In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
A considerable portion of the mathematics being created at any time is only in "discourse" between the creator and him-or-her-self.
Once the mathematics has been created, only then need it be communicated, so only then is any such discourse required. > > > > >Either you have digits 1 at finite positions only. Then your number is > > >in the list
If one has a digit at every finite position, then one has infinitely may digits, one in each of those infinitely many finite positions. At least everywhere outside Wolkenmuekenheim.
> > The list contains, by definition and by sober thinking, all digits > that have natural indices.
But not all of them in any one line. And that is the line that is missing in your list.
> > > 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
It may not be on your list, but it is on everyone else's completed list as the union of all prior lines.
> Therefore there are no actually infinite sequences at all - and there > is no Cantor-list, other than finitely defined lists.
WM may be able to bar common sense and the obvious from Wolkenmuekenheim, but elsewhere he cannot.
****** ON COUNTABILITY ******
Countability and listability are equivalent notions, at least outside of Wolkenmuekenheim:
Any set, S, is defined to be COUNTABLE if and only if there is a surjective mapping from |N onto S.
And S is othewise said to be UNCOUNTABLE.
So set S being countable is equivalent to saying that the members of S are listable in natural number order with no members of S being left out.
NOTE:members of S may appear in such a listing more than once, as long as each of them appears at least once, withut destroying countability.
So that unless WM can provide such a listing of the reals, in which each real can be proven to appear at least once, he is NOT justified in claiming that the set of reals is known to be countable.
And Cantor has provided two entirely different proofs that no such complete listings of the set of reals are possible.
Neither of which WM, nor anyone else, has yet disproved.