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Topic: Matheology § 230
Replies: 8   Last Post: Mar 24, 2013 10:42 PM

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Registered: 12/4/12
Re: Matheology § 230
Posted: Mar 24, 2013 10:45 AM
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On 3/24/2013 7:51 AM, WM wrote:
> 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
> uniquely.

Everyone is entitled to an opinion.

Any given opinion can be portrayed as someone else's
fantasy by better men than me.

But, there is nothing that is not a fantasy when the topic
of discussion is WM's theory of monotonic inclusive crayon marks.

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