> 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 .