§3. Bolzano-Weierstrass and Cauchy's criterion
233
enumeration did not present the bizarre behaviour already noted in Chap. II
one could deduce that, for n sufficiently large, the decimal expansions of order
p of Un would be independent of n, which would render Cauchy's criterion
obvious to those who believed it a priori, and wrongly, that a nonterminating
decimal expansion "evidently" represents a real number. We have already
made this remark at the end of nO 4 of Chap. II.
This may be the reason why Bolzano and Cauchy seem to have considered
their criterion as too obvious to merit a proof, but it is too late to ask them.
It might also be that they did not see its importance, not having used it
much (for what is due to Cauchy, see the proof of Theorem 14 below); their
successors were the ones who exploited this.
However it may be, these remarks show that we could have chosen Theorem 13 as axiom (IV) for JR. This would, further, be more justified, since,
in contrast to the concept of least upper bound, it has a meaning in any
metric space and, in contrast to the method of cuts, allows us to extend to
this general case the construction of JR from Q.
One can indeed, as Georg Cantor did in 1872, and as a much less famous
Frenchman, Charles Meray, did in 1869, use Cauchy sequences to define the
real numbers starting from Q. Meray remarked that the two propositions
at the base of analysis, up to then considered as "axioms", are (i) the convergence theorem for increasing sequences, (ii) Cauchy's criterion. He then
decided - like Cantor - to partition the Cauchy sequences formed by rational numbers into classes, considering two sequences (un) and (vn ) such that
Un - Vn tends to 0 as equivalent, and to define an irrational number as being
such a class: for example, V2 will be the class of all rational Cauchy sequences
> 0 such that lim u; = 2. Professor at Dijon, where he expounded these ideas
to students fresh from school, Meray published a Nouveau precis d'analyse
infinitesimal in 1872 - it was greatly enlarged after 1894 - which, as well
as containing his construction of the real numbers, grounded all of analysis
on power series (he introduced this term, and also the expression "radius of
convergence") claiming that one meets no other useful or interesting functions in mathematics or in physics, already an idea of Lagrange's. An opinion
which, in this late age, hardly risks finding unanimity since a large part of
the guild is, on the contrary, in process of attacking the continuous functions without derivatives, Cantor's bizarre sets of real numbers, everywhere
discontinuous functions that one can nevertheless integrate thanks to Emile
~rel and Henri Lebesgue at the end of the century, etc., while the physicists
are starting to meet non-analytic functions everywhere, for example in the
theory of wave propagation, while expecting better, or worse, in the century
to come. Further, Meray, who never read his contemporaries, had his own
idiosyncratic language, nor did he have the inventive genius of a Weierstrass
or of a Cantor, so his influence was negligible and his construction of the real
numbers is attributed to the latter, even in France 20 .
20 An example of the "Matthew Effect in Science" studied by the founder of the
American school of the sociology of sciences, Robert K. Merton, in an article
233
enumeration did not present the bizarre behaviour already noted in Chap. II
one could deduce that, for n sufficiently large, the decimal expansions of order
p of Un would be independent of n, which would render Cauchy's criterion
obvious to those who believed it a priori, and wrongly, that a nonterminating
decimal expansion "evidently" represents a real number. We have already
made this remark at the end of nO 4 of Chap. II.
This may be the reason why Bolzano and Cauchy seem to have considered
their criterion as too obvious to merit a proof, but it is too late to ask them.
It might also be that they did not see its importance, not having used it
much (for what is due to Cauchy, see the proof of Theorem 14 below); their
successors were the ones who exploited this.
However it may be, these remarks show that we could have chosen Theorem 13 as axiom (IV) for JR. This would, further, be more justified, since,
in contrast to the concept of least upper bound, it has a meaning in any
metric space and, in contrast to the method of cuts, allows us to extend to
this general case the construction of JR from Q.
One can indeed, as Georg Cantor did in 1872, and as a much less famous
Frenchman, Charles Meray, did in 1869, use Cauchy sequences to define the
real numbers starting from Q. Meray remarked that the two propositions
at the base of analysis, up to then considered as "axioms", are (i) the convergence theorem for increasing sequences, (ii) Cauchy's criterion. He then
decided - like Cantor - to partition the Cauchy sequences formed by rational numbers into classes, considering two sequences (un) and (vn ) such that
Un - Vn tends to 0 as equivalent, and to define an irrational number as being
such a class: for example, V2 will be the class of all rational Cauchy sequences
> 0 such that lim u; = 2. Professor at Dijon, where he expounded these ideas
to students fresh from school, Meray published a Nouveau precis d'analyse
infinitesimal in 1872 - it was greatly enlarged after 1894 - which, as well
as containing his construction of the real numbers, grounded all of analysis
on power series (he introduced this term, and also the expression "radius of
convergence") claiming that one meets no other useful or interesting functions in mathematics or in physics, already an idea of Lagrange's. An opinion
which, in this late age, hardly risks finding unanimity since a large part of
the guild is, on the contrary, in process of attacking the continuous functions without derivatives, Cantor's bizarre sets of real numbers, everywhere
discontinuous functions that one can nevertheless integrate thanks to Emile
~rel and Henri Lebesgue at the end of the century, etc., while the physicists
are starting to meet non-analytic functions everywhere, for example in the
theory of wave propagation, while expecting better, or worse, in the century
to come. Further, Meray, who never read his contemporaries, had his own
idiosyncratic language, nor did he have the inventive genius of a Weierstrass
or of a Cantor, so his influence was negligible and his construction of the real
numbers is attributed to the latter, even in France 20 .
20 An example of the "Matthew Effect in Science" studied by the founder of the
American school of the sociology of sciences, Robert K. Merton, in an article