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Topic: Matheology § 230
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Posts: 18,076
Registered: 1/29/05
Matheology § 230
Posted: Mar 24, 2013 8:51 AM
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Matheology § 230

Cantor's theory of infinite sets, developed in the late 1800's, was a
decisive advance for mathematics, but it provoked raging controversies
and abounded in paradox. One of the first books by the distinguished
French mathematician Emile Borel (1871-1956) was his Lecons sur la
Théorie des Fonctions [Borel, 1950], originally published in 1898, and
subtitled Principes de la théorie des ensembles en vue des
applications à la théorie des fonctions.
This was one of the first books promoting Cantor's theory of sets
(ensembles), but Borel had serious reservations about certain aspects
of Cantor's theory, which Borel kept adding to later editions of his
book as new appendices. The final version of Borel's book, which was
published by Gauthier-Villars in 1950, has been kept in print by
Gabay. That's the one that I have, and this book is a treasure trove
of interesting mathematical, philosophical and historical material.
One of Cantor's crucial ideas is the distinction between the
denumerable or countable infinite sets, such as the positive integers
or the rational numbers, and the much larger nondenumerable or
uncountable infinite sets, such as the real numbers or the points in
the plane or in space. Borel had constructivist leanings, and as we
shall see he felt comfortable with denumerable sets, but very
uncomfortable with nondenumerable ones.

The idea of being able to list or enumerate all possible texts in a
language is an extremely powerful one, and it was exploited by Borel
in 1927 [Tasic, 2001, Borel, 1950] in order to define a real number
that can answer every possible yes/no question!
You simply write this real in binary, and use the nth bit of its
binary expansion to answer the nth question in French.
Borel speaks about this real number ironically. He insinuates that
it's illegitimate, unnatural, artificial, and that it's an "unreal"
real number, one that there is no reason to believe in.
Richard's paradox {{s. KB090826}} and Borel's number are discussed
in [Borel, 1950] on the pages given in the list of references, but the
next paradox was considered so important by Borel that he devoted an
entire book to it. In fact, this was Borel's last book [Borel, 1952]
and it was published, as I said, when Borel was 81 years old. I think
that when Borel wrote this work he must have been thinking about his
legacy, since this was to be his final book-length mathematical
statement. The Chinese, I believe, place special value on an artist's
final work, considering that in some sense it contains or captures
that artist's soul. If so, [Borel, 1952] is Borel's "soul work." [...]
Here it is: Borel's "inaccessible numbers:" Most reals are
unnameable, with probability one. Borel's often-expressed credo is
that a real number is really real only if it can be expressed, only if
it can be uniquely defined, using a finite number of words. It's only
real if it can be named or specifed as an individual mathematical
object. [...] So, in Borel's view, most reals, with probability one,
are mathematical fantasies, because there is no way to specify them

Borel, E. [1950] Lecons sur la Théorie des Fonctions (Gabay, Paris)
pp. 161, 275.
Borel, E. [1952] Les Nombres Inaccessibles (Gauthier-Villars, Paris)
p. 21.
Tasic, V. [2001] Mathematics and the Roots of Postmodern Thought
(Oxford University Press, New York) pp. 52, 81-82.

[Gregory Chaitin: "How real are real numbers?" (2004)]

Regards, WM

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