fom
Posts:
1,037
Registered:
12/4/12
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Re: Matheology § 210
Posted:
Feb 5, 2013 11:06 AM
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On 2/5/2013 4:15 AM, WM wrote:
>
> Matheology § 210
>
> An accessible number, to Borel, is a number which can be described as
> a mathematical object. The problem is that we can only use some finite
> process to describe a real number so only such numbers are accessible.
> We can describe rationals easily enough, for example either as, say,
> one-seventh or by specifying the repeating decimal expansion 142857.
> Hence rationals are accessible. We can specify Liouville's
> transcendental number easily enough as having a 1 in place n! and 0
> elsewhere. Provided we have some finite way of specifying the n-th
> term in a Cauchy sequence of rationals we have a finite description of
> the resulting real number. However, as Borel pointed out, there are a
> countable number of such descriptions. Hence, as Chaitin writes: "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."
> [J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to
> understand"]
> http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html
>
> But how to pick this dark matter of numbers? Only accessible numbers
> can get picked. Unpickable numbers cannot appear anywhere, neither in
> mathematics nor in Cantor's lists. Therefore Cantor "proves" that the
> pickable numbers, for instance numbers that can appear as an
> antidiagonal of a defined list, i.e., the countable numbers, are
> uncountable.
It is important to state what Cantor's proof proves properly.
Cantor's proof proves that any assertion claiming to put the
real numbers in one-to-one correspondence with the natural
numbers is fallible.
In the context of the statements above, one cannot even put
Borel's accessible numbers in one-to-one correspondence with
the natural numbers.
"countable" and "uncountable" are labels that distinguish non-finite
parts of absolute infinity which cannot be put into one-to-one
correspondence with one another with "countable" referring
to any such part that can be put into one-to-one correspondence
with the natural numbers.
Since "the natural numbers" should be problematic here, it
might be best to add "by a rule".
But, then your fight is actually with Dirichlet, isn't it?
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