|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Infobox scientist
| | The objective of giving Wedding Gifts to wedding guests is to show your appreciation. A specially picked out Present shows that you place thought and consideration into the gift. Like a [http://answers.yahoo.com/search/search_result?p=customized+note&submit-go=Search+Y!+Answers customized note] with...<br><br>A extended standing tradition at weddings is a gift given by the bride and the groom. A common say about gifts that its better to give than to get may well not be agreed to by a newly wedded couple who almost certainly received some of the nicest gifts ever on their wedding day..<br><br>The purpose of giving Wedding Gifts to wedding guests is to show your appreciation. A specially picked out Present shows that you place believed and consideration into the gift. Like a customized note with the present or personalizing the present itself to thank the particular person for whatever part he or she played in the wedding is definitely a good touch that shows really very good taste and thoughtfulness. If you hate to discover further on [http://www.lglamp.com/their-the-details-that-count-down-seriously-to-the-bridal-garters/ Their The Details That Count, Down Seriously To The Bridal Garters | LG LED], we recommend many libraries people could pursue. The Wedding Gifts need to have not be really expensive so as to break your wedding spending budget but ought to be properly thought out and consider the personal taste and likes and dislikes of the person on the receiving end of the gift. Also the packaging and presentation goes a extended way in showing how a lot you care and appreciate the people in your life.<br><br>So who need to be at the receiving end of the Gift by the bride and the groom? Members of the wedding party have always topped that list, which involves practically everybody, the greatest man, maid or matron of honor, bridesmaids, ushers, groomsmen, junior bridesmaids, flower girls and ring bearers.<br><br>Although it is not mandatory, but it is a nice gesture for the bride and the groom to give gifts to each other. These days each the bride and the groom contain every set of parents on the gift giving list. It is also a nice gesture to give a gift to somebody who did something particular in the course of the occasion for instance somebody who did a particular reading at the ceremony or to step in for the brides father and stroll the bride down the aisle. It is good taste to give gifts to acknowledge the contribution these folks made and consist of them in present providing.<br><br>Gifts ought to be bought about four weeks prior to the wedding date, which will give you enough time to have them engraved and to generate personalization messages and so forth.<br><br>To assist the gifts make an even greater impression on the recipient, take specific care in packaging the gifts. Use tasteful wrapping paper and embellishments using colors that complement the wedding colors. The very best time to give your gifts is in the course of the rehearsal dinner as it is intimate and the individuals to whom you"re providing the gifts are all present at the occasion.<br><br>Last but not least need to have some present ideas? Nicely first check your wedding budget to see how significantly you have allocated to this category. You have a lot more than likely spent really a bit of your money by this stage, so it is important to take a realistic appear at the your remaining price range.<br><br>Brides usually give every member of the bridal party a piece of jewelry or a hair accessory that can be worn on the wedding day. Other gift suggestions include bud vases, picture frames, bath oils, a collectible item or a present basket.<br><br>The guys in the wedding party like to receive engraved pens, pocket watches, cigars, enterprise card holders, clocks for their desk, or important chains. Discover further on this partner paper by clicking [http://www.dlfcsj.com/their-the-details-that-count-down-to-the-bridal-garters/ marley lilly]. For the parents, a expert wedding photo in an engraved frame is excellent or a customized letter detailing your appreciation. For every other, anything that conveys your love and devotion is excellent.<br><br>The tradition of providing wedding gifts makes the wedding day a lot more unique and individual for each couple and it will be alive for as extended as there are weddings in one way or one more..<br><br>Should you cherished this informative article as well as you would want to get more info regarding government health insurance; [http://royaldeity3921.sosblogs.com click the up coming webpage], kindly pay a visit to our webpage. |
