction]], starts by assuming there is a set ''S'' whose cardinality is not an aleph. A function from the ordinals to ''S'' is constructed by successively choosing different elements of ''S'' for each ordinal. If this construction runs out of elements, then the function well-orders the set ''S''. This implies that the cardinality of ''S'' is an aleph, contradicting the assumption about ''S''. Therefore, the function maps all the ordinals one-to-one into ''S''. The function's [[Image (mathematics)|image]] is an inconsistent submultiplicity contained in ''S'', so the set ''S'' is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make ''successive'' arbitrary choices." ([[#Hallett|Hallett 1986]], pp. 169–170.)</ref>

Cantor avoided [[paradox]]es by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, [[Bertrand Russell]] treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is [[inconsistent]]. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the [[Burali-Forti paradox]] (which was just mentioned), [[Cantor's paradox]], and [[Russell's paradox]].<ref>[[#Moore1988|Moore 1988]], pp. 52–53; [[#Moore1981|Moore and Garciadiego 1981]], pp. 330–331.</ref> Russell named paradoxes after [[Cesare Burali-Forti]] and Cantor even though neither of them believed that they had found paradoxes.<ref>[[#Moore1981|Moore and Garciadiego 1981]], pp. 331, 343; [[#Purkert1989|Purkert 1989]], p. 56.</ref>

In 1908, Zermelo published [[Zermelo set theory|his axiom system for set theory]]. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the [[well-ordering theorem]].<ref>[[#Moore1982|Moore 1982]], pp. 158–160. Moore argues that the latter was his primary motivation.</ref> Zermelo had proved this theorem in 1904 using the [[axiom of choice]], but his proof was criticized for a variety of reasons.<ref>Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", [[#Moore1982|Moore 1982]], pp. 85–141.</ref> His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.<ref>[[#Moore1982|Moore 1982]], pp. 158–160. [[#Zermelo1908|Zermelo 1908]], pp. 263–264; English translation: [[#Heijenoort|van Heijenoort 1967]], p. 202.</ref>

In 1923, [[John von Neumann]] developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a [[Class (set theory)|class]] is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class.<ref>[[#Hallett|Hallett 1986]], pp. 288, 290–291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. ([[#Hallett|Hallett 1986]], p. 286.)</ref> Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem.<ref>[[#Hallett|Hallett 1986]], pp. 291–292.</ref> In 1930, Zermelo defined [[Zermelo's models and the axiom of limitation of size|models of set theory that satisfy von Neumann's axiom]].<ref>[[#Zermelo1930|Zermelo 1930]]; English translation: [[#Ewald|Ewald 1996]], pp. 1208–1233.</ref>

==Philosophy, religion, literature and Cantor's mathematics==
The concept of the existence of an [[actual infinity]] was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the [[orthodoxy]] of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.<ref name="daub295">[[#Dauben1979|Dauben 1979]], p. 295.</ref> He directly addressed this intersection between these disciplines in the introduction to his ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre'', where he stressed the connection between his view of the infinite and the philosophical one.<ref>[[#Dauben1979|Dauben 1979]], p. 120.</ref> To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications&nbsp;– he identified the [[absolute infinite]] with God,<ref>[[#Hallett|Hallett 1986]], p. 13. Compare to the writings of [[Thomas Aquinas]].</ref> and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.<ref name = "xdpfir"/> He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science.<ref>{{cite journal |last=Hedman |first=Bruce |date=1993 |title=Cantor's Concept of Infinity: Implications of Infinity for Contingence |url=https://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html |journal=Perspectives on Science and Christian Faith |volume=45 |issue=1 |pages=8–16 |access-date=5 March 2020}}</ref> [[Joseph Dauben]] has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.<ref>{{cite book |last=Dauben |first=Joseph Warren |date=1979  |title=Georg Cantor: His Mathematics and Philosophy of the Infinite |url=https://www.jstor.org/stable/j.ctv10crfh1 |location= |publisher=Princeton University Press |doi=10.2307/j.ctv10crfh1 |jstor=j.ctv10crfh1 |isbn=9780691024479|s2cid=241372960 }}</ref><ref>{{cite journal |last=Dauben |first=Joseph Warren |date=1978 |title=Georg Cantor: The Personal Matrix of His Mathematics |url=https://www.jstor.org/stable/231091 |journal=Isis |volume=69 |issue=4 |pages=548 |doi=10.1086/352113 |jstor=231091 |pmid=387662 |s2cid=26155985 |access-date=5 March 2020 |quote=The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.}}</ref>

