Georg Cantor

Georg Cantor
File:Georg Cantor.jpg
Born	(1845-03-03)March 3, 1845 St. Petersberg, Russia
Died	January 6, 1918(1918-01-06) (aged 72) Halle, Germany
Alma mater	ETH Zurich,University of Berlin
Known for	Set theory
Scientific career
Fields	Mathematics
Institutions	University of Halle

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician. He is best known as the creator of set theory. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well–ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities." He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's theory of transfinite numbers was originally regarded as so counter–intuitive — even shocking — that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L.E.J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics, and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan," a "renegade" and a "corrupter of youth." Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries, but these bouts can now be seen as probable manifestations of a bipolar disorder.

The harsh criticism has been matched by international accolades. In 1904 the Royal Society of London awarded Cantor its Sylvester Medal, the highest honor it can confer. Today, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift. Cantor believed his theory of transfinite numbers had been communicated to him by God. In the words of David Hilbert: "No one shall expel us from the Paradise that Cantor has created."

Life

Youth and studies

Cantor was born in 1845 in the Western merchant colony in St. Petersburg, Russia, and brought up in the city until he was eleven. Georg was the eldest of six children. He was an outstanding violinist, having inherited his parents' considerable musical and artistic talents.Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of St. 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 Federal Polytechnic Institute in Zurich, today the ETH Zurich and began studying mathematics.

After his father's death in 1863, leaving a substantial inheritance, Cantor shifted his studies to the University of Berlin, attending lectures by Weierstrass, Kummer, and Kronecker, and befriending his fellow student Hermann Schwarz. He spent the summer of 1866 at the University of Göttingen, then and later a very important center for mathematical research. In 1867, Berlin granted him the Ph.D. for a thesis on number theory, De aequationibus secundi gradus indeterminatis.

Teacher and researcher

After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory.

In 1874, Cantor married Vally Guttmann. They had six children, the last 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 befriended two years earlier while on Swiss holiday.

Cantor was promoted to Extraordinary Professor in 1872, and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor very much desired a chair at a more prestigious university, in particular at Berlin, then the leading German university. However, Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with having Cantor as a colleague. Worse yet, Kronecker, a well–established figure within the mathematical community and Cantor's former professor, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever 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 Dedekind, Heinrich 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, apprently as a result of Dedekind's refusal to accept the chair at Halle. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag–Leffler's journal Acta Mathematica. But in 1885, Mittag–Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta. 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 wrote 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."

Cantor then sharply curtailed his relationship and correspondence with Mittag–Leffler, displaying a tendency to interpret well–intentioned criticism as a deeply personal affront.

Cantor suffered his first known bout of depression in 1884. This emotional crisis led him to apply to lecture on philosophy rather than on mathematics. Every one of the 52 letters Cantor wrote to Gösta Mittag-Leffler that year attacked Kronecker. Cantor soon recovered, but a passage from one of these letters is revealing of the damage to his 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."

Although he performed some valuable work after 1884, he never attained again the high level of his remarkable papers of 1874–84. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker. While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful that they were its cause. Rather, his posthumous diagnosis of bipolarity has been accepted as the root cause of his erratic mood.

Cantor believed that Francis Bacon wrote the plays attributed to Shakespeare. During his 1884 illness, he began an intense study of Elizabethan literature in an attempt to prove his Bacon authorship thesis. He eventually published two pamphlets, in 1896 and 1897, setting out his thinking about Bacon and Shakespeare.

In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891; 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 he felt towards Kronecker, Cantor invited him to address the meeting, but Kronecker was unable to do so because his spouse was dying at the time.

Late years

Cantor's youngest son died suddenly in 1899 (while Cantor was delivering a lecture on Shakespeare and Bacon), and this tragedy finally drained Cantor of much of his passion for mathematics. 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. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Russell's paradox, and Cantor'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 doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, and suffered from poverty, even hunger, during World War I. The public celebration of his 70th birthday was cancelled because of the war. He died in the sanatorium where he had spent the final year of his life.

Work

An illustration of Cantor's diagonal argument for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the list of sequences above.

Cantor's work between 1874 and 1884 is the origin of set theory. He first pointed out that given any set A, the set of all possible subsets of A, called the power set of A, exists. He then proved that the power set of an infinite set A has a size greater than the size of A (this fact is now known as Cantor's theorem). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs the transfinite cardinal and ordinal numbers, and their peculiar arithmetic. His notation for the cardinal numbers was the Hebrew letter $\aleph$ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter omega.

