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) .