> 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.