546 19. LOGIC AND SET THEORYwe now call it) might be easily describable, Cantor needed to discuss the general
case. He needed the abstract concept of "sethood." Cantor felt compelled to dig
to the bottom of this matter and soon abandoned trigonometric series to write a
series of papers on "infinite linear point-manifolds."
Early on, he noticed the possibility of the transfinite. If the nth-level derived
set is P^n\ the nesting of these sets allows the natural definition of the derived set of
infinite order p(°°' as the intersection of all sets of finite order. But then one could
consider derived sets even at the transfinite level: the derived set of p(°°) could be
defined as p(°°+l) = (p(°°)). Cantor had discovered the infinite ordinal numbers.
However, he did not at first recognize them as numbers, but rather regarded them
as "symbols of infinity" (see Ferreiros, 1995).
Cardinal numbers. Cantor was not only an analyst, however. He had written his
dissertation under Kronecker and Kummer on number-theoretic questions. Only
two years after he wrote his first paper in trigonometric series, he noticed that his
set-theoretic principles led to another interesting conclusion. The set of algebraic
real or complex numbers is a countable set (as we would now say in the famil-
iar language that we owe to Cantor), but the set of real numbers is not. Cantor
had proved this point to his satisfaction in a series of exchanges of letters with
Dedekind.^6 Hence there must exist transcendental numbers. This second hierar-
chy of sets led to the concept of a cardinal number, two sets being of the same
cardinality if they could be placed in one-to-one correspondence. To establish such
correspondences, Cantor allowed himself certain powers of defining sets and func-
tions that went beyond what mathematicians had been used to seeing. The result
was a controversy that lasted some two decades.
Grattan-Guinness (2000, p. 125) has pointed out that Cantor emphasized five
different aspects of point sets: their topology, dimension, measure, cardinality, and
ordering. In the end, point-set topology was to become its own subject, and dimen-
sion theory became part of both algebraic and point-set topology. Measure theory
became an important part of modern integration theory and had equally important
applications to the theory of probability and random variables. Cardinality and or-
dering remained as an essential core of set theory, and the study of sets in relation
to their complexity rather than their size became known as descriptive set theory.
Although descriptive set theory produces its own questions, it had at first a
close relation to measure theory, since it was necessary to specify which sets could
be measured. Borel was very careful about this procedure, allowing that the kinds
of sets one could clearly define would have to be obtained by a finite sequence of
operations, each of which was either a countable union or a countable intersection
or a complementation, starting from ordinary open and closed sets. Ultimately
those of a less constructive disposition than Borel honored him with the creation of
(^6) There are two versions of this proof, one due to Cantor and one due to Dedekind, but both
involve getting nested sequences of closed intervals that exclude, one at a time, the elements of
any given sequence {an } of numbers. The intersection of the intervals must then contain a number
not in the sequence. In his private speculations, Luzin noted that Cantor was actually assuming
more than the mere existence of the countable set {áç}· In order to construct a point not in
it, one had to know something about each of its elements, enough to find a subinterval of the
previous closed interval that would exclude the next element. On that basis, he concluded that
Cantor had proved that there was no effective enumeration of the reals, not that the reals were
uncountable. Luzin thus raised the question of what it could mean for an enumeration to "exist"
if it was not effective. He too delved into philosophy to find out the meaning of "existence."