Debate among mathematicians grew out of opposing views in the [[philosophy of mathematics]] regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.<ref name="daub225">[[#Dauben1979|Dauben 1979]], p. 225</ref> Mathematicians from three major schools of thought ([[Constructivism (mathematics)|constructivism]] and its two offshoots, [[intuitionism]] and [[finitism]]) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that [[nonconstructive proof]]s such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that [[constructive proof]]s are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.<ref name="daub266">[[#Dauben1979|Dauben 1979]], p. 266.</ref> Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.<ref>{{Cite journal |last=Snapper |first=Ernst |year=1979|url=<!-- http://math.boisestate.edu/~tconklin/MATH547/Main/Exhibits/Three%20Crises%20in%20Math%20A.pdf -->http://www2.gsu.edu/~matgtc/three%20crises%20in%20mathematics.pdf|title=The Three Crises in Mathematics: Logicism, Intuitionism and Formalism|journal=Mathematics Magazine|volume=524|issue=4|pages=207–216|access-date=2 April 2013|archive-url=https://web.archive.org/web/20120815055019/http://www2.gsu.edu/~matgtc/three%20crises%20in%20mathematics.pdf|archive-date=15 August 2012|url-status=dead|doi=10.1080/0025570X.1979.11976784}}</ref> Mathematicians such as [[L.&nbsp;E.&nbsp;J. Brouwer]] and especially [[Henri Poincaré]] adopted an [[intuitionism|intuitionist]] stance against Cantor's work.  Finally, [[Ludwig Wittgenstein|Wittgenstein]]'s attacks were finitist: he believed that Cantor's diagonal argument conflated the [[intension]] of a set of cardinal or real numbers with its [[Extension (semantics)|extension]], thus conflating the concept of rules for generating a set with an actual set.{{sfn|Rodych|2007}}

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.<ref name = "nuozkv"/> In particular, [[neo-scholasticism|neo-Thomist]] thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".<ref>{{Cite journal |last=Davenport|year=1997 |first=Anne A. |title=The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century|journal=Isis|volume=88|issue=2|pages=263–295|jstor=236574|doi=10.1086/383692|s2cid=154486558 }}</ref> Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:<ref name = "daub7785"/> "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.".<ref>{{Harvnb|Cantor|1932|p=404}}. Translation in [[#Dauben1977|Dauben 1977]], p. 95.</ref> Prominent neo-scholastic German philosopher Konstantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}}

Cantor also believed that his theory of transfinite numbers ran counter to both [[materialism]] and [[determinism]]&nbsp;– and was shocked when he realized that he was the only faculty member at Halle who did ''not'' hold to deterministic philosophical beliefs.<ref name="daub296">[[#Dauben1979|Dauben 1979]], p. 296.</ref>

