- PHILOSOPHIES OF MATHEMATICS 553
numbers on this one relation and nine axioms, together with a symbolic logic that
he had developed. The work of Cantor, Frege, and Peano attracted the notice of a
young student at Cambridge, Bertrand Russell, who had written his thesis on the
philosophy of Leibniz. Russell saw in this work confirmation that mathematics is
merely a prolongation of formal logic. This view, that mathematics can be deduced
from logic without any new axioms or rules of inference, is now called logicism.
Godel's work was partly inspired by it, and can be interpreted as a counterar-
gument to its basic thesis—that mathematics can be axiomatized. Logicism had
encountered difficulties still earlier, however. Even the seemingly primitive notion
of membership in a set turned out to require certain caveats.
3.1. Paradoxes. In 1897 Peano's assistant Cesare Burali-Forti (1861-1931), ap-
parently unintentionally, revealed a flaw in the ordinal numbers.^15 To state the
problem in the clear light of hindsight, if two ordinal numbers satisfy ÷ < y, then
÷ G y, but y £ x. In that case, what are we to make of the set of all ordinal
numbers? Call this set A. Like any other ordinal number, it has a successor A + 1
and A G A +1. But since A +1 is an ordinal number, we must also have A +1 G A,
and hence A < A + 1 and A + 1 < A. This was the first paradox of uncritical set
theory, but others were to follow.
The most famous paradox of set theory arose in connection with cardinal num-
bers rather than ordinal numbers. Cantor had defined equality between cardinal
numbers as the existence of a one-to-one correspondence between sets representing
the cardinal numbers. Set  has larger cardinality than set A if there is no function
/ : A —>  that is "onto," that is, such that every element of  is f(x) for some
÷ G A. Cantor showed that the set of all subsets of A, which we denote 2 A, is always
of larger cardinality than A, so that there can be no largest cardinal number. If
/ : A -* 2A, the set C = {t G A : t <£. f{t)} is a subset of A, hence an element of
2 A, and it cannot be f(x) for any ÷ G A. For if C = f(x), we ask whether ÷ G C
or not. If ÷ G C, then ÷ G /(x) and so by definition of C, ÷ ö C. On the other
hand, if ÷ ö C, then ÷ ö /(x), and again by definition of C, ÷ G C. Since the whole
paradox results from the assumption that C = /(x) for some x, it follows that no
such ÷ exists, that is, the mapping / is not "onto." This argument was at first
disputed by Russell, who wrote in an essay entitled "Recent work in the philosophy
of mathematics" (1901) that "the master has been guilty of a very subtle fallacy."
Russell thought that there was a largest set, the set of all sets. In a later reprint
of the article he added a footnote explaining that Cantor was right.^16 Russell's
first attempt at a systematic exposition of mathematics as he thought it ought to
be was his 1903 work Principles of Mathematics. According to Grattan-Guinness
(2000, p. 311), Russell removed his objection to Cantor's proof and published his
paradox in this work, but kept the manuscript of an earlier version, made before
he was able to work out where the difficulty lay.
To explain Russell's paradox, consider the set of all sets. We must, by its
definition, believe it to be equal to the set of all its subsets. Therefore the mapping
f(E) = Å should have the property that Cantor says no mapping can have. Now
(^15) Moore [1982, p. 59) notes that Burali-Forti himself did not see any paradox and (p. 53) that
the difficulty was known earlier to Cantor.
(^16) Moore (1982, p. 89) points out that Zermelo had discovered Russell's paradox two years before
Russell discovered it and had written to Hilbert about it. Zermelo, however, did not consider it
a very troubling paradox. To him it meant only that no set should contain all of its subsets as
elements.