Georg Ferdinand Ludwig Philipp Cantor was
a German mathematician, best known as the
inventor of set theory, which has become a
fundamental theory in mathematics. Cantor
established the importance of one-to-one correspondence
between the members of two sets, defined infinite
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 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. Some Christian theologians saw
Cantor's work as a challenge to the uniqueness
of the absolute infinity in the nature of
God – 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: Poincaré referred to his 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." 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".
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,
though some have explained these episodes
as probable manifestations of a bipolar disorder.
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. It
has been suggested that Cantor believed his
theory of transfinite numbers had been communicated
to him by God. David Hilbert defended it from
its critics by famously declaring: "No one
shall expel us from the Paradise that Cantor
has created."
Life
Youth and studies
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 violinist. His grandfather Franz
Böhm was a well-known musician and soloist
in a Russian imperial orchestra. 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 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 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.
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 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.
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
had met 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 desired
a 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. Kronecker, who headed mathematics
at Berlin until his death in 1891, became
increasingly uncomfortable with the prospect
of having Cantor as a colleague, perceiving
him as a "corrupter of youth" for teaching
his ideas to a younger generation of mathematicians.
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 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
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. 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 then
curtailed his relationship and correspondence
with Mittag-Leffler, writing to a third party:
Had Mittag-Leffler had his way, I should have
to wait until the year 1984, which to me seemed
too great a demand! ... But of course I never
want to know anything again about Acta Mathematica.
Cantor suffered his first known bout of depression
in 1884. Criticism of his work weighed on
his mind: every one of the fifty-two letters
he wrote to Mittag-Leffler in 1884 mentioned
Kronecker. A passage from one of these letters
is revealing of the damage to Cantor's self-confidence:
... I don't know when I shall return to the
continuation of my scientific work. At the
moment I can do absolutely nothing with it,
and limit myself to the most necessary duty
of my lectures; how much happier I would be
to be scientifically active, if only I had
the necessary mental freshness.
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 Shakespeare;
this ultimately resulted in two pamphlets,
published in 1896 and 1897.
Cantor recovered soon thereafter, and subsequently
made further important contributions, including
his famous diagonal argument and 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 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. Soon after that second hospitalization,
Cantor's youngest son Rudolph died suddenly,
and this tragedy drained Cantor of much of
his passion for mathematics. 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. Although Ernst Zermelo demonstrated
less than a day later that König's proof
had failed, Cantor remained shaken, and momentarily
questioning God. 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. He did
not abandon mathematics completely, however,
lecturing on the paradoxes of set theory 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, living in poverty
and suffering from malnourishment during World
War I. 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. 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 and "the
infinite". By proving that there are many
possible sizes for infinite sets, Cantor established
that set theory was not trivial, and it needed
to be studied. Set theory has come to play
the role of a foundational theory in modern
mathematics, in the sense that it interprets
propositions about mathematical objects from
all the traditional areas of mathematics 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 sizes. He was also the first
to appreciate the importance of one-to-one
correspondences in set theory. He used this
concept to define finite and infinite sets,
subdividing the latter into denumerable sets
and uncountable sets.
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 subsets
of A. He later proved that the size of the
power set of A is strictly larger than the
size of A, even when A is an infinite set;
this result soon became known as Cantor's
theorem. Cantor developed an entire theory
and arithmetic of infinite sets, called cardinals
and ordinals, which extended the arithmetic
of the natural numbers. His notation for the
cardinal numbers was the Hebrew letter with
a natural number subscript; for the ordinals
he employed the Greek letter ω. This notation
is still in use today.
The Continuum hypothesis, introduced by Cantor,
was presented by David Hilbert as the first
of his twenty-three open problems in his famous
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,
Hurwitz and 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 analysis. Heine proposed that Cantor solve
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 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 nth derived set
Sn 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 S1 as its set of zeros, where
S1 is the set of limit points of S. If Sk+1
is the set of limit points of Sk, then he
could construct a trigonometric series whose
zeros are Sk+1. Because the sets Sk were closed,
they contained their Limit points, and the
intersection of the infinite decreasing sequence
of sets S, S1, S2, S3,... formed a limit set,
which we would now call Sω, and then he noticed
that Sω would also have to have a set of
limit points Sω+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 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 Cantor's 1874 article, "Über eine Eigenschaft
des Inbegriffes aller reellen algebraischen
Zahlen". 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 equinumerous. 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
is more complex than the more elegant 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 a1, a2, a3, .... In other words,
the real algebraic numbers are countable.
