No one shall be able to drive us from the paradise that Cantor created
David Hilbert, 1925
Whatever the remedy adopted (for resolving the paradoxes of set the-
ory), we can promise ourselves the joy of a doctor called to witness
an interesting pathological case.
Henri Poincare, 1908
In 1873, the German mathematician Georg Cantor discovered that the set of
algebraic reals is countable. A few weeks later he was able to demonstrate that the
set of all real numbers is uncountable. A new mathematical discipline was born: set
theory. In the course of the next two decades, Cantor developed the fundamental
concepts of this new discipline; the concepts of equipotent sets, order-isomorphic
structures, cardinals and ordinals are all due to him.
Generally speaking, set theory is the study of collections of objects. This view
was expressed by Cantor in his famous definition of a set:
By a "set" we mean any collection M into a whole of definite, distinct
objects m (which are called the "elements" of M) of our perception
or of our thought.
If an element m belongs to a set M, we write m G M. It is also quite common
to say in this case that m is a member of M, and to refer to G as the membership
relation. As we shall see, all relevant facts about sets can be expressed in terms of
the membership relation.
The career of set theory has been impressive. In the first half of the 20th cen-
tury, the new fields of set-theoretic topology, theory of real functions, and functional
analysis evolved. Each of these disciplines is strongly rooted in set theory, albeit not
exclusively. Even more importantly, set theory can be regarded as the foundation
of all mathematics. It is possible to interpret the other branches of mathematics as
the study of sets. This seems at first glance to be an implausible claim. For most
people the real number \f2 is just a single object, perhaps a point on the real line,
but certainly not a collection of other objects.
EXERCISE 1(X): If you already know some methods of reinterpreting mathe-
matics in terms of set theory, use it to rewrite the Fundamental Theorem of Calculus
in the language of set theory.
Nevertheless, in the sections ahead, we shall present compelling evidence for the
possibility of founding all of mathematics on set theory.