It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 ''Grundlagen'', he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz.<ref>{{Cite journal|last=Newstead|first=Anne|date=2009|title=Cantor on Infinity in Nature, Number, and the Divine Mind|journal=American Catholic Philosophical Quarterly|volume=83|issue=4|pages=533–553|doi=10.5840/acpq200983444|url=https://philarchive.org/rec/NEWQOI}}</ref> In making these claims, Cantor may have been influenced by [[Friedrich Adolf Trendelenburg|F. A. Trendelenburg]], whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's ''Ethica''. Trendelenburg was also the examiner of Cantor's ''[[Habilitation#Germany|Habilitationsschrift]]''.<ref>{{Cite journal|last=Newstead|first=Anne|date=2009|title=Cantor on Infinity in Nature, Number, and the Divine Mind|journal=American Catholic Philosophical Quarterly|volume=84 |issue=3 |pages=535}}</ref><ref>{{Cite journal|last=Ferreiros|first=Jose|date=2004|title=The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions|journal=Science in Context|volume=17 |issue=1–2 |pages=49–83|doi=10.1017/S0269889704000055|pmid=15359485|s2cid=19040786|url=http://philsci-archive.pitt.edu/17321/1/Ferreir%C3%B3s%202004.pdf |archive-url=https://web.archive.org/web/20200921124322/http://philsci-archive.pitt.edu/17321/1/Ferreir%C3%B3s%202004.pdf |archive-date=2020-09-21 |url-status=live}}</ref>

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as [[Tilman Pesch]] and [[Joseph Hontheim]],<ref>[[#Dauben1979|Dauben 1979]], p. 144.</ref> as well as theologians such as Cardinal [[Johann Baptist Franzelin]], who once replied by equating the theory of transfinite numbers with [[pantheism]].<ref name="daub77102">[[#Dauben1977|Dauben 1977]], p. 102.</ref> Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's.{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}} Cantor even sent one letter directly to [[Pope Leo XIII]] himself, and addressed several pamphlets to him.<ref name="daub7785">[[#Dauben1977|Dauben 1977]], p. 85.</ref>

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this [[Metaphysics|metaphysical]] system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom."<ref>[[#Dauben1977|Dauben 1977]], pp. 91–93.</ref> These ideas parallel those of [[Edmund Husserl]], whom Cantor had met in Halle.<ref>On Cantor, Husserl, and [[Gottlob Frege]], see Hill and Rosado Haddock (2000).</ref>

Meanwhile, Cantor himself was fiercely opposed to [[infinitesimal]]s, describing them as both an "abomination" and "the [[cholera]] [[bacillus]] of mathematics".<ref name="Infinitesimal" />

Cantor's 1883 paper reveals that he was well aware of the [[Controversy over Cantor's theory|opposition]] his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."<ref name="daub96">"[[#Dauben1979|Dauben 1979]], p. 96.</ref>

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of [[contradiction]] and defined in terms of previously accepted concepts. He also cites Aristotle, [[René Descartes]], [[George Berkeley]], [[Gottfried Leibniz]], and [[Bernard Bolzano]] on infinity. Instead, he always strongly rejected [[Immanuel Kant]]'s philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical [[Philistinism|Philistine]] who knew so little mathematics."<ref>Russell, Bertrand ''The Autobiography of Bertrand Russell'', George Allen and Unwin Ltd., 1971 (London), vol. 1, p. 217.</ref>

==Cantor's ancestry==
[[File:Blackboard Georg Cantor (11-line V O building 24).jpg|thumb|The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor", [[Vasilievsky Island]], Saint-Petersburg.]]
Cantor's paternal grandparents were from [[Copenhagen]] and fled to Russia from the disruption of the [[Napoleonic Wars]]. There is very little direct information on them.<ref>''E.g.'', Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.</ref> Cantor's father, Georg Waldemar Cantor, was educated in the [[Lutheranism|Lutheran]] mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education.<ref name="pi87">[[#Purkert|Purkert and Ilgauds 1985]], p. 15.</ref> Cantor's mother, Maria Anna Böhm, was an [[Austro-Hungarian]] born in Saint Petersburg and baptized [[Roman Catholic Church|Roman Catholic]]; she converted to [[Protestantism]] upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:
{{Blockquote|Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ... ("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...")<ref name="pi87"/>}}

which could be read to imply that she was of [[Jewish]] ancestry.<ref>For more information, see: [[#Dauben1979|Dauben 1979]], p. 1 and notes; [[#Guinness1971|Grattan-Guinness 1971]], pp. 350–352 and notes; [[#Purkert|Purkert and Ilgauds 1985]]; the letter is from {{harvnb|Aczel|2000|pp=93–94}}, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.</ref>

According to biographer [[Eric Temple Bell]], Cantor was of Jewish descent, although both parents were baptized.<ref>[[Men of Mathematics|Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré]], 1937, E. T. Bell</ref> In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions ([[Annals of Science]] 27, pp.&nbsp;345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).

