On 4 Mai, 20:31, Dan <dan.ms.ch...@gmail.com> wrote:
> For the record , language is man-made . Mathematics is not .Its > universality is proof enough.
Mathematics is, in large parts, discourse between mathematicians. That requires a language that can be spoken and understood, i.e., a finite language. > > >Either you have digits 1 at finite positions only. Then your number is > >in the list, since all finite positions are covered. (You cannot find > >that line. This is the same as: For every n there are infinitely many > >naturals m > n. But you cannot find those which are larger than all n. > >(Since there are not "all" n.)) > >Or you have digits at larger than all finite positions. Then you > >cannot replace them and cannot apply Cantor's argument. > > Ok . Let's say I accept your argument . We can't say that 0.11111.... > is not in the list with finite digits only . Finitude does not permit > us to make that distinction .
The list contains, by definition and by sober thinking, all digits that have natural indices. There is no choice to accept it or not. > > I accept your argument . There are no naturals larger than all n. > And , since we index numbers in our list by naturals, > > 1. 0.1 > 2. 0.11 > 3. 0.111 > 4............ > 5. > . > . > > There are no indices larger than all n .
> Now , since 0.11111..... is on our list , the question is: > At what index n is 0.11111..... on our list?
It is not in our list since it has no chance to surpass all entries of the list. It has only a finite definition like 1/9 or any other nmae that you can write here in order to identify it.
> Since there are no numbers larger than all naturals, 0.11111.... is > not on the list .
Correct. And it will not gain a place in the list by any digit substitution. It is a limit that has no decimal representation like sqrt(2) and, by the way, like every real number. You cannot identify any real number, not even 0, by writing only its sequence of digits. You dod not get ready, since the sequence is infinite, and the reader does not know, after having read 0.000000000000000000000000000 and so on, for a large part of the day, whether it will continue with zeros or not. Every real needs a finite definition. When using terminating reals we implicitly use the finite definition that only zeros will follow.
Therefore there are no actually infinite sequences at all - and there is no Cantor-list, other than finitely defined lists.