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 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: "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 uniquely. {{Dies wurde bereits

in KB091209 ausführlicher dargelegt:

http://www.hs-augsburg.de/~mueckenh/KB/KB%20001-200.pdf

}}

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)]

http://arxiv.org/abs/math.HO/0411418
Gruß, WM

## Lesen sie die antworten