In a letter written to [[Paul Tannery]] in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p.&nbsp;306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde...."  ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.")<ref>Tannery, Paul (1934) ''Memoires Scientifique 13 Correspondance'', Gauthier-Villars, Paris, p. 306.</ref>
In addition, Cantor's maternal great uncle,<ref>[[#Dauben1979|Dauben 1979]], p. 274.</ref>  [[Josef Böhm]], a Hungarian violinist, has been described as Jewish,<ref>Mendelsohn, Ezra (ed.) (1993) [http://trove.nla.gov.au/work/11440061?versionId=13425381 ''Modern Jews and their musical agendas''], [[Oxford University Press]], p. 9.</ref> which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.<ref>''Ismerjük oket?: zsidó származású nevezetes magyarok arcképcsarnoka'', István Reményi Gyenes Ex Libris, (Budapest 1997), pages 132–133</ref>

In a letter to [[Bertrand Russell]], Cantor described his ancestry and self-perception as follows:
{{Blockquote|Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a ''regular just Germain'', for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.<ref>Russell, Bertrand. ''Autobiography'', vol. I, p. 229. In English in the original; italics also as in the original.</ref>}}
 
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
{{Blockquote|More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew&nbsp;..."<ref name="pi87"/>}}

==Biographies==
Until the 1970s, the chief academic publications on Cantor were two short monographs by [[Arthur Moritz Schönflies]] (1927)&nbsp;– largely the correspondence with Mittag-Leffler&nbsp;– and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by [[Eric Temple Bell]]'s ''[[Men of Mathematics]]'' (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the [[history of mathematics]]"; and as "one of the worst".<ref>[[#Guinness1971|Grattan-Guinness 1971]], p. 350.</ref> Bell presents Cantor's relationship with his father as [[Oedipal]], Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell&nbsp;– including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.<ref>[[#Guinness1971|Grattan-Guinness 1971]] (quotation from p. 350, note), [[#Dauben1979|Dauben 1979]], p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)</ref> A critique of Bell's book is contained in [[Joseph Dauben]]'s biography.<ref>[[#Dauben1979|Dauben 1979]]</ref> Writes Dauben:
{{Blockquote|Cantor devoted some of his most vituperative correspondence, as well as a portion of the ''Beiträge'', to attacking what he described at one point as the '[[infinitesimal]] Cholera bacillus of mathematics', which had spread from Germany through the work of [[Carl Johannes Thomae|Thomae]], [[Paul du Bois-Reymond|du Bois Reymond]] and [[Otto Stolz|Stolz]], to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and [[Giuseppe Veronese|Veronese]] was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.<ref>Dauben, J.: The development of the Cantorian set theory, pp.~181–219. See pp.216–217. In Bos, H.; Bunn, R.; Dauben, J.; [[Grattan-Guinness]], I.; Hawkins, T.; Pedersen, K. From the calculus to set theory, 1630–1910. An introductory history. Edited by I. Grattan-Guinness. Gerald Duckworth & Co. Ltd., London, 1980.</ref>}}

==See also==
{{Portal|Biography|Philosophy|Mathematics}}
* [[Absolute infinite]]
* [[Aleph number]]
* [[Cardinality of the continuum]]
* [[Cantor medal]]&nbsp;– award by the [[Deutsche Mathematiker-Vereinigung]] in honor of Georg Cantor
* [[Cardinal number]]
* [[Continuum hypothesis]]
* [[Countable set]]
* [[Derived set (mathematics)]]
* [[Epsilon numbers (mathematics)]]
* [[Factorial number system]]
* [[Pairing function]]
* [[Transfinite number]]
* [[List of things named after Georg Cantor]]