Cantor was also the first to appreciate the value of one-to-one correspondences (hereinafter denoted "1–to–1") for set theory. He defined finite and infinite sets, breaking down the latter into denumerable and nondenumerable sets, and proved that the set of all rational numbers is denumerable, but that the set of all real numbers is not and hence is strictly bigger.

Number theory and function theory

Cantor's first 10 papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one 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.

Set theory

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen" ("On a Characteristic Property of All Real Algebraic Numbers"). The paper, published in Crelle's Journal thanks to Dedekind's support (and despite Kronecker's opposition), was the first to formulate a mathematically rigorous proof that there was more than one kind of infinity. This demonstration is a centerpiece of his legacy as a mathematician, helping lay the groundwork for both calculus and the analysis of the continuum of real numbers. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). In the process, he also became the first to invoke the notion of a 1–to–1 correspondence, though he did not use that phrase. He then proved that the real numbers were not countable, albeit employing a proof more complex than the remarkably elegant and justly celebrated diagonal argument he set out in 1891.

Liouville had established the existence of transcendental numbers in 1851, and Cantor's paper established that they are nondenumerable (that is, not countable). The logic is as follows: Cantor had shown that union of two denumerable sets must be denumerable. The set of all real numbers is equal to the union of the set of algebraic numbers with the set of transcendental numbers (that is, real numbers must be either algebraic or transcendental). The 1874 paper showed that the algebraic numbers (that is, the roots of polynomial equations with integer coefficients), were denumerable. In contrast, Cantor had also just shown that the real numbers were not denumerable. If transcendental numbers were denumerable, then the result of their union with algebraic numbers would also be denumerable. Since their union (which equals the set of all real numbers) is nondenumerable, it logically follows that the transcendentals must be nondenumerable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval.

In 1874, Cantor 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 stronger result: there exists a 1–to–1 correspondence between the points on the unit line segment and all of the points in a p–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!") 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, which again displeased Kronecker. Cantor wanted to withdraw the paper, but Dedekind persuaded him not to do so; moreover, Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

This paper made precise the notion of a 1–to–1 correspondence, and defined denumerable sets as sets which can be put into a 1–to–1 correspondence with the natural numbers. Cantor introduces 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. He then proves that the rational numbers have the smallest infinite power, and that Rⁿ has the same power as R. Moreover, countably many copies of R have the same power as R. While he made free use of countable 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.

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. By agreeing to publish these articles, the editor displayed courage, because of the growing opposition to Cantor's ideas, led by Kronecker. Kronecker 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.

The fifth paper in this series, "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 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.

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."

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, Descartes, Berkeley, Leibniz, and Bolzano on infinity.

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). His inability to prove the continuum hypothesis caused Cantor considerable anxiety but, with the benefit of hindsight, is entirely understandable: a 1940 result by Gödel and a 1963 one by 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").

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. Edmund Husserl was his Halle colleague and friend from 1886 to 1901. While Husserl later made his reputation in philosophy, his doctorate was in mathematics and supervised by Weierstrass' student Leo Königsberger. Cantor also wrote on the theological implications of his mathematical work; for instance, he identified the Absolute Infinite with God. To Cantor, the philosophical implications of his mathematical views were intrinsically linked — in part because 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.

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. (The English translation is Cantor 1955.) 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 sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schroeder 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 Ph.D. thesis; hence the name Cantor-Schroeder-Bernstein theorem.

Paradoxes of set theory

Around this time, the first discussion of set–theoretic paradoxes began to appear, some of which implied fundamental problems with Cantor's entire program of set theory. 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 Hilbert. Criticism mounted to the point where Cantor launched counter–arguments in 1903, intended to defend the basic tenets of his set theory.

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any sets A, the cardinal number of the power set of A > cardinal number of A (Cantor's theorem again). This paradox, together with Burali–Forti's, led Cantor to formulate his concept of limitation of size, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Today they would be called 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 Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the 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.

Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures delivered at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard both expressed their admiration for Cantor's set theory. At that Congress, Cantor also renewed his friendship and correspondence with Dedekind. Charles Peirce in America also praised Cantor's set theory. In 1905, Cantor began a correspondence, later published, with his British admirer and translator Philip Jourdain, on the history of set theory and on Cantor's religious ideas.

