Re: Cantor and the binary tree

Virgil said:
> 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.
>
Virgil, you're totally convoluted. Try to poke a hole in the argument I just
posted regarding insertion of nodes in ANY tree. Sorry, the prior existence of
leaf nodes is not a ruse that's available for you in this one.
--
Smiles,

Tony
.