==Notes==
{{Reflist}}

==References==
* {{Cite journal |last=Dauben |first=Joseph W. |author-link=Joseph Dauben |jstor=2708842 |year=1977|title=Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite|journal=Journal of the History of Ideas|volume=38|number=1|pages=85–108|ref=Dauben1977|doi=10.2307/2708842}}
* {{Cite book |last=Dauben |first=Joseph W.|year=1979|title=[Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite|place=Boston|publisher=Harvard University Press|isbn=978-0-691-02447-9|ref=Dauben1979|url-access=registration|url=https://archive.org/details/georgcantorhisma0000daub}}
* {{Cite conference |last=Dauben |first=Joseph|orig-year=1993|year=2004|url=https://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf |archive-url=https://web.archive.org/web/20180123072518/http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf |archive-date=2018-01-23 |url-status=live |title=Georg Cantor and the Battle for Transfinite Set Theory |conference=Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.)|pages=1–22 |ref=Dauben2004}} Internet version published in ''Journal of the ACMS'' 2004.  Note, though, that Cantor's Latin quotation described in this article as ''a familiar passage from the Bible'' is actually from the works of Seneca and has no implication of divine revelation.
* {{Cite book |editor-last=Ewald|editor-first=William B.|year=1996|title=From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics|place=New York|publisher=Oxford University Press|isbn=978-0-19-853271-2|ref=Ewald}}
* {{Cite journal |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |year=1971|title=Towards a Biography of Georg Cantor|doi=10.1080/00033797100203837|journal=Annals of Science|volume=27|issue=4|pages=345–391|ref=Guinness1971}}
* {{Cite book |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |year=2000|title=The Search for Mathematical Roots: 1870–1940|publisher=Princeton University Press|isbn=978-0-691-05858-0|ref=Guinness2000}}
* {{Cite book |last=Hallett |first=Michael|title=Cantorian Set Theory and Limitation of Size|publisher=Oxford University Press|place=New York|year=1986|isbn=978-0-19-853283-5|ref=Hallett}}
* {{Cite book |first=Gregory H.|last=Moore|year=1982|title=Zermelo's Axiom of Choice: Its Origins, Development & Influence|publisher=Springer|isbn=978-1-4613-9480-8|ref=Moore1982}}
* {{cite journal | last=Moore | first=Gregory H. | title=The roots of Russell's paradox | journal=Russell: The Journal of Bertrand Russell Archives | publisher=Johns Hopkins University Press | volume=8 | issue=1 | date=2023-04-19 | issn=1913-8032 | pages=46–56 | url=https://muse.jhu.edu/article/882076/summary |ref=Moore1988| access-date=2025-03-28}}
* {{Cite journal |first1=Gregory H.|last1=Moore|first2=Alejandro|last2=Garciadiego|year=1981|title=Burali-Forti's Paradox: A Reappraisal of Its Origins|journal=Historia Mathematica|volume=8|issue=3|pages=319–350|doi=10.1016/0315-0860(81)90070-7|ref=Moore1981|doi-access=free}}
* {{Cite book |last=Purkert|first=Walter|chapter=Cantor's Views on the [[Foundations of Mathematics]]|editor-last1=Rowe|editor-first1=David E.|editor-last2=McCleary|editor-first2=John|title=The History of Modern Mathematics, Volume 1|pages=[https://archive.org/details/historyofmodernm0000symp/page/49 49–65]|publisher=Academic Press|year=1989|isbn=978-0-12-599662-4|ref=Purkert1989|url=https://archive.org/details/historyofmodernm0000symp/page/49}}
* {{Cite book |last1=Purkert |first1=Walter |last2=Ilgauds |first2=Hans Joachim|year=1985|title=Georg Cantor: 1845–1918|publisher=[[Birkhäuser Verlag|Birkhäuser]]|isbn=978-0-8176-1770-7|ref=Purkert}}
*{{Cite book |last=Suppes |first=Patrick|year=1972|orig-year=1960|title=Axiomatic Set Theory|place=New York|publisher=Dover|isbn=978-0-486-61630-8|ref=Suppes|url=https://archive.org/details/axiomaticsettheo00supp_0}} Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
* {{Cite journal |first=Ernst|last=Zermelo|year=1908|title=Untersuchungen über die Grundlagen der Mengenlehre I|journal=Mathematische Annalen|volume=65|issue=2|pages= 261–281|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065&DMDID=DMDLOG_0018|doi=10.1007/bf01449999|s2cid=120085563|ref=Zermelo1908}}
* {{Cite journal |first=Ernst|last=Zermelo|title=Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf |archive-url=https://web.archive.org/web/20040628014704/http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf |archive-date=2004-06-28 |url-status=live|journal=[[Fundamenta Mathematicae]]|volume=16|pages=29–47|year=1930|ref=Zermelo1930|doi=10.4064/fm-16-1-29-47|doi-access=free}}
* {{Cite book |last=van&nbsp;Heijenoort|first=Jean|year=1967|publisher=Harvard University Press|title=From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931|isbn = 978-0-674-32449-7|ref=Heijenoort}}