Cantor's ancestry

Cantor's paternal grandparents were from Copenhagen, and fled the disruption of the Napoleonic Wars. Cantor himself called them "Israelites", but there is no direct evidence on whether his grandparents, were practicing Jews. In point of fact, Jakob Cantor, his grandfather, gave his children saint's names. Further, several of his grandmother's relatives were in the Czarist civil service, an institution that did not encourage non–Christian self–identification. Cantor's father, Georg Woldemar Cantor, was educated in the 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 Austrian born in St. Petersburg and baptized Roman Catholic but converted to Protestantism upon marriage. She may, however, have been of Jewish descent.

Cantor was not himself Jewish by faith, but has nevertheless been called variously German, Jewish, Russian, and Danish.

Historiography

Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönfliess (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". Bell presents Cantor's relationship with his father as Oedipal; Cantor's differences with Kronecker as a quarrel between two Jews, and fills the picture with stereotypes; and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan–Guinness (1971) found that none of these 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 St Petersburg by unknown parents.

See also

• Cantor dust
• Cantor function
• Cantor set
• Cantor's back-and-forth method
• Cantor's diagonal argument
• Cantor's theorem
• Cantor's paradox
• Cantor–Bernstein–Schroeder theorem
• Heine–Cantor theorem
• Cantor's first uncountability proof
• Continuum hypothesis
• Countable set
• Uncountable set
• naive set theory
• one-to-one correspondence
• Cardinality
  • Cardinal number
  • Aleph number
• Ordinal number
• well-order
• Controversy over Cantor's theory
• Infinity
• Cantor medal—award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.
• Baconian theory which Cantor subscribed to

References

Older sources on Cantor's life should be treated with some caution. See the section on historiography above.

Primary literature in English

• Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0486600451
• Ewald, William B. (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. ISBN 978-0198532712

Primary literature in German

• Cantor, Georg (1932). Gesammelte Abhandlungen mathematischen und philosophischen inhalts. Almost everything that Cantor wrote.

Secondary literature

• Aczel, Amir D. (2000). The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind. New York: Four Walls Eight Windows Publishing. ISBN 0760777780. A popular treatment of infinity, in which Cantor is frequently mentioned.
• Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. The definitive biography to date. ISBN 978-0-691-02447-9
• Dauben, Joseph (1993, 2004). "Georg Cantor and the Battle for Transfinite Set Theory" in Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA) (pp. 1–22). Internet version published in Journal of the ACMS 2004.
• Grattan–Guinness, Ivor (1971). Towards a Biography of Georg Cantor. Annals of Science 27:345–391.
• Grattan–Guinness, Ivor (2000). The Search for Mathematical Roots: 1870–1940. Princeton University Press. ISBN 978-0691058580
• Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN 0-19-853283-0
• Halmos, Paul (1998, 1960). Naive Set Theory. Springer.
• Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. Three chapters and 18 index entries on Cantor.
• Johnson, Phillip E. (1972). The Genesis and Development of Set Theory. The Two-Year College Mathematics Journal 3.1:55–62.
• Moore, A.W. (1995, April). A brief history of infinity. Scientific American.4:112–116.
• Penrose, Roger (2004). The Road to Reality. Alfred A. Knopf. Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
• Purkert, Walter & Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 0-8176-1770-1
• Reid, Constance (1996). Hilbert. New York: Springer–Verlag.
• Rucker, Rudy (2005, 1982). Infinity and the Mind. Princeton University Press. Deals with similar topics to Aczel, but in more depth.
• Suppes, Patrick (1972, 1960). Axiomatic Set Theory. New York: Dover. 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.
• Wallace, David Foster (2003). Everything and More: A Compact History of Infinity. New York: W.W. Norton and Company.
• Weir, Alan (1998). Naive Set Theory is Innocent!. Mind 107.428:763–98.

External links

• O'Connor, J. J., and Robertson, E.F. MacTutor archive. The following are the source for much of this entry:
  • O'Connor, John J.; Robertson, Edmund F., "Georg Cantor", MacTutor History of Mathematics Archive, University of St Andrews
  • A history of set theory. Mainly devoted to Cantor's accomplishment.
• Georg Cantor at the Mathematics Genealogy Project
• Selections from Cantor's philosophical writing.
• Text of the 1891 diagonal proof.
• Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech.
• Encyclopedia Britannica: Georg Cantor.
• Grammar school Georg–Cantor Halle(Saale): Georg-Cantor-Gynmasium Halle