Georg Cantor: Difference between revisions
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*{{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 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. |
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* {{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|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}}. |
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* {{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|journal=[[Fundamenta Mathematicae]]|volume=16|pages=29–47|year=1930|ref=Zermelo1930|doi=10.4064/fm-16-1-29-47}}. |
* {{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|journal=[[Fundamenta Mathematicae]]|volume=16|pages=29–47|year=1930|ref=Zermelo1930|doi=10.4064/fm-16-1-29-47}}. |
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* {{cite book|last=Wolfram|first=Stephen|authorlink=Stephen Wolfram|title=A New Kind of Science|url=https://www.wolframscience.com/nks/|publisher=Wolfram Media, Inc.|year=2002|pages=893,901-902,1127-1128,1153-1154|isbn=1-57955-008-8}} |
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* {{Cite book |last=van 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}}. |
* {{Cite book |last=van 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}}. |
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Revision as of 15:46, 5 January 2021
Georg Cantor | |
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Born | Georg Ferdinand Ludwig Philipp Cantor March 3, 1845 |
Died | January 6, 1918 | (aged 72)
Nationality | German |
Alma mater | |
Known for | Set theory |
Spouse |
Vally Guttmann (m. 1874) |
Awards | University of Halle |
Thesis | De aequationibus secundi gradus indeterminatis (1867) |
Doctoral advisor |
Georg Ferdinand Ludwig Philipp Cantor (
Cantor's theory of
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".[8] 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".[9] 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,[10] though some have explained these episodes as probable manifestations of a bipolar disorder.[11]
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.[12] David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."[13][14]
Life of Georg Cantor
Youth and studies
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Georg_Cantor3.jpg/220px-Georg_Cantor3.jpg)
Georg Cantor was born in 1845 in the western merchant colony of
Teacher and researcher
Cantor submitted his
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, Cantor spent much time in mathematical discussions with Richard Dedekind, 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.
In 1881, Cantor's Halle colleague
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.[24] 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.[25] 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."[26]
Cantor suffered his first known bout of depression in May 1884.[18][27] 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:
... 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.[28]
This crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of
Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891.[19] 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[19] 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, which was held in Zürich, Switzerland, in 1897.[19]
Later years and death
After Cantor's 1884 hospitalization, there is no record that he was in any
In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the
Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[33] 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.[18]
Mathematical work
Cantor's work between 1874 and 1884 is the origin of
In one of his earliest papers,
Cantor developed important concepts in
Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter (aleph) with a natural number subscript; for the ordinals he employed the Greek letter 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 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.[14] 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 Zurich 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
Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics".[40] Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.[41]
Set theory
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Diagonal_argument_2.svg/250px-Diagonal_argument_2.svg.png)
The beginning of set theory as a branch of mathematics is often marked by the publication of
Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers[47] as a sequence a1, a2, a3, .... 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 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 – 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 – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.[48] 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.[49]
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 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.[50] 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,[51] 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 numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the 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
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/220px-Passage_with_the_set_definition_of_Georg_Cantor.png)
In 1895 and 1897, Cantor published a two-part paper in
One-to-one correspondence
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/220px-Bijection.svg.png)
Cantor's 1874
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 "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 sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite 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 mapping between the unit interval and the unit square was not a 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.[54] Nevertheless, Cantor never again submitted anything to Crelle.