==Bibliography==
:''Older sources on Cantor's life should be treated with caution. See section [[#Biographies|§ Biographies]] above.''

===Primary literature in English===
* {{Cite book |last=Cantor |first=Georg |year=1955|orig-year=1915|url=https://archive.org/details/contributionstot003626mbp|title=Contributions to the Founding of the Theory of Transfinite Numbers|editor=[[Philip Jourdain]]|place=New York|publisher=Dover Publications|isbn=978-0-486-60045-1|ref=Cantor1955}}.

===Primary literature in German===
* {{Cite journal |last=Cantor |first=Georg |year=1874|url=http://gdz.sub.uni-goettingen.de/download/PPN243919689_0077/PPN243919689_0077___LOG_0014.pdf |archive-url=https://web.archive.org/web/20171007095818/http://gdz.sub.uni-goettingen.de/download/PPN243919689_0077/PPN243919689_0077___LOG_0014.pdf |archive-date=2017-10-07 |url-status=live|title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen|journal=[[Journal für die Reine und Angewandte Mathematik]]|volume=1874|issue=77|pages=258–262|doi=10.1515/crll.1874.77.258|s2cid=199545885 }}
* {{Cite journal | last = Cantor | first = Georg | title = Ein Beitrag zur Mannigfaltigkeitslehre | journal = Journal für die Reine und Angewandte Mathematik
| volume = 1878 | issue = 84 | pages = 242–258 | url = http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002156806 | year = 1878| doi = 10.1515/crelle-1878-18788413 }}
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (1) | journal=Mathematische Annalen | volume=15 | issue=1 | pages=1–7 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0015&DMDID=DMDLOG_0004 | year=1879 | doi=10.1007/bf01444101| s2cid=179177510 | ref={{harvid|Cantor|1879}} }}
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (2) | journal=Mathematische Annalen | volume=17 | issue=3 | pages=355–358 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0017&DMDID=DMDLOG_0043 | year=1880 | doi=10.1007/bf01446232| s2cid=179177438 }}
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (3) | journal=Mathematische Annalen | volume=20 | issue=1 | pages=113–121 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0020&DMDID=DMDLOG_0015 | year=1882 | doi=10.1007/bf01443330| s2cid=177809016 }}
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (4) | journal=Mathematische Annalen | volume=21 | issue=1 | pages=51–58 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0021&DMDID=DMDLOG_0008 | year=1883 | doi=10.1007/bf01442612| s2cid=179177480 }}
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (5) | journal=Mathematische Annalen | volume=21 | issue=4 | pages=545–591 | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0021&DMDID=DMDLOG_0051 | year=1883 | doi=10.1007/bf01446819 | s2cid=121930608 | ref={{harvid|Cantor|1883}} }} Published separately as: ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre''.
