Date: May 2, 2013 1:51 PM
Subject: Re: Matheology § 258
On May 2, 11:55 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> 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.
> Regards, WM
You can substitute in any expression "the first digit of 1/9" with
"0.1111....." with "1" and it wouldn't make any difference .
You can substitute all the digit expansions in Cantor's argument with
formulas , and it wouldn't make any difference.
In what way you choose to write down the number (whether as a
fraction, or as a digit expansion , etc. ) is of no relevance .
Because it's still THE SAME NUMBER . It still has THE SAME
DIGITS ,even if you can't write them down .
The way YOU CHOOSE to write a number is as relevant to Cantor's
argument as the color of the ink in your pen .
You've ran out of red ink, and because it's written in a typewriter
(it has a weird font , 1/9 , not 0.1111.... like you would expect),
you say it's not the same number) .
A single valid name is good enough . The fact that we choose ,from
one of the possible names for an object ("2+2" , "1+3" ... ) , one of
them to call "its value" is of no fundamental relevance . I should
know . I work with infinite lists of numbers all the time .