| | name = Georg Cantor
| |
| | image = Georg Cantor2.jpg
| |
| | image_size = 225px
| |
| | birth_name = Georg Ferdinand Ludwig Philipp Cantor
| |
| | birth_date = {{Birth date|1845|3|3}}
| |
| | birth_place = [[Saint Petersburg]], [[Russian Empire]]
| |
| | death_date = {{Death date and age|1918|1|6|1845|3|3}}
| |
| | death_place = [[Halle (Saale)|Halle]], [[Province of Saxony]], [[German Empire]]
| |
| | residence = [[Russian Empire]] (1845–1856),<br>[[German Empire]] (1856–1918)
| |
| | nationality = [[Germany|German]]
| |
| | ethnicity = Jewish
| |
| | field = [[Mathematics]]
| |
| | work_institutions = [[University of Halle]]
| |
| | alma_mater = [[ETH Zurich]], [[University of Berlin]]
| |
| | thesis_title = De aequationibus secundi gradus indeterminatis
| |
| | thesis_year = 1867
| |
| | doctoral_advisor = [[Ernst Kummer]]<br>[[Karl Weierstrass]]
| |
| | doctoral_students = [[Alfred Barneck]]
| |
| | known_for = [[Set theory]]
| |
| | prizes =
| |
| }}
| |
|
| |
| '''Georg Ferdinand Ludwig Philipp Cantor''' ({{IPAc-en|ˈ|k|æ|n|t|ɔr}} {{respell|KAN|tor}}; {{IPA-de|ˈɡeɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantɔʁ|lang}}; {{OldStyleDate|March 3|1845|February 19}} – January 6, 1918<ref>[[#Guinness2000|Grattan-Guinness 2000]], p. 351</ref>) was a [[Germans|German]] [[mathematician]], best known as the inventor 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. In fact, 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 of which he was well aware.<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>
| |
| | |
| Cantor's theory of [[transfinite number]]s was originally regarded as so counter-intuitive – even shocking – that it encountered [[Controversy over Cantor's theory|resistance]] from mathematical contemporaries such as [[Leopold Kronecker]] and [[Henri Poincaré]]<ref>[[#Dauben2004|Dauben 2004]], p. 1.</ref> and later from [[Hermann Weyl]] and [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]], while [[Ludwig Wittgenstein]] raised [[philosophical objections to Cantor's theory|philosophical objections]]. 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. 120, 143.</ref> – on one occasion equating the theory of transfinite numbers with [[pantheism]]<ref name = "daub77102"/> – a proposition that Cantor vigorously rejected.
| |
| | |
| The objections to Cantor's work were occasionally fierce: Poincaré referred to his ideas as a "grave disease" infecting the discipline of [[mathematics]],<ref name="daub266">[[#Dauben1979|Dauben 1979]], p. 266.</ref> and 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 even 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".<ref name = "Rodych"/> Cantor's recurring bouts of [[clinical depression|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"/> It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.<ref name = "xdpfir">[[#Dauben2004|Dauben 2004]], pp. 8, 11, 12–13.</ref> | |
| [[David Hilbert]] defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."<ref>{{Citation |surname=Hilbert|given=David|year=1926|url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002270641|title=Über das Unendliche|journal=Mathematische Annalen|volume=95|pages=161–190 (170) |doi=10.1007/BF01206605}}.</ref><ref name="encomium"/>
| |
| | |
| ==Life==
| |
| | |
| ===Youth and studies===
| |
| [[File:Georg Cantor3.jpg|thumb|left|Cantor, ca. 1870.]]
| |
| Cantor was born in the western merchant colony in [[Saint Petersburg]], [[Russia]], and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding [[violin]]ist. His grandfather [[Franz Böhm (musician)|Franz Böhm]] (1788–1846) (the violinist [[Joseph Böhm]]'s brother) was the well-known musician and the soloist in the Russian empire in an 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 winters milder 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 1862, Cantor entered the [[University of Zürich]]. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies 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 [[Georg-August University of Göttingen|University of Göttingen]], then and later a center for mathematical research.
