
Matheology § 230
Posted:
Mar 24, 2013 8:51 AM


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 (18711956) 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 GauthierVillars 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 booklength 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 oftenexpressed 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.
Borel, E. [1950] Lecons sur la Théorie des Fonctions (Gabay, Paris) pp. 161, 275. Borel, E. [1952] Les Nombres Inaccessibles (GauthierVillars, Paris) p. 21. Tasic, V. [2001] Mathematics and the Roots of Postmodern Thought (Oxford University Press, New York) pp. 52, 8182.
[Gregory Chaitin: "How real are real numbers?" (2004)] http://arxiv.org/abs/math.HO/0411418
Regards, WM

