Re: Cantor and the binary tree



Virgil wrote:

> > Nevertheless. The nodes are the only chances for paths to separate /
> > distinguish themselves from others. Therefore there are only countably
> > many fifferent paths.
>
> Non sequitur.

How then can they be distinguished?
>
> This no more proves countability of the set of paths than the fact that
> lists of bits can represent all reals in [0,1] proves the reals
> couontable. The set of all such lists, like the set of all those maximal
> paths, is uncountable.

By construction of the tree we see that all binary representations
which differ at a finite position are represented by bunches (=
non-empty subsets) of paths in the tree. Further we see that they can
distinguish themselves only at nodes or branches. The occasions to do
so are countable.

Regards, WM

.

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