Early history by
Purchotius (1730), presenting
Aristotle's
Categories The basic notion of grouping objects has existed since at least the
emergence of numbers, and the notion of treating sets as their own objects has existed since at least the
Tree of Porphyry in 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics; however,
Bernard Bolzano's
Paradoxes of the Infinite (
Paradoxien des Unendlichen, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on
Galileo's paradox, and introduced
one-to-one correspondence of infinite sets, for example between the
intervals [0,5] and [0,12] by the relation 5y = 12x. However, he resisted saying these sets were
equinumerous, and his work is generally considered to have been uninfluential in mathematics of his time. Before mathematical set theory, basic concepts of
infinity were considered to be in the domain of philosophy (see:
Infinity (philosophy) and ''''). Since the 5th century BC, beginning with Greek philosopher
Zeno of Elea in the West (and early
Indian mathematicians in the East), mathematicians had struggled with the concept of infinity. With the
development of calculus in the late 17th century, philosophers began to generally distinguish between potential and
actual infinity, wherein mathematics was only considered in the latter.
Carl Friedrich Gauss famously stated: Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics.Development of mathematical set theory was motivated by several mathematicians.
Bernhard Riemann's lecture
On the Hypotheses which lie at the Foundations of Geometry (1854) proposed new ideas about
topology. His lectures also introduced the concept of basing mathematics in terms of sets or
manifolds in the sense of a
class (which he called
Mannigfaltigkeit) now called
point-set topology. The lecture was published by
Richard Dedekind in 1868, along with Riemann's paper on
trigonometric series (which presented the
Riemann integral). The latter was the starting point for a movement in
real analysis of the study of “seriously”
discontinuous functions. A young
Georg Cantor entered into this area, which led him to the study of
point-sets. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with
equivalence relations,
partitions of sets, and
homomorphisms. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.
Naive set theory , 1894 Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by
Georg Cantor titled
On a Property of the Collection of All Real Algebraic Numbers. In his paper, he developed the notion of
cardinality, comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all
real numbers is
uncountable, that is, one cannot put all real numbers in a list. This theorem is proved using
Cantor's first uncountability proof, which differs from the more familiar proof using his
diagonal argument. 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 a theory of
transfinite numbers, 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). Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter 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 (see: ''
Controversy over Cantor's theory''). , Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and was famously constructing the real numbers in 1872 using
Dedekind cuts. Cantor and Dedekind were in correspondence about set theory, especially in the 1870s. However, Dedekind's algebraic style only began to find followers in the 1890s. Cantor also worked with
Giuseppe Peano in developing the
Peano axioms, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the
epsilon symbol for
set membership. Possibly most prominently,
Gottlob Frege began to develop his
Foundations of Arithmetic. In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept
horse in the barn. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or Nx: Fx ), relying on
Hume's principle. , 1936 However, Frege's work was short-lived, as it was found by
Bertrand Russell that his axioms lead to a
contradiction. Specifically, Frege's
Basic Law V (now known as the
axiom schema of unrestricted comprehension). According to
Basic Law V, for any sufficiently well-defined
property, there is the set of all and only the objects that have that property. The contradiction, called
Russell's paradox, is shown as follows:Let
R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If
R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : \text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R This came around a time of several
paradoxes or counter-intuitive results. For example, that the
parallel postulate cannot be proved, the existence of
mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with
Peano arithmetic. The result was a
foundational crisis of mathematics. ==Basic concepts and notation==