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 prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[10]
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by
Absolute infinite, well-ordering theorem, and paradoxes
In 1883, Cantor divided the infinite into the transfinite and the
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 + 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.[57] In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought".[58]
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.[59] 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.[60] In 1932, Zermelo criticized the construction in Cantor's proof.[61]
Cantor avoided
In 1908, Zermelo published 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.[64] Zermelo had proved this theorem in 1904 using the axiom of choice, but his proof was criticized for a variety of reasons.[65] 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.[66]
In 1923,
Philosophy, religion, literature and Cantor's mathematics
The concept of the existence of an
Debate among mathematicians grew out of opposing views in the
Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[6] In particular, 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".[76] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[77] "... 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."[78]
Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.[79]
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.[80] In making these claims, Cantor may have been influenced by FA Trendelenburg, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's Ethica. FA Trendelenburg was also the examiner of Cantor's Habilitationsschrift.[81][82]
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,[83] as well as theologians such as Cardinal Johann Baptist Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[7] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[77]
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 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."[84] These ideas parallel those of Edmund Husserl, whom Cantor had met in Halle.[85]
Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".[40]
Cantor's 1883 paper reveals that he was well aware of the 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."[86]
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
Cantor's ancestry
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 his grandparents.[88] Cantor was sometimes called Jewish in his lifetime,[89] but has also variously been called Russian, German, and Danish as well.
Jakob Cantor, Cantor's grandfather, gave his children Christian
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 ...[90]
("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...") which could be read to imply that she was of Jewish ancestry.[91]
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
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 ..."[90]
It is also later said in the same document:
Also efforts for a long time by the librarian Josef Fischer, one of the best experts on Jewish genealogy in Denmark, charged with identifying Jewish professors, that Georg Cantor was of Jewish descent, finished without result. [Something seems to be wrong with this sentence, but the meaning seems clear enough.] In Cantor's published works and also in his Nachlass there are no statements by himself which relate to a Jewish origin of his ancestors. There is to be sure in the Nachlass a copy of a letter of his brother Ludwig from 18 November 1869 to their mother with some unpleasant antisemitic statements, in which it is said among other things: ...[90]
(the rest of the quote is finished by the very first quote above). In Men of Mathematics, Eric Temple Bell described Cantor as being "of pure Jewish descent on both sides", although both parents were baptized. 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. 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 by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 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.")[92]
In addition, Cantor's maternal great uncle,
In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows:
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.[96]
Biographies
Until the 1970s, the chief academic publications on Cantor were two short monographs by
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 Thomae, du Bois Reymond and 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 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.[100]
See also
- Cantor algebra
- Cantor cube
- Cantor function
- Deutsche Mathematiker-Vereinigungin honor of Georg Cantor
- Cantor space
- Cantor's back-and-forth method
- Cantor–Bernstein theorem
- Heine–Cantor theorem
- Pairing function
Notes
- ^ Grattan-Guinness 2000, p. 351.
- ^ The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
- ^ Dauben 2004, p. 1.
- ISBN 9780691024479.
- ^ a b Dauben 2004, pp. 8, 11, 12–13.
- ^ a b Dauben 1977, p. 86; Dauben 1979, pp. 120, 143.
- ^ a b Dauben 1977, p. 102.
- ^ Dauben 2004, p. 1; Dauben 1977, p. 89 15n.
- ^ a b Rodych 2007.
- ^ Arthur Moritz Schönfliesblamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
- mental illnessas "cyclic manic-depression".
- ^ a b Dauben 1979, p. 248.
- ^ 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.")
- ^ ISBN 978-0-387-04999-1.
- ^ ru: The musical encyclopedia (Музыкальная энциклопедия).
- ^ "Georg Cantor (1845-1918)". www-groups.dcs.st-and.ac.uk. Retrieved September 14, 2019.
- ISBN 978-3764317706.
- ^ a b c d e "Cantor biography". www-history.mcs.st-andrews.ac.uk. Retrieved October 6, 2017.
- ^ OCLC 41497065.
- ^ O'Connor, John J; Robertson, Edmund F (1998). "Georg Ferdinand Ludwig Philipp Cantor". MacTutor History of Mathematics.
- ^ Dauben 1979, p. 163.
- ^ Dauben 1979, p. 34.
- ^ Dauben 1977, p. 89 15n.
- ^ Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
- ^ Dauben 1979, p. 138.
- ^ Dauben 1979, p. 139.
- ^ a b Dauben 1979, p. 282.
- ^ Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
- ^ Dauben 1979, pp. 281–283.
