On Dec 16, 10:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 16 Dez., 22:01, Rupert <rupertmccal...@yahoo.com> wrote: > > > > And all mathematics that has been done by Cantor can be discussed in > > > any natural (i.e. countable) language. Further Cantor's diagonal > > > argument works solely with digits (i.e., nodes). A proof that nobody > > > can distinguish more than countable many paths in the Binary Tree is a > > > contradiction of uncountability. > > > It's not. > > Why then should we be able to distinguish the diagonal from the > entries of the list?
Suppose that you list all the real numbers that are definable in some countable language. You must then be working in some metalanguage in which you can talk about the countable language. The anti-diagonal number can be defined in this metalanguage.
> Why then do we need the diagonal argument at all?
It is necessary to show that the real numbers are uncountable.
> If we are ready to believe that there are undefinable and > undistinguishable reals, we need no diagonal argument. We could simply > believe.
No, the question of whether the real numbers are countable or uncountable is something which requires a proof one way or the other. Cantor had to think hard about this question before he got the answer.
> Cantor would be very sad seing that his famous proof is > completely superfluous. >