- SET THEORY 551
when referring to an object that was defined effectively, through a finite number of
uses of well-defined operations on a given set of primitive objects.
After reading Lebesgue's opinion, Hadamard was sure that the essential dis-
tinction was between what is determined and what is described. He compared the
situation with the earlier debate over the allowable definitions of a function. But,
he said, uniqueness was not an issue. If one could say "For each x, there exists a
number satisfying Let y be this number," surely one could also say "For each
x, there exists an infinity of numbers satisfying— Let y be one of these numbers."
But he put his finger squarely on one of the paradoxes of set theory (the Burali-
Forti paradox, discussed in the next section). "It is the very existence of the set W
that leads to a contradiction... the general definition of the word set is incorrectly
applied." (Question to ponder: What is the definition of the word set?)
The validity and value of the axiom of choice remained a puzzle for some time.
It leads to short proofs of many theorems whose statements are constructive. For
example, it proves the existence of a nonzero translation-invariant Borel measure on
any locally compact Abelian group. Since such a measure is provably unique (up to
a constant multiple), there ought to be effective proofs of its existence that do not
use the axiom of choice (and indeed there are). One benefit of the 1905 debate was
a clarification of equivalent forms of the axiom of choice and an increased awareness
of the many places where it was being used. A list of important theorems whose
proof used the axiom was compiled for Luzin's seminar in Moscow in 1918. The
list showed, as Luzin wrote in his journal, that "almost nothing is proved without
it." Luzin was horrified, and spent some restless nights pondering the situation.
The axiom of choice is ubiquitous in modern analysis; almost none of functional
analysis or point-set topology would remain if it were omitted entirely (although
weaker assumptions might suffice). It is fortunate, therefore, that its consistency
with, and independence of, the other axioms of set theory has been proved. How-
ever, the consequences of this axiom are suspiciously strong. In 1924 Alfred Tarski
(1901-1983) and Stefan Banach (1892-1945) deduced from it that any two sets A
and JB in ordinary three-dimensional Euclidean space, each of which contains some
ball, can be decomposed into pairwise congruent subsets. This means, for example,
that a cube the size of a grain of salt (set A) and a ball the size of the Sun (set
B) can be written as disjoint unions of sets A\,..., An and Si,..., Bn respectively
such that Ai is congruent to Bi for each i. This result (the Banach-Tarski paradox)
is very difficult to accept. It can be rationalized only by realizing that the notion of
existence in mathematics has no metaphysical content. To say that the subsets Ai,
Bi "exist" means only that a certain formal statement beginning 3... is deducible
from the axioms of set theory.
2.5. Doubts about set theory. The powerful and counterintuitive results ob-
tained from the axiom of choice naturally led to doubts about the consistency of set
theory. Since it was being inserted under the rest of mathematics as a foundation,
the consistency question became an important one. A related question was that of
completeness. Could one provide a foundation for mathematics, that is, a set of
basic objects and rules of proof, that would allow any meaningful proposition to
be proved true or false? The two desirable qualities are in the abstract opposed to
each other, just as avoiding disasters and avoiding false alarms are opposing goals.