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Georg Cantor

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   Georg Cantor
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   Georg Cantor

   Georg Ferdinand Ludwig Philipp Cantor ( March 3, 1845, St. Petersburg,
   Russia – January 6, 1918, Halle, Germany) was a German mathematician
   who 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 work encountered resistance from mathematical contemporaries
   such as Leopold Kronecker and Henri Poincaré, and later from Hermann
   Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical
   objections. His 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.

   Nowadays, 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. In the words
   of David Hilbert: "No one shall expel us from the Paradise that Cantor
   has created."

Life

   The ancestry of Cantor's father, Georg Woldemar Cantor, is not entirely
   clear. He was born between 1809 and 1814 in Copenhagen, Denmark, and
   brought up in a Lutheran German mission in St. Petersburg. Georg
   Cantor's father was a Danish man of Lutheran religion. Biographies
   dispute his family's ancestral origins; they often point to a potential
   Jewish background, perhaps of Iberian origin. His mother, Maria Anna
   Böhm, was born in St. Petersburg and came from an Austrian Roman
   Catholic family. She had converted to Protestantism upon marriage.
   Georg Cantor was the eldest of six children. The father was very devout
   and instructed all his children thoroughly in religious affairs.
   Throughout the rest of his life Georg Cantor held to the Christian
   (Lutheran) faith.

   The father was a broker on the St Petersburg Stock Exchange. Cantor, an
   outstanding violinist, inherited his parents' considerable musical and
   artistic talents.

   When Cantor's father 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, following his father's
   wishes, Cantor entered the Federal Polytechnic Institute in Zurich,
   today the ETH Zurich and began studying mathematics.

   After his father's death in 1863, 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
   a summer at the University of Göttingen, then and later a very
   important centre for mathematical research. In 1867, Berlin granted him
   the Ph.D. for a thesis on number theory, De aequationibus secundi
   gradus indeterminatis. After teaching one year 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 an inheritance from his father. During his
   honeymoon in Switzerland, Cantor spent much time in mathematical
   discussions with Richard Dedekind, whom he befriended two years earlier
   while on another 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, and his colleague Hermann Schwarz were not
   agreeable to having Cantor as a colleague. Worse yet, Kronecker, who
   was peerless among German mathematicians while he was alive,
   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. This episode is revealing of
   Halle's lack of standing among German mathematics departments. Wangerin
   was eventually appointed, but he was never close to Cantor.

   In 1884, Cantor suffered his first known bout of depression. This
   emotional crisis led him to apply to lecture on philosophy rather than
   on mathematics. Every one of the 52 letters Cantor wrote to
   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 whether they were its cause, which was
   probably bipolar disorder.

   In 1888, he 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. On Cantor,
   Husserl, and Frege, see Hill and Rosado Haddock (2000). Cantor also
   wrote on the theological implications of his mathematical work; for
   instance, he identified the Absolute Infinite with God.

   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, chaired its first meeting in Halle in 1891,
   and was elected its first president. This is strong evidence that
   Kronecker's attitude had not been fatal to his reputation. Setting
   aside the animosity he felt towards Kronecker, Cantor invited him to
   address the meeting; Kronecker was unable to do so because his spouse
   was dying at the time.

   After the 1899 death of his youngest son, 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, lecturing on the
   paradoxes of set theory (eponymously attributed to Burali-Forti,
   Russell, and Cantor himself) 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

   Cantor was the originator of set theory, 1874-84. He was the first to
   see that infinite sets come in different sizes, as follows. He first
   showed 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 with a natural number
   subscript; for the ordinals he employed the Greek letter omega.

   Cantor was 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. There exists a 1-to-1
   correspondence between any denumerable set and the set of all natural
   numbers; all other infinite sets are nondenumerable. He 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. The cardinality of
   the natural numbers is aleph-null; that of the reals is larger, and is
   at least aleph-one (the latter being the next smallest cardinal after
   aleph-null).

   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 Eduard 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.

   Cantor's 1874 paper, "On a Characteristic Property of All Real
   Algebraic Numbers", marks the birth of set theory. It was published in
   Crelle's Journal, despite Kronecker's opposition, thanks to Dedekind's
   support. Previously, all infinite collections had been (silently)
   assumed to be of "the same size"; Cantor was the first to show that
   there was more than one kind of infinity. In doing so, he became the
   first to invoke the notion of a 1-to-1 correspondence, albeit not
   calling it such. He then proved that the real numbers were not
   denumerable, employing a proof more complex than the remarkably elegant
   and justly celebrated diagonal argument he first set out in 1891.

   The 1874 paper also showed that the algebraic numbers, i.e., the roots
   of polynomial equations with integer coefficients, were denumerable.
   Real numbers that are not algebraic are transcendental. Liouville had
   established the existence of transcendental numbers in 1851. Since
   Cantor had just shown that the real numbers were not denumerable and
   that the union of two denumerable sets must be denumerable, it
   logically follows from the fact that a real number is either algebraic
   or transcendental 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 wrote famously (and in French) "I see it, but I don't
   believe it!" This astonishing result 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 R^n 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 Godel 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 1882, the rich mathematical correspondence between Cantor and
   Dedekind came to an end. Cantor also began another important
   correspondence, with Mittag-Leffler in Sweden, and soon began to
   publish in Mittag-Leffler's journal Acta Mathematica. But in 1885,
   Mittag-Leffler asked Cantor to withdraw a 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."

   Thus ended his correspondence with Mittag-Leffler, as did Cantor's
   brilliant development of set theory over the previous 12 years.
   Mittag-Leffler had meant well, but this incident reveals how even
   Cantor's most brilliant contemporaries often failed to appreciate his
   work.

   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.

   Around this time, the set-theoretic paradoxes began to rear their
   heads. 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. Curiously, Cantor was highly critical
   of Burali-Forti's paper.

   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, ^fact check needed 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.
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