Virgil
Posts:
4,661
Registered:
1/6/11
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Re: Matheology � 210
Posted:
Feb 5, 2013 6:40 PM
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In article
<f5ed3ead-f52a-447b-afbb-f2662b24ec8a@j4g2000vby.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 5 Feb., 17:06, fom <fomJ...@nyms.net> wrote:
> > 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.
>
> Same is valid for the one-to-one correspondence of the natural numbers
> with the natural numbers.
What is "fallible" about "for all n in |N, n <--> n" ?
Unless one is tied up in WMytheology, nothing is wrong with it.
>
> >
> > In the context of the statements above, one cannot even put
> > Borel's accessible numbers in one-to-one correspondence with
> > the natural numbers.
>
> That is correct because one cannot even put all natutal numbers in
> correspondence with all natural numbers.
>
> For every line n of the list
>
> 1 - 1
> 2 - 1, 2
> 3 - 1, 2, 3
What is "fallible" about "for all n in |N, n <--> n" ?
> ...
>
> we can state that there are not all natural numbers.
You can state that the moon is made of green cheese, but stating
something does not make it true.
> Since there is
> never more than one line required to gather all numbers that appear in
> any two lines of finite index and since there are only lines with
> finite index, we can conclude that not all natural numbers are in the
> list.
Show us that there is a natural number missing from
"for all n in |N, n <--> n"
>
> "All" is different from "every", when "all" is understood in the
> matheological sense of more than all finite numbers.
Only in WMytheology.
> >
> > "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.
>
> This correspondence is as impossible, as I have shown above
That WM has claimed something does not mean that he has shown it.
and usually seems to mean he has not SHOWN it at all.
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