* {{cite journal | author=Georg Cantor | title=Ueber unendliche, lineare Punktmannichfaltigkeiten (6) | journal=Mathematische Annalen | volume=23 | issue=4 | pages=453–488 | url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0023?tify=%7B%22pages%22%3A%5B472%2C473%5D%2C%22view%22%3A%22info%22%7D | year=1884 | doi=10.1007/BF01446598| s2cid= 179178052}}
* {{cite journal | author=Georg Cantor | title=Ueber eine elementare Frage der Mannigfaltigkeitslehre | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume=1 | pages=75–78 | year=1891 |url=http://gdz.sub.uni-goettingen.de/pdfcache/PPN37721857X_0001/PPN37721857X_0001___LOG_0029.pdf |archive-url=https://web.archive.org/web/20180101100326/http://gdz.sub.uni-goettingen.de/pdfcache/PPN37721857X_0001/PPN37721857X_0001___LOG_0029.pdf |archive-date=2018-01-01 |url-status=live}}
* {{Cite journal |last=Cantor |first=Georg |year=1895 |url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN235181684_0046&divID=LOG_0044&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN235181684_0046%7C&targetFileName=PPN235181684_0046_LOG_0044.pdf& |title=Beiträge zur Begründung der transfiniten Mengenlehre (1) |journal=Mathematische Annalen |volume=46 |issue=4 |pages=481–512 |doi=10.1007/bf02124929 |s2cid=177801164 |url-status=dead |archive-url=https://web.archive.org/web/20140423224341/http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN235181684_0046&divID=LOG_0044&pagesize=original&pdfTitlePage=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fpdftitle%2F%3FmetsFile%3DPPN235181684_0046%7C&targetFileName=PPN235181684_0046_LOG_0044.pdf& |archive-date=23 April 2014 }}
* {{Cite journal |last=Cantor |first=Georg |year=1897|title=Beiträge zur Begründung der transfiniten Mengenlehre (2)|journal=Mathematische Annalen|volume=49|issue=2|pages=207–246|doi=10.1007/bf01444205 |s2cid=121665994 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0049&DMDID=DMDLOG_0024&L=1}}
* {{Cite web |last=Cantor |first=Georg |year=1932 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN237853094&DMDID=DMDLOG_0001&L=1 |title=Gesammelte Abhandlungen mathematischen und philosophischen inhalts |editor=[[Ernst Zermelo]] |publisher=Springer |location=Berlin |url-status=dead |archive-url=https://web.archive.org/web/20140203234213/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN237853094&DMDID=DMDLOG_0001&L=1 |archive-date=3 February 2014 }}. Almost everything that Cantor wrote. Includes excerpts of his correspondence with [[Richard Dedekind|Dedekind]] (p.&nbsp;443–451) and [[Adolf Fraenkel|Fraenkel's]] Cantor biography (p.&nbsp;452–483) in the appendix.

