Georg Cantor

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

. He created
set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
, which has become a fundamental theory in mathematics. Cantor established the importance of
one-to-one correspondence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
between the members of two sets, defined
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...
and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. 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 he was well aware of. Cantor's theory of transfinite numbers 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 and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections to Cantor's theory, philosophical objections. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God.#Dauben2004, Dauben 2004, pp. 8, 11, 12–13. 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#Dauben1977, Dauben 1977, p. 86; #Dauben1979, Dauben 1979, pp. 120, 143. – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries,#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. though some have explained these episodes as probable manifestations of a bipolar disorder.#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". 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. David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."

# Life of Georg Cantor

## Youth and studies

Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. Cantor, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (musician), Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg Bourse, Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. In 1862, Cantor entered the ETH Zurich, Swiss Federal Polytechnic. After receiving a substantial inheritance upon his father's death in June 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 University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctorate degree in 1867.

# Mathematical work

Cantor's work between 1874 and 1884 is the origin of
set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Axiomatic set theory, Set theory has come to play the role of a foundations of mathematics, foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, Mathematical analysis, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics. In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite set, finite and infinite sets, subdividing the latter into countable set, denumerable (or countably infinite) sets and uncountable set, nondenumerable sets (uncountably infinite sets). Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875, is Nowhere dense set, nowhere dense, but has the same cardinality as the set of all real numbers, whereas the Rational number, rationals are everywhere dense, but countable. He also showed that all countable dense total order, linear orders without end points are order-isomorphic to the rational numbers. Cantor introduced fundamental constructions in set theory, such as the power set of a set ''A'', which is the set of all possible subsets of ''A''. He later proved that the size of the power set of ''A'' is strictly larger than the size of ''A'', even when ''A'' is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and 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 $\aleph$ (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 address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of
set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
and on Cantor's religious ideas. This was later published, as were several of his expository works.

## Number theory, trigonometric series and ordinals

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to Mathematical analysis, analysis. Heine proposed that Cantor solve Open problem, an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a Function (mathematics), function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices ''n'' in the ''n''th derived set (mathematics), derived set ''S''''n'' of a set ''S'' of zeros of a trigonometric series. Given a trigonometric series f(x) with ''S'' as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had ''S''1 as its set of zeros, where ''S''1 is the set of limit points of ''S''. If ''S''''k+1'' is the set of limit points of ''S''''k'', then he could construct a trigonometric series whose zeros are ''S''''k+1''. Because the sets ''S''''k'' were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets ''S'', ''S''1, ''S''2, ''S''3,... formed a limit set, which we would now call ''S''''ω'', and then he noticed that ''S''ω would also have to have a set of limit points ''S''ω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ''ω'', ''ω'' + 1, ''ω'' + 2, ... Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as Sequence space, convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.

## Set theory

The beginning of set theory as a branch of mathematics is often marked by the publication of Georg Cantor's first set theory article, Cantor's 1874 paper, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerosity, equinumerous (that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from Cantor's diagonal argument, diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844. Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers as a sequence ''a''1, ''a''2, ''a''3, .... 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. 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. Between 1879 and 1884, Cantor published a series of six articles in ''Mathematische Annalen'' that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finitism, finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.#Dauben1977, Dauben 1977, p. 89. 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 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 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. 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'' equinumerous, equivalent to a subset of ''B'' and ''B'' equivalent to a subset of ''A'', then ''A'' and ''B'' are equivalent. Ernst Schröder (mathematician), Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein (mathematician), Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem.

### One-to-one correspondence

Cantor's 1874 Crelle's Journal, Crelle paper was the first to invoke the notion of a Bijection, 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far Mathematical jargon#stronger, stronger result: for any positive integer ''n'', there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space, ''n''-dimensional space. About this discovery Cantor wrote to Dedekind: "" ("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, 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 sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that ''n''-dimensional Euclidean space R''n'' has the same power as the real numbers R, as does a countably infinite Cartesian product, product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his Map (mathematics), mapping between the unit interval and the unit square was not a continuous function, continuous one. This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

### Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is ''exactly'' aleph-one, rather than just ''at least'' aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to mathematical proof, prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety. The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen (mathematician), Paul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").

# Cantor's ancestry

Cantor's paternal grandparents were from Copenhagen and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on them. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheranism, Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education.#Purkert, Purkert and Ilgauds 1985, p. 15. Cantor's mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic Church, Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating: ("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. According to biographers Eric Temple Bell, Cantor was of Jewish descent, 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 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.") In addition, Cantor's maternal great uncle, a Hungarian violinist Josef Böhm, has been described as Jewish, which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community. In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows: There were documented statements, during the 1930s, that called this Jewish ancestry into question:

# Biographies

Until the 1970s, the chief academic publications on Cantor were two short monographs by Arthur Moritz 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". Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents. A critique of Bell's book is contained in Joseph Dauben's biography.#Dauben1979, Dauben 1979 Writes Dauben:

* Absolute Infinite * Aleph number * Cardinality of the continuum * Cantor algebra * Cantor cube * Cantor distribution * Cantor function * Cantor medal – award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor * Cantor normal form theorem * Cantor space * Cantor tree surface * Cantor's back-and-forth method * Cantor's diagonal argument * Cantor's intersection theorem * Cantor's isomorphism theorem * Cantor's first set theory article * Cantor's paradox * Cantor's theorem * Knaster–Kuratowski fan, Cantor's leaky tent * Perfect set property, Cantor–Bendixson theorem * Cantor–Dedekind axiom * Schröder–Bernstein theorem, Cantor–Schröder–Bernstein * Cantor–Bernstein theorem * Cantor set * Cardinal number * Continuum hypothesis * Countable set * Derived set (mathematics) * Epsilon numbers (mathematics) * Factorial number system * Heine–Cantor theorem * Pairing function * Smith–Volterra–Cantor set * Transfinite number

# References

* . * . * Internet version published in ''Journal of the ACMS'' 2004. Note, though, that Cantor's Latin quotation described in this article as ''a familiar passage from the Bible'' is actually from the works of Seneca and has no implication of divine revelation. * . * . * . * . * . * . * . * . * . *. 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 section #Biographies, § Biographies above.''

* .

## Primary literature in German

* * . * * * * * Published separately as: ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre''. * * * * . 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

* . . A popular treatment of infinity, in which Cantor is frequently mentioned. * * . Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory. * . * * . Three chapters and 18 index entries on Cantor. * * Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Min

''American Catholic Philosophical Quarterly'', 83 (4): 532–553, https://doi.org/10.5840/acpq200983444. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in the ''Pantheismusstreit''. Brief mention is made of Cantor's learning from F.A.Trendelenburg. * . Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary Theoretical physics, theoretical physicist. * . Deals with similar topics to Aczel, but in more depth. * . * Leonida Lazzari, ''L'infinito di Cantor''. Editrice Pitagora, Bologna, 2008.