mathematician. Kronecker exerted his inf luence from his chair at the
prestigious University of Berlin, while Cantor was relegated to the minor
leagues at the University of Halle. Kronecker was an old-school mathe-
matician who took Gauss at his word on the subject of infinity, and he did
his best to denigrate Cantor’s works. This helped spark numerous out-
breaks of depression and paranoia in Cantor, who spent much of his later
life in mental institutions. It did not help matters that Cantor proclaimed
his mathematics to be a message from God, and that his other interests
included attempting to convince the world that Francis Bacon wrote the
works of Shakespeare.
Nonetheless, between periods of confinement, Cantor produced works
of stunning brilliance, results which changed the direction of mathemat-
ics. Sadly, he died in the mental institution where he had spent much of
his adult life. Just as the greatness of Mozart and van Gogh became ap-
parent after their deaths, so did the work of Cantor. Hilbert described
transfinite arithmetic, one of Cantor’s contributions, as “the most aston-
ishing product of mathematical thought, one of the most beautiful reali-
zations of human activity in the domain of the purely intelligible.”^4 Hilbert
continued by declaring that “No one shall expel us from the paradise
which Cantor has created for us.”^5 One can only wonder how Cantor’s life
would have differed if Hilbert, rather than Kronecker, had been the one
holding down the chair at the University of Berlin.
Another Visit to Hilbert’s Hotel
One of Cantor’s great discoveries was that there were infinite sets whose
cardinality was greater than that of the positive integers—infinite sets
that could not be matched one-to-one with the positive integers. Such a
set is the collection of all people with infinitely long names.
An infinitely long name is a sequence of letters A through Z and
blanks—one letter or blank for each of the positive integers. Some peo-
ple, such as “Georg Cantor,” have names consisting mostly of blanks—
the first letter is a G, the second letter an E,... , the sixth letter a
blank, the twelfth letter an R, and letters thirteen, fourteen,... (the
three dots stand for “on and on forever,” or some such phrase) are all
blanks. Some people, such as “AAAAAAAAA... ,” have names consist-
ing exclusively of letters—every letter of her name is an A. Of course, it
takes her quite a while to fill out the registration card at Hilbert’s Hotel,
but we’ll dispense with that problem for the time being.
The collection of all people with infinitely long names cannot be
matched one-to-one with the integers. To see that this is the case, as-
The Mea sure of All Things 19