| |
| | |
| ===Teacher and researcher===
| |
| In 1867, Cantor completed his dissertation, on number theory, at the University of Berlin. After teaching briefly in a Berlin girls' school, Cantor 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>{{cite web |author=O'Connor, John J, and 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 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 two years earlier while on [[Switzerland|Swiss]] holiday.
| |
| | |
| Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a [[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. 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. Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.
| |
| | |
| In 1881, Cantor's Halle colleague [[Eduard Heine]] died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to [[Richard Dedekind|Dedekind]], [[Heinrich M. Weber]] and [[Franz Mertens]], in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
| |
| | |
| In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.<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 "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party:
| |
| | |
| {{quote|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 1884.<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:
| |
| | |
| {{quote|... 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 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|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 famous [[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. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
| |
| | |
| In 1890, Cantor was instrumental in founding the ''[[Deutsche Mathematiker-Vereinigung]]'' and chaired its first meeting in Halle in 1891, where he first introduced his [[Cantor's diagonal argument|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.
| |
| | |
| ===Late years===
| |
| 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 (while 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, even 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 in 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 ''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 attend 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 this did not come about. The following year, St. Andrews awarded Cantor an [[Honorary degree|honorary doctorate]], but illness precluded his receiving the degree in person.
| |
| | |
| Cantor retired in 1913, living in poverty and suffering from 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. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.
| |
| | |
| ==Mathematical work==
| |
| Cantor's work between 1874 and 1884 is the origin of [[set theory]].<ref name="Johnson p. 55">{{Citation |surname=Johnson|given= 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|page=55}}.</ref> Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of [[Aristotle]].<ref name ="uncontrov">This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later.</ref> 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>{{citation|title=Axiomatic Set Theory|first=Patrick|last=Suppes|authorlink=Patrick Suppes|year=1972|publisher=Dover|isbn=9780486616308|page=1|url=http://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, 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]]s (nondenumerable 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]] is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable.
| |
| | |
| 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> ([[aleph number|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 [[Hilbert's problems|twenty-three open problems]] in his famous 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">{{Citation |surname=Reid|given= Constance|year=1996|title=Hilbert|place=New York|publisher=Springer-Verlag|isbn=0-387-04999-1|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 Zurich in 1897, [[Adolf Hurwitz|Hurwitz]] and [[Jacques Hadamard|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 difficult problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices ''n'' in the ''n''th derived set ''S''<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 point]]s, 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+1, ω+2, ...<ref>{{Citation|last1=Cooke|first1=Roger|title=Uniqueness of trigonometric series and descriptive set theory, 1870–1985|journal=Archive for History of Exact Sciences|volume=45|pages=281|year=1993|doi=10.1007/BF01886630|issue=4|postscript=.}}</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>{{citation|author=Katz, Karin Usadi and [[Mikhail Katz|Katz, Mikhail G.]] |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}}></ref> Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.<ref>{{citation|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}}.</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 Cantor's 1874 article,<ref name="Johnson p. 55"/> "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").<ref>Cantor 1874. English translation: [[#Ewald|Ewald 1996]], pp. 840–843.</ref> This article 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 – see {{Citation |surname=Moore|given= A.W.|date=April 1995|title=A brief history of infinity|journal=Scientific American|volume=272|issue=4|pages=112–116 (114)|url=http://math123.net/hchs/MathDept/MathTalks/2012-13/b-h-inf.pdf}}.</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]]. [[Cantor's first uncountability proof|His proof]] is more complex than the more elegant [[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>, .... 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 – 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.<ref>For more details on Cantor's article, see [[Cantor's first uncountability proof]] and {{Citation |surname=Gray|given=Robert|year=1994|url=http://mathdl.maa.org/images/upload_library/22/Ford/Gray819-832.pdf |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|pages=819–832}}. 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>'' = ''[[e (mathematical constant)|e]] / n''). Call this set ''T’''. Then ''T'' = ''T’'' ∪ {''t<sub>n</sub>''} = ''T’'' ∪ {''t''<sub>2''n''-1</sub>} ∪ {''t''<sub>2''n''</sub>}. The set of reals '''R''' = ''T'' ∪ {''a<sub>n</sub>''} = ''T’'' ∪ {''t<sub>n</sub>''} ∪ {''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 [[disjoint sets]]: ''T’'' and two countable sets. A one-to-one correspondence between ''T'' and '''R''' is given by the function: ''f''(''t'') = ''t'' if ''t'' ∈ ''T’'', ''f''(''t''<sub>2''n''-1</sub>) = ''t<sub>n</sub>'', and ''f''(''t''<sub>2''n''</sub>) = ''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 (Cantor 1932, p. 142).</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 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, 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.
