- SET THEORY 547
the Borel sets, which is the smallest class that contains all closed subsets and also
contains the complement of any of its sets and the union of any countable collection
of its sets. This class can be "constructed" only by a transfinite induction.
Set theory, although it was an attempt to provide a foundation of clear and
simple principles for all of mathematics, soon threw up its own unanswered math-
ematical questions. The most important was the continuum question. Cantor had
shown that the set of all real numbers could be placed in one-to-one correspon-
dence with the set of all subsets of the integers. Since he denoted the cardinality
of the integers as No and the cardinality of the real numbers as c (where c stands
for "continuum"), the question naturally arose whether there was any subset of
the real numbers that had a cardinality between these two. Cantor struggled for
a long time to settle this issue. One major theorem of set theory, known as the
Cantor-Bendixson theorem,^7 after Ivar Bendixson (1861-1935), asserts that every
closed set is the union of a countable set and a perfect set, one equal to its derived
set. Since it is easily proved that a nonempty perfect subset of the real numbers
has cardinality c, it follows that every uncountable closed set contains a subset of
cardinality c. Thus a set of real numbers having cardinality between tt 0 and c can-
not be a closed set. Many mathematicians, especially the Moscow mathematicians
after the arrival of Luzin as professor in 1915, worked on this problem. Luzin's stu-
dents Pavel Sergeevich Aleksandrov (1896-1982) and Mikhail Yakovlevich Suslin
(1894-1919) proved that any uncountable Borel set must contain a nonempty per-
fect subset, and so must have cardinality c. Indeed, they proved this fact for a
slightly larger class of sets called analytic sets. Luzin then proved that a set was a
Borel set if and only if the set and its complement were both analytic sets.
The problem of the continuum remained open until 1938, when Kurt Godel
(1906-1978) partially closed it by showing that set theory is consistent with the
continuum hypothesis and the axiom of choice,^8 provided that it is consistent
without them. Closure came to this question in 1963, when Paul Cohen (b. 1934—
like Cantor, he began his career by studying uniqueness of trigonometric series
representations) showed that the continuum hypothesis and the axiom of choice
are independent of the other axioms of set theory.
2.3. The reception of set theory. If Venn believed that probability was an
unwarranted intrusion of mathematics into philosophy, there were many mathe-
maticians who believed that set theory was an equally unwarranted intrusion of
philosophy into mathematics. One of those was Cantor's teacher Leopold Kro-
necker. Although Cantor was willing to consider the existence of a transcendental
number proved just because the real numbers were "too numerous" to be exhausted
by the algebraic numbers, Kronecker preferred a more constructivist approach. His
most famous utterance,^9 and one of the most famous in all the history of mathe-
matics, is: "The good Lord made the integers; everything else is a human creation."
("Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.")
That is, the only infinity he admitted was the series of positive integers 1,2,
(^7) Ferreiros (1995) points out that it was the desire to prove this theorem adequately, in 1882,
that really led Cantor to treat transfinite ordinal numbers as numbers. He was helped toward
this discovery by Dedekind's pointing out to him the need to use finite ordinal numbers to define
finite cardinal numbers.
(^8) Godel actually included four additional assumptions in his consistency proof, one of the other
two being that there exists a set that is analytic but is not a Borel set.
(^9) He made this statement at a meeting in Berlin in 1886 (see Grattan-Guiness, 2000, p. 122).