Re: Cantor and the binary tree

Martin Shobe said:
> On 20 Jun 2005 04:09:53 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >> 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.
>
> What is it you are smoking? Can I have some? Even in finite binary
> trees, the number of paths is greater than the number of nodes where
> branching occurs.
>
> Martin
>
>
Excuse me Martin, but maybe you should have some of what I am smoking. Every
path ends in a leaf node, which are half the nodes in the tree. You start with
one node that represents the root path. For each pair of nodes, you create a
new path. A finite tree with n levels (including the root) has (2^n)-1 nodes,
(2^n)-2 branches, and only 2^(n-1), or (2^n)/2 paths, as denoted by its leaf
nodes. This relationship is preserved through infinity, even in the absence of
identifiable leaf nodes.
--
Smiles,

Tony
.

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