| |
| | |
| 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>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]] had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. [[Felix Bernstein]] supplied a correct proof in his 1898 PhD thesis; hence the name [[Cantor–Bernstein–Schroeder theorem]].
| |
| | |
| ====One-to-one correspondence====
| |
| {{Main|Bijection}}
| |
| [[File:Bijection.svg|thumb|A bijective function.]]
| |
| Cantor's 1874 [[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 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 famously wrote to Dedekind: "''Je le vois, mais je ne le crois pas''!" ("I see it, but I don't believe it!")<ref>{{Citation |surname=Wallace|given= David Foster|year=2003|title=Everything and More: A Compact History of Infinity|place=New York|publisher=W.W. Norton and Company|isbn=0-393-00338-8|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|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|Gödel]] and a 1963 one by [[Paul Cohen (mathematician)|Paul Cohen]] together imply that the continuum hypothesis can neither be 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>
| |
| | |
| ====Paradoxes of set theory====
| |
| Discussions of set-theoretic [[paradox]]es began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program.<ref>[[#Dauben1979|Dauben 1979]], pp. 240–270; see especially pp. 241, 259.</ref> In an 1897 paper on an unrelated topic, [[Cesare Burali-Forti]] set out the first such paradox, the [[Burali-Forti paradox]]: the [[ordinal number]] of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to [[David Hilbert|Hilbert]]. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.<ref name="daub248"/>
| |
| | |
| In 1899, Cantor discovered his eponymous [[Cantor's paradox|paradox]]: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set ''A'', the cardinal number of the power set of ''A'' is strictly larger than the cardinal number of ''A'' (this fact is now known as [[Cantor's theorem]]). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called ''[[limitation of size]]'',<ref>[[#Hallett|Hallett 1986]].</ref> according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as [[Class (set theory)|proper classes]].
| |
| | |
| One common view among mathematicians is that these paradoxes, together with [[Russell's paradox]], demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for [[Ernst Zermelo|Zermelo]] and others to produce [[axiomatic set theory|axiomatizations]] of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the [[von Neumann universe|iterative hierarchy]], which can be seen as explaining the idea of limitation of size. Some also question whether the [[Gottlob Frege|Fregean]] formulation of [[naive set theory]] (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.<ref>{{Citation |surname=Weir|given=Alan|year=1998|title=Naive Set Theory is Innocent!|journal=Mind|volume=107|number=428|doi=10.1093/mind/107.428.763|pages=763–798}} p. 766: "...it may well be seriously mistaken to think of Cantor's ''Mengenlehre'' [set theory] as naive..."</ref>
| |
| | |
| ==Philosophy, religion, 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 – 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"/>
| |
| | |
| 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"/> 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>{{Citation |surname=Snapper|given=Ernst |year=1979|title=[<!-- http://math.boisestate.edu/~tconklin/MATH547/Main/Exhibits/Three%20Crises%20in%20Math%20A.pdf -->http://www2.gsu.edu/~matgtc/three%20crises%20in%20mathematics.pdf The Three Crises in Mathematics: Logicism, Intuitionism and Formalism]|journal=Mathematics Magazine|volume=524|pages=207–216}}.</ref> Mathematicians such as [[Luitzen Egbertus Jan Brouwer|Brouwer]] and especially [[Henri Poincaré|Poincaré]] adopted an [[Intuitionism|intuitionist]] stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."<ref name= "daub266"/> 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.<ref name="Rodych">Rodych 2007</ref>
| |
| | |
| 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>{{Citation |surname=Davenport|year=1997|given=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}}.</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"/>
| |
| | |
| {{quote|... 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>Cantor 1932, p. 404. Translation in [[#Dauben1977|Dauben 1977]], p. 95.</ref>}}
| |
| | |
| 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.<ref name="daub296">[[#Dauben1979|Dauben 1979]], p. 296.</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 Johannes Franzelin]], who once replied by equating the theory of transfinite numbers with [[pantheism]].<ref name="daub77102">[[#Dauben1977|Dauben 1977]], p. 102.</ref> 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 famous 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 infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".