Cantor starts his second construction with
any sequence of real numbers. Using this sequence,
he constructs nested intervals whose intersection
contains a real number not in the sequence.
Since every sequence of real numbers can be
used to construct a real not in the sequence,
the real numbers cannot be written as a sequence
– that is, the real numbers are not countable.
By applying his construction to the sequence
of real algebraic numbers, Cantor produces
a transcendental number. Cantor points out
that his constructions prove more – namely,
they provide a new proof of Liouville's theorem:
Every interval contains infinitely many transcendental
numbers. Cantor's next article contains a
construction that proves the set of transcendental
numbers has the same "power" 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 Kronecker,
who admitted mathematical concepts only if
they could be constructed in a finite number
of steps from the natural numbers, which he
took as intuitively given. For Kronecker,
Cantor's hierarchy of infinities was inadmissible,
since accepting the concept of actual infinity
would open the door to paradoxes which would
challenge the validity of mathematics as a
whole. Cantor also introduced the Cantor set
during this period.
The fifth paper in this series, "Grundlagen
einer allgemeinen Mannigfaltigkeitslehre",
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.
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 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 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
Cantor's 1874 Crelle paper was the first to
invoke the notion of a 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 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. About
this discovery Cantor famously wrote to Dedekind:
"Je le vois, mais je ne le crois pas!" 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 "power" or "equivalence"
of sets: two sets are equivalent if there
exists a 1-to-1 correspondence between them.
Cantor defined countable sets as sets which
can be put into a 1-to-1 correspondence with
the natural numbers, and proved that the rational
numbers are denumerable. He also proved that
n-dimensional Euclidean space Rn has the same
power as the real numbers R, as does a countably
infinite product of copies of R. While he
made free use of countability as a concept,
he did not write the word "countable" until
1883. Cantor also discussed his thinking about
dimension, stressing that his mapping between
the unit interval and the unit square was
not a continuous one.
This paper displeased Kronecker, and Cantor
wanted to withdraw it; however, Dedekind persuaded
him not to do so and 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. Cantor believed the
continuum hypothesis to be true and tried
for many years to prove it, in vain. His inability
to prove the continuum hypothesis caused him
considerable anxiety.
The difficulty Cantor had in proving the continuum
hypothesis has been underscored by later developments
in the field of mathematics: 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.
Paradoxes of set theory
Discussions of set-theoretic paradoxes began
to appear around the end of the nineteenth
century. Some of these implied fundamental
problems with Cantor's set theory program.
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 set A, the
cardinal number of the power set of A is strictly
larger than the cardinal number of A. This
paradox, together with Burali-Forti's, led
Cantor to formulate a concept called 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. Such collections later became known as
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
is really a faithful interpretation of the
Cantorian conception.
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. 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. To Cantor, his mathematical views were
intrinsically linked to their philosophical
and theological implications – he identified
the Absolute Infinite with God, 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.
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. Mathematicians from
three major schools of thought 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 proofs
such as Cantor's diagonal argument are sufficient
proof that something exists, holding instead
that constructive proofs 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. 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. Mathematicians
such as Brouwer and especially Poincaré adopted
an 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." Finally, 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, thus conflating the concept
of rules for generating a set with an actual
set.
Some Christian theologians saw Cantor's work
as a challenge to the uniqueness of the absolute
infinity in the nature of God. In particular,
Neo-Thomist thinkers saw the existence of
an actual infinity that consisted of something
other than God as jeopardizing "God's exclusive
claim to supreme infinity". Cantor strongly
believed that this view was a misinterpretation
of infinity, and was convinced that set theory
could help correct this mistake:
... 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.
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.
