550 19. LOGIC AND SET THEORYit is consistent to assume that the real numbers can be expressed as a countable
union of countable sets.^11
Zermelo made this axiom explicit and showed its connection with ordinal num-
bers. The problem then was either to justify the axiom of choice, or to find a more
intuitively acceptable substitute for it, or to find ways of doing without such "non-
effective" concepts. A debate about this axiom took place in 1905 in the pages of
the Comptes rendus of the French Academy of Sciences, which published a number
of letters exchanged among Hadamard, Borel, Lebesgue, and Baire.^12 Borel had
raised objections to Zermelo's proof that every set could be well-ordered on the
grounds that it assumed an infinite number of enumerations. Hadamard thought
it an important distinction that in some cases the enumerations were all indepen-
dent, as in Cantor's proof above, but in others each depended for its definition
on other enumerations having been made in correspondence with a smaller ordinal
number. He agreed that the latter should not be used transfinitely. Borel had
objected to using the axiom of choice nondenumeratively, but Hadamard thought
that this usage brought no further damage, once a denumerable infinity of choices
was allowed. He also mentioned the distinction due to Jules Tannery (1848-1910)
between describing an object and defining it. To Hadamard, describing an object
was a stronger requirement than defining it. To supply an example for him, we
might mention a well-ordering of the real numbers, which is defined by the phrase
itself, but effectively indescribable. Hadamard noted Borel's own work on analytic
continuation and pointed out how it would change if the only power series admitted
were those that could be effectively described. The difference, he said, belongs to
psychology, not mathematics.
Hadamard received a response from Baire, who took an even more conserva-
tive position than Borel. He said that once an infinite set was spoken of, "the
comparison, conscious or unconscious, with a bag of marbles passed from hand to
hand must disappear completely."^13 The heart of Baire's objection was Zermelo's
supposition that to each (nonempty) subset of a set Ì there corresponds one of its
elements." As Baire said, "all that it proves, as far as I am concerned, is that we
do not perceive a contradiction" in imagining any set well-ordered.
Responding to Borel's request for his opinion, Lebesgue gave it. As far as he
was concerned, Zermelo had very ingeniously shown how to solve problem A (to
well-order any set) provided one could solve problem  (to choose an element from
every nonempty subset of a given set). He remarked, probably with some irony,
that, "Unfortunately, problem  is not easy to resolve, it seems, except for the sets
that we know how to well-order." Lebesgue mentioned a concept that was to play a
large role in debates over set theory, that of "effectiveness," roughly what we would
call constructibility. He interpreted Zermelo's claim as the assertion that a well-
ordering exists (that word again!) and asked a question, which he said was "hardly
new": Can one prove the existence of a mathematical object without defining it?
One would think not, although Zermelo had apparently proved the existence of a
well-ordering (and Cantor had proved the existence of a transcendental number)
without describing it. Lebesgue and Borel preferred the verb to name (nommer)
(^11) Not every countable union of countable sets is uncountable, however; the rational numbers
remain countable, because an explicit counting function can be constructed.
(^12) These letters were translated into English and published by Moore (1982, pp. 311-320).
(^13) Luzin said essentially the same in his journal: What makes the axiom of choice seem reasonable
is the picture of reaching into a set and helping yourself to an element of it.