# FOM: Cinq Lettres and the Borel universe

Walter Felscher walter.felscher at uni-tuebingen.de
Sat Dec 13 10:57:24 EST 1997

A.

Mr. Kanovei on Dec 6th drew attention to the 'Cinq Lettres'
exchanged between Borel, Hadamard, Baire, Lebesgue in
December 1904 and printed in Bull.Soc.Math.Fr. 33 (1905)
261-273; Mr.Simpson later pointed out that they were
reprinted in a Note to the 2nd edition of Borel's Lecons sur
la The/orie des Fonctions from 1914.  Apparently, they were
caused by an inquiry of Borel's, sent to the others, asking
for their opinion about Zermelo's recently published (first)
proof of the well ordering theorem.

While the first letters are concerned with the psychology of
successive versus simultaneous choices, Hadamard's second
letter isolates the disagreement about the validity of
Zermelo's proof in the question whether it is possible to

d/emontrer l'existence d'un e/\tre mathe/matique sans
le de/finir

[prove the existence of a mathematical object without
defining it]

where "de/finir" is explained as to state a "proprie\te
explicitly to an analogy with Kronecker's 'constructivist'
attitudes.

B.

Mr. Simpson, in two notes on the Borel universe from Dec 6th
and Dec 11th, wrote referring to Mr.Kanovei's remarks

The French analysts were deeply disturbed by set-theoretic
pathology flowing from the existence of a well-ordering
of the reals.  They considered various ways to remedy
this pathology.  One of the remedies that they considered
was to modify the foundations by DISCARDING ALL SETS OF
REALS EXCEPT BOREL SETS.

and

However, I can well understand why thoughtful mathematicians
*might* be offended.  Borel, Baire and Lebesgue were not
only offended but horrified by Zermelo's theorem on the
existence of a well-ordering of the reals, because (1) no
specific well-ordering was exhibited, (2) the existence
of such a well-ordering gives rise to pathology such as
non-measurable sets and, later and more dramatically,
the Banach-Tarski paradox.  That's why these great
mathematicians proposed to restrict attention to Borel
sets: in order to avoid such pathology.

[Of course, the Banach-Tarski paradoxical decomposition was
discovered only 20 years afterwards.]

Mr. Simpson's formulations, e.g. his "these great mathematicians",
give the impression as if he were referring to the Cinq Lettres
as the source for calling his witnesses from history.

Having just returned from reading the Cinq Lettres in the
library, I can assure that these do not contain any
intimation of proposals to restrict attention to Borel sets.

What reference is Mr. Simpson then basing his claims on ?

There is, at a much later page in the 2nd edition of Borel's
Lecons, the Note VI on perfect and measurable sets, and in
section IV of it Borel also calls "ensembles bien de/finies"
the sets "mesurables-B" (the original ensembles mesurables
of Borel, called mesurables-B by Lebesgue when he invented
his more general measurable sets). But also there Borel in
no way proposes to restrict attention to the ensembles
mesurables-B .

C.

It seems to me that Mr.Simpson's (and Mr.Friedman's)
program to study the realm of Borel sets, say in the
framework of TBU_0, is a most interesting mathematical
enterprise. Being so, it should not need dubious historical
references as crown's witnesses - just as NSA did not need
the incomplete quotations from Leibniz for such purpose.

W.F.