On 11 Jun., 14:13, Rainer Rosenthal <r.rosent...@web.de> wrote: > WM wrote: > > My proof rests upon the fact that after the set of all finite paths of > > the form 0.1, 0.11, 0.111, ... has been constructed, there is no > > chance to construct the path 0.111... in addition. > > Why not start with path 0.111... and then add those other paths > 0.1, 0.11, 0.111, ...?
This is really a splendid idea! In this manner all real numbers of the unit interval can be inserted into the infinite binary tree, 1/3 and 3/137 and even 1/pi. In addidion the tree can be completed by all the "terminating paths" (those with tails 000...).
Alas, we have used for construction a countable set of paths (because all real numbers you know or can construct by means of Cantor's diagonal method form a countable set).
This leads to the result that there are not uncountably many paths in the tree, unless they sneak in during / after construction. Refusing such we magic, we find that there are no infinite paths at all. And this implies that we cannot start off with 0.111....
However, your contribution to this problem is of high value and I appreciate it very much.