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,
as well as theologians such as Cardinal Johannes
Franzelin, who once replied by equating the
theory of transfinite numbers with pantheism.
Cantor even sent one letter directly to Pope
Leo XIII himself, and addressed several pamphlets
to him.
Cantor's philosophy on the nature of numbers
led him to affirm a belief in the freedom
of mathematics to posit and prove concepts
apart from the realm of physical phenomena,
as expressions within an internal reality.
The only restrictions on this metaphysical
system are that all mathematical concepts
must be devoid of internal contradiction,
and that they follow from existing definitions,
axioms, and theorems. This belief is summarized
in his famous assertion that "the essence
of mathematics is its freedom." These ideas
parallel those of Edmund Husserl, whom Cantor
had met in Halle.
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 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's ancestry
Cantor's paternal grandparents were from Copenhagen,
and fled to Russia from the disruption of
the Napoleonic Wars. There is very little
direct information on his grandparents. Cantor
was sometimes called Jewish in his lifetime,
but has also variously been called Russian,
German, and Danish as well.
Jakob Cantor, Cantor's grandfather, gave his
children Christian saints' 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 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
George Woldemar's origin or education. His
mother, Maria Anna Böhm, was an Austro-Hungarian
born in Saint Petersburg and baptized Roman
Catholic; she converted to Protestantism upon
marriage. However, there is a letter from
Cantor's brother Louis to their mother, stating:
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 ...
("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.
There were documented statements, during the
1930s, that called this Jewish ancestry into
question:
More often [i.e., than the ancestry of the
mother] the question has been discussed of
whether Georg Cantor was of Jewish origin.
About this it is reported in a notice of the
Danish genealogical Institute in Copenhagen
from the year 1937 concerning his father:
"It is hereby testified that Georg Woldemar
Cantor, born 1809 or 1814, is not present
in the registers of the Jewish community,
and that he completely without doubt was not
a Jew ..."
It is also later said in the same document:
Also efforts for a long time by the librarian
Josef Fischer, one of the best experts on
Jewish genealogy in Denmark, charged with
identifying Jewish professors, that Georg
Cantor was of Jewish descent, finished without
result. [Something seems to be wrong with
this sentence, but the meaning seems clear
enough.] In Cantor's published works and also
in his Nachlass there are no statements by
himself which relate to a Jewish origin of
his ancestors. There is to be sure in the
Nachlass a copy of a letter of his brother
Ludwig from 18 November 1869 to their mother
with some unpleasant antisemitic statements,
in which it is said among other things: ...
(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 that he was unable
to find evidence of Jewish ancestry..
In a letter written by Georg Cantor to Paul
Tannery in 1896, 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..."
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:
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.
Historiography
Until the 1970s, the chief academic publications
on Cantor were two short monographs by Schönflies
– largely the correspondence with Mittag-Leffler
– and Fraenkel. 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, 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,
and fills the picture with stereotypes. Grattan-Guinness
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.
See also
Cantor algebra
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
References
Dauben, Joseph W., "Georg Cantor and Pope
Leo XIII: Mathematics, Theology, and the Infinite",
Journal of the History of Ideas 38: 85–108,
JSTOR 2708842 .
Dauben, Joseph W., Georg Cantor: his mathematics
and philosophy of the infinite, Boston: Harvard
University Press, ISBN 978-0-691-02447-9 .
Dauben, Joseph, "Georg Cantor and the Battle
for Transfinite Set Theory", Proceedings of
the 9th ACMS Conference, pp. 1–22  . Internet
version published in Journal of the ACMS 2004.
Ewald, William B., ed., From Immanuel Kant
to David Hilbert: A Source Book in the Foundations
of Mathematics, New York: Oxford University
Press, ISBN 978-0-19-853271-2 .
Grattan-Guinness, Ivor, "Towards a Biography
of Georg Cantor", Annals of Science 27: 345–391,
doi:10.1080/00033797100203837 .
Grattan-Guinness, Ivor, The Search for 
Mathematical Roots: 1870–1940, Princeton
University Press, ISBN 978-0-691-05858-0 .