===Secondary literature===
* {{Cite book |last1=Aczel|first1=Amir D.|author1-link=Amir Aczel|year=2000|title=The Mystery of the Aleph: Mathematics, the Kabbala, and the Search for Infinity|place=New York|publisher=Four Walls Eight Windows Publishing}}. {{isbn|0-7607-7778-0}}. A popular treatment of infinity, in which Cantor is frequently mentioned.
* {{Cite journal |last=Dauben |first= Joseph W.|date=June 1983|title=Georg Cantor and the Origins of Transfinite Set Theory|journal=Scientific American|volume=248|issue=6|pages=122–131|doi=10.1038/scientificamerican0683-122|bibcode=1983SciAm.248f.122D}}
* {{Cite book |last=Ferreirós |first=José |year=2007|title=Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought|place=Basel, Switzerland|publisher=Birkhäuser}}. {{isbn|3-7643-8349-6}} Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
* {{Cite book |last=Halmos |first=Paul |author-link=Paul Halmos |year=1998|orig-year=1960|title=Naive Set Theory|place=New York & Berlin|publisher=Springer}}. {{isbn|3-540-90092-6}}
* {{Cite journal |last1=Hilbert|first1=David|author1-link=David Hilbert|year=1926|url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002270641|title=Über das Unendliche|journal=Mathematische Annalen|volume=95|pages=161–190|doi=10.1007/BF01206605|s2cid=121888793}}
* {{Cite book |last1=Hill |first1= C. O. |last2=Rosado Haddock |first2=G. E.|year=2000|title=Husserl or Frege? Meaning, Objectivity, and Mathematics|place=Chicago|publisher=Open Court}}. {{isbn|0-8126-9538-0}} Three chapters and 18 index entries on Cantor.
* {{Cite book |last=Meschkowski |first=Herbert|year=1983|title=Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influence, in German)|publisher= Vieweg, Braunschweig}}
* Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind"[https://www.pdcnet.org/acpq/content/acpq_2009_0083_0004_0533_0553], ''American Catholic Philosophical Quarterly'', '''83''' (4): 532–553, https://doi.org/10.5840/acpq200983444. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in the ''Pantheismusstreit''. Brief mention is made of Cantor's learning from F.A.Trendelenburg.
* {{Cite book |last=Penrose |first=Roger |author-link=Roger Penrose |year=2004|title=The Road to Reality|publisher=Alfred A. Knopf}}. {{isbn|0-679-77631-1}} Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary [[Theoretical physics|theoretical physicist]].
* {{Cite book |last=Rucker |first=Rudy |author-link=Rudy Rucker |year=2005|orig-year=1982|title=Infinity and the Mind|publisher=Princeton University Press}}. {{isbn|0-553-25531-2}} Deals with similar topics to Aczel, but in more depth.
* {{Cite encyclopedia |last=Rodych |first=Victor|year=2007|title=Wittgenstein's Philosophy of Mathematics |url=http://plato.stanford.edu/entries/wittgenstein-mathematics/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor=Edward N. Zalta |publisher=Metaphysics Research Lab, Stanford University}}.
* Leonida Lazzari, ''L'infinito di Cantor''. Editrice Pitagora, Bologna, 2008.

==External links==
* {{Wikiquote-inline}}
* {{Commons category-inline}}
* {{Internet Archive author |sname=Georg Cantor}}
* {{MacTutor|id=Cantor}}
* {{MacTutor|class=HistTopics|id = Beginnings_of_set_theory|title = A history of set theory}} Mainly devoted to Cantor's accomplishment.
* [https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor Georg Cantor], britannica.com
* ''Stanford Encyclopedia of Philosophy'': [http://plato.stanford.edu/entries/set-theory/ Set theory] by [[Thomas Jech]]. [https://plato.stanford.edu/entries/settheory-early/ The Early Development of Set Theory] by José Ferreirós.
* "Cantor infinities", analysis of Cantor's 1874 article, ''[http://www.bibnum.education.fr/mathematiques/theorie-des-nombres/cantor-et-les-infinis BibNum]'' <small>(for English version, click 'à télécharger')</small>. There is an error in this analysis. It states Cantor's Theorem 1 correctly: Algebraic numbers can be counted. However, it states his Theorem 2 incorrectly: Real numbers cannot be counted. It then says: "Cantor notes that, taken together, Theorems 1 and 2 allow for the redemonstration of the existence of non-algebraic real numbers …" This existence demonstration is [[non-constructive]]. Theorem 2 stated correctly is: Given a sequence of real numbers, one can determine a real number that is not in the sequence. Taken together, Theorem 1 and this Theorem 2 produce a non-algebraic number. Cantor also used Theorem 2 to prove that the real numbers cannot be counted. See [[Cantor's first set theory article]] or [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf Georg Cantor and Transcendental Numbers] {{Webarchive|url=https://web.archive.org/web/20220121155859/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf |date=21 January 2022 }}.

{{Classical logic}}
{{Fractals}}
{{Set theory}}
{{Infinity}}
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