Re: Cantor and the binary tree

In article <1117021029.341934.48040@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
> > In article <1116958479.555107.284630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >
> >
> > > >
> > > > > .
> > > > > 0 1
> > > > > 0 1 0 1
> > > > > ..................
> > >
> > > Any path is an infinite sequence of bits which by multiplying with 2^-n
> > > and summing up establishes an infinite series representing a real
> > > number. Every combination of countably many bits is realized by
> > > definition.
> >
> > But such an "infinite" binary tree is not a list, so this has nothing to
> > say about the validity of Cantor's theorem.
>
> The tree is not a list. Therefore I choose it. Nevertheless it has to
> say much about infinite sets and Cantor's theorem, namely that the set
> of nodes and the set of paths are equivalent sets.

The set of leaf nodes and the set of finite or terminating paths are
equivalent, but infinite or non-terminating paths have no leaf nodes,
so that there is no equivalence.

> The set of nodes is
> countable and the set of paths is equivalent to the set of reals in
> (0,1). That's quite a lot of information, isn't it?

GIGO!

WM's "proof" disproved"

WM conflates bounded paths, having terminal or leaf nodes with unbounded
unending paths which have no terminal or leaf nodes, but contain
infinitely many intermediate nodes.

1) Each number of (0,1) is given by an UNENDING path stretching over
infinitely many nodes (bits).

2) All nodes (bits) of the tree belong to a countable set.

3) A node can only exist within a path.

4) Any node increases the number of ENDING paths, having terminal or
leaf nodes, by 1 from 1 coming in, to 2> going out. 2 - 1 = 1.

5) Any node increases the number of nodes by 1, but have absolutely
nothing to do with the number of unending paths.


All unending paths in an unending binary tree contain infinitely many
nodes.

The number of leaf nodes exactly equals the number of ending or finite
paths in any finite binary tree (in which all paths end).

Considering the binary tree whose root is "." and each branch is
indicated by a "0" or a "1", each leaf node, and therefore each path, is
represented by a terminating binary fraction , but each unending path is
represented by a non-terminating binary fraction.

There are moreof the non-terminating than of the terminating.

So WM is wrong yet again.
.