Hallett, Michael, Cantorian Set Theory and
Limitation of Size, New York: Oxford University
Press, ISBN 0-19-853283-0 .
Purkert, Walter; Ilgauds, Hans Joachim, Georg
Cantor: 1845–1918, Birkhäuser, ISBN 0-8176-1770-1 .
Suppes, Patrick, Axiomatic Set Theory, New
York: Dover, ISBN 0-486-61630-4  . 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 section above.
Primary literature in English
Cantor, Georg [1915], Philip Jourdain, ed.,
Contributions to the Founding of the Theory
of Transfinite Numbers, New York: Dover, ISBN 978-0-486-60045-1 .
Primary literature in German
Cantor, Georg, "Ueber eine Eigenschaft des
Inbegriffes aller reellen algebraischen Zahlen",
Journal für die Reine und Angewandte Mathematik
77: 258–262 .
Cantor, Georg. "Grundlagen einer allgemeinen
Mannigfaltigkeitslehre". Mathematische Annalen
21: 545–586. .
Cantor, Georg. "Beiträge zur Begründung
der transfiniten Mengenlehre". Mathematische
Annalen 46: 481–512. 
Cantor, Georg. "Beiträge zur Begründung
der transfiniten Mengenlehre". Mathematische
Annalen 49: 207–246. 
Cantor, Georg, Ernst Zermelo, ed., Gesammelte
Abhandlungen mathematischen und philosophischen
inhalts, Berlin: Springer . Almost everything
that Cantor wrote. Includes excerpts of his
correspondence with Dedekind and Fraenkel's
Cantor biography in the appendix.
Secondary literature
Aczel, Amir D., The Mystery of the Aleph:
Mathematics, the Kabbala, and the Search for
Infinity, New York: Four Walls Eight Windows
Publishing . ISBN 0-7607-7778-0. A popular
treatment of infinity, in which Cantor is
frequently mentioned.
Dauben, Joseph W., "Georg Cantor and the Origins
of Transfinite Set Theory", Scientific American
248: 122–131 
Ferreirós, José, Labyrinth of Thought: A
History of Set Theory and Its Role in Mathematical
Thought, Basel, Switzerland: Birkhäuser .
ISBN 3-7643-8349-6 Contains a detailed treatment
of both Cantor's and Dedekind's contributions
to set theory.
Halmos, Paul, Naive Set Theory, New York & Berlin:
Springer  . ISBN 3-540-90092-6
Hilbert, David. "Über das Unendliche". Mathematische
Annalen 95: 161–190. doi:10.1007/BF01206605. 
Hill, C. O.; Rosado Haddock, G. E., Husserl
or Frege? Meaning, Objectivity, and Mathematics,
Chicago: Open Court . ISBN 0-8126-9538-0
Three chapters and 18 index entries on Cantor.
Meschkowski, Herbert, Georg Cantor, Leben,
Werk und Wirkung, Wieveg, Braunschweig 
Penrose, Roger, The Road to Reality, Alfred
A. Knopf . ISBN 0-679-77631-1 Chapter 16
illustrates how Cantorian thinking intrigues
a leading contemporary theoretical physicist.
Rucker, Rudy, Infinity and the Mind, Princeton
University Press  . ISBN 0-553-25531-2 Deals
with similar topics to Aczel, but in more
depth.
Rodych, Victor, "Wittgenstein's Philosophy
of Mathematics", in Edward N. Zalta, The Stanford
Encyclopedia of Philosophy .
External links
O'Connor, John J.; Robertson, Edmund F., "Georg
Cantor", MacTutor History of Mathematics archive,
University of St Andrews .
O'Connor, John J.; Robertson, Edmund F., "A
history of set theory", MacTutor History of
Mathematics archive, University of St Andrews .
Mainly devoted to Cantor's accomplishment.
Georg Cantor at the Mathematics Genealogy
Project
Stanford Encyclopedia of Philosophy: Set theory
by Thomas Jech.
Grammar school Georg-Cantor Halle: Georg-Cantor-Gynmasium
Halle
Poem about Georg Cantor