- ^ Dauben 1979, p. 283.
- ^ For a discussion of König's paper see Dauben 1979, pp. 248–250. For Cantor's reaction, see Dauben 1979, pp. 248, 283.
- ^ Dauben 1979, pp. 283–284.
- ^ Dauben 1979, p. 284.
- ^ JSTOR 3026799.
- ISBN 9780486616308.
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.
- ^ Cantor 1874
- ^ 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".
- ^ The Cantor Set Before Cantor Mathematical Association of America
- S2CID 122744778.
- ^ S2CID 119250310.
- S2CID 123157068. Archived from the original(PDF) on February 15, 2013.
- ^ This follows closely the first part of Cantor's 1891 paper.
- ^ Cantor 1874. English translation: Ewald 1996, pp. 840–843.
- .
- ^ For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., Suppes 1972.
- ^ Liouville, Joseph (May 13, 1844). A propos de l'existence des nombres transcendants.
- coefficients.
- JSTOR 2975129.. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
- pairwise disjoint sets: T0 and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T0, f(t2n-1) = tn, and f(t2n) = an. 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 (Cantor 1879, p. 4).
- ^ Dauben 1977, p. 89.
- ^ Cantor 1883.
- ^ Cantor (1895), Cantor (1897). The English translation is Cantor 1955.
- ISBN 978-0-393-00338-3.
- ^ 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.
- ^ 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.
- ^ Cantor 1883, pp. 587–588; English translation: Ewald 1996, pp. 916–917.
- ^ Hallett 1986, pp. 41–42.
- ^ Moore 1982, p. 42.
- ^ 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.
- ^ Hallett 1986, pp. 166–169.
- ^ 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 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 1986, pp. 169–170.)
- ^ Moore 1988, pp. 52–53; Moore and Garciadiego 1981, pp. 330–331.
- ^ Moore and Garciadiego 1981, pp. 331, 343; Purkert 1989, p. 56.
- ^ Moore 1982, pp. 158–160. Moore argues that the latter was his primary motivation.
- ^ Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", Moore 1982, pp. 85–141.
- ^ Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202.
- ^ 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 1986, p. 286.)
- ^ Hallett 1986, pp. 291–292.
- ^ Zermelo 1930; English translation: Ewald 1996, pp. 1208–1233.
- ^ Dauben 1979, p. 295.
- ^ Dauben 1979, p. 120.
- ^ Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
- ^ Dauben 1979, p. 225
- ^ Dauben 1979, p. 266.
- doi:10.1080/0025570X.1979.11976784. Archived from the original(PDF) on August 15, 2012. Retrieved April 2, 2013.
- S2CID 154486558.
- ^ a b Dauben 1977, p. 85.
- ^ Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
- ^ Dauben 1979, p. 296.
- .
- ^ Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind". American Catholic Philosophical Quarterly. 84 (3): 535.
- PMID 15359485.
- ^ Dauben 1979, p. 144.
- ^ Dauben 1977, pp. 91–93.
- ^ On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
- ^ "Dauben 1979, p. 96.
- ^ Russell, Bertrand The Autobiography of Bertrand Russell, George Allen and Unwin Ltd., 1971 (London), vol. 1, p. 217.
- ^ E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
- Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book1896–97, "List of Jewish Celebrities of the Nineteenth Century", p. 119; this list has a star against people with one Jewish parent, but Cantor is not starred.
- ^ a b c d Purkert and Ilgauds 1985, p. 15.
- ^ For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert and Ilgauds 1985; the letter is from 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.
- ^ Tannery, Paul (1934) Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, p. 306.
- ^ Dauben 1979, p. 274.
- ^ Mendelsohn, Ezra (ed.) (1993) Modern Jews and their musical agendas, Oxford University Press, p. 9.
- ^ Ismerjük oket?: zsidó származású nevezetes magyarok arcképcsarnoka, István Reményi Gyenes Ex Libris, (Budapest 1997), pages 132–133
- ^ Russell, Bertrand. Autobiography, vol. I, p. 229. In English in the original; italics also as in the original.
