Virgil
Posts:
4,479
Registered:
1/6/11
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Re: Matheology 203
Posted:
Feb 6, 2013 4:31 PM
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In article
<63c3dd2b-cf74-407d-a750-c6a6981b1421@fv9g2000vbb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 6 Feb., 10:08, Virgil <vir...@ligriv.com> wrote:
>
> > > Everything of 0.111... that can be defined by sequences of 1's, is in
> > > the list. The finite definition "s" or "o.111..." is not in the list,
> > > but finite definitions have nothing to do with Cantor's diagonal
> > > proof.
> > > Is that really exceeding the capacity of your brain?
> >
> > It certainly seems beyond the capacity of WM's.
>
> With no doubt.
> >
> > 0.111... is a finite definition for Sum_(n in |N) 1/b^n,
>
> which is another finite definition of 1/9 and notr available for
> diagonalization.
>
> Try to diagonalize:
>
> 1 divided by 9
> 1 divided by circumference of the unit circle
> series of Gregory-Leibniz
> basis of the logarithms
> Euler's constant
> andsoon
If the nth term in that sequence can be expressed in some n-ary form
with n >= 4, , say, decimal form, accurate to n+1 n-ary places, then
forming an anti-diagonal is trivial.
>
> Nothing else is required by Cantor with the only exception, that for
> every n the finite initial segment a_n1, ..., a_nn of entry number n
> has to be expanded by digits. But as everybody knows, these finite
> initial segments belong to a countable subset of the countable set of
> rational numbers.
The nth term need not be expanded beyond n+1 digit accuracy in order to
have a well defined antidiagonal.
>
> Regards, WM
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