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 (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.
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 (for example, numbers and functions) from all the traditional areas of mathematics (such as
algebra,
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
sizes. He was the first to appreciate the importance of
one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory: two sets are said to have the same "size" if there exists a 1-to-1 correspondence between them. He used this concept to define
finite and
infinite sets, subdividing the latter into
denumerable (or countably infinite) sets and
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, but has the same cardinality as the set of all real numbers, whereas the
rationals are everywhere dense, but countable. He also showed that all countable dense
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
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 \aleph (
ℵ,
aleph) with a natural number subscript; for the ordinals he employed the Greek letter \omega (,
omega). 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 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 Zürich 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 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 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 for the existence of
uncountable sets. The sequence at the bottom cannot occur anywhere in the infinite list of sequences above. The publication of
Cantor's 1874 paper This paper was the first to provide a rigorous proof that there is more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be
equinumerous (that is, of "the same size", or having the same number of elements). Cantor proved that real numbers and the positive
integers cannot be put in 1-to-1 correspondence and are thus not equinumerous. In other words, the real numbers are not
countable. His proof differs from the
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 adds that his constructions provide more – namely, 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. Analysis of the correspondence between Cantor and Dedekind later showed that the proof for the countability of the algebraic numbers originated with Dedekind, who also substantially simplified Cantor's original proof of the uncountability of the reals. Cantor failed to acknowledge Dedekind's contributions in his 1874 article. 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 are 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, he extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, Cantor published a paper containing his "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 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 laying out 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–Schröder theorem.
One-to-one correspondence Cantor's 1874
Crelle paper was the first to invoke the notion of
1-to-1 correspondence, though he did not use that term. He then began looking for 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
stronger result: for any positive integer
n, there exists a 1-to-1 correspondence between the points on the unit line segment and all the points in an
n-dimensional space. Of this discovery, Cantor wrote to Dedekind: "" ("I see it, but I don't believe it!") The result 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 precisely defined the concept of a 1-to-1 correspondence and introduced 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 is a 1-to-1 correspondence between them. Cantor defined
countable sets (or denumerable sets) as sets that can be put into 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, but 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 in vain for many years to prove it. His inability to prove the continuum hypothesis caused him considerable anxiety.
Absolute infinite, well-ordering theorem, and paradoxes In 1883, Cantor divided the infinite into the transfinite and the
absolute. The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α+1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the
well-ordering principle "every set can be well-ordered" and called it a "law of thought". Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an
aleph. First he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof. Cantor avoided
paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast,
Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is
inconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the
Burali-Forti paradox,
Cantor's paradox, and
Russell's paradox. Russell named paradoxes after
Cesare Burali-Forti and Cantor even though neither of them believed they had found paradoxes. In 1908, Zermelo published
his axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the
well-ordering theorem. Zermelo had proved this theorem in 1904 using the
axiom of choice, but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets. In 1923,
John von Neumann developed an axiom system that eliminates the paradoxes with an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann wrote that a
class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and postulated the axiom "A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets". This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined
models of set theory that satisfy von Neumann's axiom. ==Philosophy, religion, literature and Cantor's mathematics==