| |
| | |
| Cantor's 1883 paper reveals that he was well aware of the [[Controversy over Cantor's theory|opposition]] his ideas were encountering:
| |
| | |
| {{quote|... 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|Descartes]], [[George Berkeley|Berkeley]], [[Gottfried Leibniz|Leibniz]], and [[Bernard Bolzano|Bolzano]] on infinity.
| |
| | |
| ==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.]]
| |
| "Very little is known for sure about the origin and education of George Woldemar Cantor."<ref name="pi87">[[#Purkert|Purkert and Ilgauds 1985]], p. 15.</ref> 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.<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 was sometimes called Jewish in his lifetime,<ref>For example, [[Jewish Encyclopedia]], art. "Cantor, Georg"; [[Jewish Year Book]] 1896–97, "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.</ref> but has also variously been called Russian, German, and Danish as well.
| |
| | |
| Jakob Cantor, Cantor's grandfather, gave his children [[Christianity|Christian]] [[saint]]s' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they [[converted]] to Christianity. 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. His 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:
| |
| {{quote|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 ...<ref name="pi87"/>}}
| |
| | |
| ("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.<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 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>
| |
| | |
| There were documented statements, during the 1930s, that called this Jewish ancestry into question:
| |
| {{quote|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 ..."<ref name="pi87"/>}}
| |
| | |
| It is also later said in the same document:
| |
| {{quote|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: ...<ref name="pi87"/>}}(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.")<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> a Hungarian violinist [[Josef Böhm]], 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:
| |
| {{quote|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>}}
| |
| | |
| ==Historiography==
| |
| Until the 1970s, the chief academic publications on Cantor were two short monographs by [[Arthur Moritz Schönflies|Schönflies]] (1927) – largely the correspondence with Mittag-Leffler – 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, and fills the picture with stereotypes. 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 – 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>[[#Dauben79|Dauben 1979]]</ref>
| |
| | |
| ==See also==
| |
| {{Portal|Set theory|Logic}}
| |
| * [[Cantor cube]]
| |
| * [[Cantor function]]
| |
| * [[Cantor medal]] – award by the [[Deutsche Mathematiker-Vereinigung]] in honor of Georg Cantor.
| |
| * [[Cantor set]]
| |
| * [[Cantor space]]
| |
| * [[Cantor's back-and-forth method]]
| |
| * [[Controversy over Cantor's theory]]
| |
| * [[Heine–Cantor theorem]]
| |
| * [[Infinity]]
| |
| * [[List of German inventors and discoverers]]
| |
| * [[Pairing function]]
| |
| | |
| ==Notes==
| |
| {{Reflist|colwidth=30em}}
| |
| | |
| ==References==
| |
| * {{Citation |surname=[[Joseph Dauben|Dauben]]|jstor=2708842|given= Joseph W.|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}}.
