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Matheology § 187
Posted:
Jan 9, 2013 1:42 AM
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Matheology § 187
Whatever the choice of language, there will only be a countable
infinity of possible texts, since these can be listed in size order,
and among texts of the same size, in alphabetical order. {{Here is a
simple example:
0
1
00
01
10
11
000
...
}}
This has the devastating consequence that there are only a
denumerable infinitely of such "accessible" reals, and therefore 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 infinity of
reals, a set of reals of measure zero, an infinitesimal subset of the
set of all possible {{interesting question: what are /possible/
properties of /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. {{How can we pick? By
picking it, a real number would be finitely defined already. That
means an undefined real number can never be picked mathematically. And
with finger or beak nobody could succed.}}
[Gregory Chaitin: "How real are real numbers?" (2004)]
http://arxiv.org/abs/math.HO/0411418
The enumeration of all rational numbers is tantamount to an infinite
sum of units. One gets the divergent sequence of all finite cardinal
numbers and maintains that a limit exists. That is a mistake. The fact
that we can count up to every number does not imply that we can count
all numbers. After every finite cardinal number there are infinitel
many but not after all. In a similar way it is impossible to sum all
terms of the series SUM 1/2^n. But contrary to a diverging sequence,
the sequence of partial sums of this series deviates from 1 less and
less. Therefore 1 can be called the limit.
Regards, WM
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