In article <email@example.com>, Dan <firstname.lastname@example.org> wrote:
> > Wittgenstein ... > Should have known better than to attempt to misquote Wittgenstein .He > has so very few valid passages . > For the record , language is man-made . Mathematics is not .Its > universality is proof enough. We don't restrict the electromagnetic > spectrum because we can't see uv's. We shouldn't restrict the infinite > due to absence of sense organs for it. Mathematics shall model the > world, not man's ineptitude in sensing it. > > >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 . > > 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? > Is it at 1? > 0.1 not = 0.11111...... > Is it at 2? > 0.11 not = 0.11111..... > Is it at 3? > 0.111 not = 0.111111..... > > If a number is on the list , it has to have at least one position k > of the list occupied by the number . > 0.1 has position 1 > 0.11 has position 2 > 0.111 has position 3 . > > If 0.1111 ..... has position k , then k is larger than ALL N .But we > already know, numbers larger than all naturals don't exist. > Since there are no numbers larger than all naturals, 0.11111.... is > not on the list .