Les nombres inaccessibles (Collection de monographies sur la théorie des fonctions)

Les nombres inaccessibles (Collection de monographies sur la théorie des fonctions) Summary:

By Emile Borel

Publisher: Gauthier-Villars
Number Of Pages: 141
Publication Date: 1952
ISBN-10 / ASIN: B001OS402Q
ISBN-13 / EAN:

from Chaitin's "How Real are the Reals": 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 specified as an individual
mathematical object. And in order to do this we must necessarily employ
some particular language, e.g., French. Whatever the choice of language,
there will only be a countable innity of possible texts, since these can be
listed in size order, and among texts of the same size, in alphabetical order.
This has the devastating consequence that there are only a denumerable
infinity of such "accessible" reals, and therefore, as we saw in Sec. 2.2, the
set of accessible reals has measure zero. So, in Borel's view, most reals, with probability one, are mathematical
fantasies, because there is no way to specify them uniquely. Most reals are
inaccessible to us, and will never, ever, be picked out as individuals using any
conceivable mathematical tool, because whatever these tools may be they
could always be explained in French, and therefore can only individualize"
a countable innity of reals, a set of reals of measure zero, an infinitesimal
subset of the set of all possible reals. Pick a real at random, and the probability is zero that it's accessible|
the probability is zero that it will ever be accessible to us as an individual
mathematical object.

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