- ^ Grattan-Guinness 1971, p. 350.
- ^ Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
- ^ Dauben 1979
- 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.
References
- JSTOR 2708842..
- Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9..
- Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Internet version published in Journal of the ACMS 2004.
- Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2..
- ..
- ISBN 978-0-691-05858-0..
- Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN 978-0-19-853283-5..
- Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer. ISBN 978-1-4613-9480-8..
- Moore, Gregory H. (1988). "The Roots of Russell's Paradox". Russell: The Journal of Bertrand Russell Studies. 8: 46–56. ..
- Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. ..
- Purkert, Walter (1989). "Cantor's Views on the ISBN 978-0-12-599662-4..
- Purkert, Walter; Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. ISBN 978-0-8176-1770-7..
- Suppes, Patrick (1972) [1960]. Axiomatic Set Theory. New York: Dover. ISBN 978-0-486-61630-8.. 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.
- Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. S2CID 120085563..
- Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF). ..
- ISBN 1-57955-008-8.
- van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 978-0-674-32449-7..
Bibliography
- Older sources on Cantor's life should be treated with caution. See section #Biographies above.
Primary literature in English
- Cantor, Georg (1955) [1915]. ISBN 978-0-486-60045-1..
Primary literature in German
- Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). S2CID 199545885.
- Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die Reine und Angewandte Mathematik. 1878 (84): 242–258. ..
- Georg Cantor (1879). "Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen. 15 (1): 1–7. S2CID 179177510.
- Georg Cantor (1880). "Ueber unendliche, lineare Punktmannichfaltigkeiten (2)". Mathematische Annalen. 17 (3): 355–358. S2CID 179177438.
- Georg Cantor (1882). "Ueber unendliche, lineare Punktmannichfaltigkeiten (3)". Mathematische Annalen. 20 (1): 113–121. S2CID 177809016.
- Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (4)". Mathematische Annalen. 21 (1): 51–58. S2CID 179177480.
- Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
- Georg Cantor (1891). "Ueber eine elementare Frage der Mannigfaltigkeitslehre" (PDF). Jahresbericht der Deutschen Mathematiker-Vereinigung. 1: 75–78.
- Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. S2CID 177801164. Archived from the originalon April 23, 2014.
- Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. S2CID 121665994.
- Cantor, Georg (1932). Fraenkel'sCantor biography (p. 452–483) in the appendix.
Secondary literature
- ISBN 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned.
- Dauben, Joseph W. (June 1983). "Georg Cantor and the Origins of Transfinite Set Theory". Scientific American. 248 (6): 122–131. .
- Ferreirós, José (2007). Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought. Basel, Switzerland: Birkhäuser.. ISBN 3-7643-8349-6Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
- ISBN 3-540-90092-6
- S2CID 121888793.
- Hill, C. O.; Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court.. ISBN 0-8126-9538-0Three chapters and 18 index entries on Cantor.
- Meschkowski, Herbert (1983). Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influence, in German). Vieweg, Braunschweig.
- Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind"[1], 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.
- ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
- ISBN 0-553-25531-2Deals with similar topics to Aczel, but in more depth.
- Rodych, Victor (2007). "Wittgenstein's Philosophy of Mathematics". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University..
- Leonida Lazzari, L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.
External links
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- Works by or about Georg Cantor at Internet Archive
- O'Connor, John J.; Robertson, Edmund F., "Georg Cantor", MacTutor History of Mathematics Archive, University of St Andrews
- O'Connor, John J.; Robertson, Edmund F., "A history of set theory", MacTutor History of Mathematics Archive, University of St Andrews Mainly devoted to Cantor's accomplishment.
- Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech. The Early Development of Set Theory by José Ferreirós.
- Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gymnasium Halle
- Poem about Georg Cantor
- "Cantor infinities", analysis of Cantor's 1874 article, BibNum (for English version, click 'à télécharger'). 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 Georg Cantor and Transcendental Numbers.