| |
| * {{Citation |surname=Dauben|given= Joseph W.|year=1979|title=Georg Cantor: his mathematics and philosophy of the infinite|place=Boston|publisher=Harvard University Press|isbn=978-0-691-02447-9|ref=Dauben1979}}.
| |
| * {{Citation |surname=Dauben|given= Joseph|year=1993, 2004|url=http://www.acmsonline.org/journal/2004/Dauben93.htm |chapter=Georg Cantor and the Battle for Transfinite Set Theory|title=Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA)|pages=1–22|ref=Dauben2004}}. Internet version published in Journal of the ACMS 2004.
| |
| * {{Citation |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}}.
| |
| * {{Citation |surname=[[Ivor Grattan-Guinness|Grattan-Guinness]]|given=[[Ivor Grattan-Guinness|Ivor]]|year=1971|title=Towards a Biography of Georg Cantor|doi=10.1080/00033797100203837|journal=Annals of Science|volume=27|pages=345–391|ref=Guinness1971}}.
| |
| * {{Citation |surname=[[Ivor Grattan-Guinness|Grattan-Guinness]]|given=[[Ivor Grattan-Guinness|Ivor]]|year=2000|title=The Search for Mathematical Roots: 1870–1940|publisher=Princeton University Press|isbn=978-0-691-05858-0|ref=Guinness2000}}.
| |
| * {{Citation |surname=Hallett|given=Michael|title=Cantorian Set Theory and Limitation of Size|publisher=Oxford University Press|place=New York|year=1986|isbn=0-19-853283-0|ref=Hallett}}.
| |
| * {{Citation |surname=Purkert|given=Walter|surname2=Ilgauds|given2=Hans Joachim|year=1985|title=Georg Cantor: 1845–1918|publisher=[[Birkhäuser Verlag|Birkhäuser]]|isbn=0-8176-1770-1|ref=Purkert}}.
| |
| *{{Citation |surname=Suppes|given=Patrick|year=1972, 1960|title=Axiomatic Set Theory|place=New York|publisher=Dover|isbn= 0-486-61630-4|ref=Suppes}}. 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.
| |
| | |
| ==Bibliography==
| |
| :''Older sources on Cantor's life should be treated with caution. See [[#Historiography|Historiography section]] above.''
| |
| ; Primary literature in English:
| |
| * {{Citation |surname=Cantor|given=Georg|year=1955|origyear=1915|url=http://www.archive.org/details/contributionstot003626mbp|title=Contributions to the Founding of the Theory of Transfinite Numbers|editor=[[Philip Jourdain]]|place=New York|publisher=Dover|isbn=978-0-486-60045-1|ref=Cantor1955}}.
| |
| | |
| ; Primary literature in German:
| |
| * {{Citation |surname=Cantor|given=Georg|year=1874|url=http://bolyai.cs.elte.hu/~badam/matbsc/11o/cantor1874.pdf|title=Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen|journal=[[Journal für die Reine und Angewandte Mathematik]]|volume=77|pages=258–262}}.
| |
| * {{Citation |surname=Cantor |given=Georg |year=1932 <!---|url=http://web.archive.org/web/20070630234312/http://philosophons.free.fr/philosophes/cantor1932.pdf| format=PDF---> |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}}. Almost everything that Cantor wrote. Includes excerpts of his correspondence with [[Richard Dedekind|Dedekind]] (p.443-451) and [[Adolf Fraenkel|Fraenkel's]] Cantor biography (p.452-483) in the appendix.
| |
| | |
| ; Secondary literature:
| |
| * {{Citation |surname=Aczel|given=Amir D.|year=2000|title=The Mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind|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.
| |
| * {{Citation |surname=Dauben|given= Joseph W.|year=1983|title=Georg Cantor and the Origins of Transfinite Set Theory|journal=Scientific American|volume=248|issue=6|month=June|pages=122–131}}
| |
| * {{Citation |surname=Ferreirós|given=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.
