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Topic: Matheology § 258
Replies: 6   Last Post: May 2, 2013 10:26 PM

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mueckenh@rz.fh-augsburg.de

Posts: 15,259
Registered: 1/29/05
Re: Matheology § 258
Posted: May 2, 2013 4:59 PM
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On 2 Mai, 16:03, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 1 Mai, 23:31, Dan <dan.ms.ch...@gmail.com> wrote:
>
> >> > Yes, that is true. But (and please read this very attentively!):
> >> > Cantor's argument requires the existence of the complete sequence
> >> > 0.111.... in digits:

>
> >> > You can see this easily here:
>
> >> > The list
>
> >> > 0.0
> >> > 0.1
> >> > 0.11
> >> > 0.111
> >> > ...

>
> >> > when replacing 0 by 1 has an anti-diagonal, the FIS of which are
> >> > always in the next line. So the anti-diagonal is not different from
> >> > all lines, unless it has an infinite sequence of 1's. But, as we just
> >> > saw, this is impossible.

>
> >> I see no significant difference between referring to a mathematical
> >> object by a formula and referring to it by 'writing it down' .

>
> > But Cantor's argument is invalid, in this special case, unless it can
> > produce 0.111... with actually infinitely many 1's, i.e. more than
> > every finite number of 1's.

>
> > It does not matter whether 1/9 exists as a fraction or whether it
> > exísts in the ternary system as 0.01. In order to differ from every
> > entry of my list Cantor's argument needs to produce, digit by digit,
> > the infinite sequence. And that does not exist.

>
> Not at all;
> you accept that for any naturals n,m, (n/m)^2 =/= 2,
> and that because you reason that any particular choice
> leads to a contradiction.  You do not worry in that situation
> that you need to check infinitely many cases.


I have not to check infinitely many cases. I have to check exactly one
case. I assume no common divisor and prove a common divisor.
>
> Just reason in the same way here.
>

Cantor needs the digits - all. No exception allowed. But there does
not exist a sequence for 1/9 having only natural powers of 10. Every
natural power of 10 can be reflected. 0.111... cannot be reflected.

Regards, WM



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