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.