| |
| * {{Citation |surname=[[Paul Halmos|Halmos]]|given=[[Paul Halmos|Paul]]|year=1998, 1960|title=Naive Set Theory|place=New York & Berlin|publisher=Springer}}. ISBN 3-540-90092-6
| |
| * {{Citation |surname=Hill|given= C. O.|surname2=Rosado Haddock|given2=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.
| |
| * {{Citation |surname=Meschkowski|given= Herbert|year=1983|title=Georg Cantor, Leben, Werk und Wirkung (George Cantor, Life, Work and Influence, in German)|publisher= Wieveg, Braunschweig}}
| |
| * {{Citation |surname=[[Roger Penrose|Penrose]]|given=[[Roger Penrose|Roger]]|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]].
| |
| * {{Citation |surname=[[Rudy Rucker|Rucker]]|given=[[Rudy Rucker|Rudy]]|year=2005, 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.
| |
| * {{Citation |surname=Rodych|given=Victor|year=2007|chapter=[http://plato.stanford.edu/entries/wittgenstein-mathematics/ Wittgenstein's Philosophy of Mathematics]|title=The Stanford Encyclopedia of Philosophy|editor=Edward N. Zalta}}.
| |
| | |
| ==External links==
| |
| {{Wikiquote}}
| |
| {{commons category}}
| |
| * {{MacTutor|id=Cantor}}
| |
| * {{MacTutor|class=HistTopics|id = Beginnings_of_set_theory|title = A history of set theory}} Mainly devoted to Cantor's accomplishment.
| |
| * {{MathGenealogy |id=29561}}
| |
| * Stanford Encyclopedia of Philosophy: [http://plato.stanford.edu/entries/set-theory/ Set theory] by [[Thomas Jech]].
| |
| * Grammar school Georg-Cantor Halle (Saale): [http://www.cantor-gymnasium.de Georg-Cantor-Gynmasium Halle]
| |
| | |
| {{Authority control|PND=118518887|LCCN=n/82/252409|VIAF=39412881}}
| |
| | |
| {{Featured article}}
| |
| <!--Metadata: see [[Wikipedia:Persondata]]-->
| |
| | |
| {{Logic}}
| |
| {{Set theory}}
| |
| {{Infinity}}
| |
| | |
| {{Persondata
| |
| |NAME = Cantor, Georg Ferdinand Ludwig Philipp
| |
| |ALTERNATIVE NAMES = Cantor, Georg
| |
| |SHORT DESCRIPTION = Mathematician who originated [[set theory]].
| |
| |DATE OF BIRTH = 3 March 1845
| |
| |PLACE OF BIRTH = [[Saint Petersburg]], [[Russia]]
| |
| |DATE OF DEATH = 6 January 1918
| |
| |PLACE OF DEATH = [[Halle, Saxony-Anhalt]], [[Germany]]
| |
| }}
| |
| | |
| {{DEFAULTSORT:Cantor, Georg}}
| |
| [[Category:People from Saint Petersburg]]
| |
| [[Category:German mathematicians]]
| |
| [[Category:German logicians]]
| |
| [[Category:Set theorists]]
| |
| [[Category:19th-century German writers]]
| |
| [[Category:20th-century German writers]]
| |
| [[Category:19th-century German mathematicians]]
| |
| [[Category:20th-century mathematicians]]
| |
| [[Category:German philosophers]]
| |
| [[Category:19th-century philosophers]]
| |
| [[Category:20th-century philosophers]]
| |
| [[Category:Martin Luther University of Halle-Wittenberg faculty]]
| |
| [[Category:ETH Zurich alumni]]
| |
| [[Category:German Lutherans]]
| |
| [[Category:People with bipolar disorder]]
| |
| [[Category:Baltic-German people]]
| |
| [[Category:1845 births]]
| |
| [[Category:1918 deaths]]
| |
| | |
| {{Link FA|sl}}
| |
| {{Link FA|vi}}
| |