Date: Dec 13, 2012 7:50 AM
Subject: Re: On the infinite binary Tree

On 13 Dez., 11:42, Zuhair <> wrote:
> On Dec 13, 9:56 am, WM <> wrote:> > Ah I see, so you are imposing another condition on the definition of a
> > > path,
> > No, that is *the* definition of a path in a Binary Tree.
> Actually it is not. There is no need at all to stipulate that a path
> must begin by 0. It is a fixed
> definition.

MY Binary Tree contains the paths of real numbers of the unit
interval. Of course a path starts at the root node, And as you wanted
to contradict MY argument concerning the set of real numbers, other
paths would be completely meaningless. You have made a mistake but
don't want to confess it. That's all.
> I already showed you that the number of paths in a second degree
> binary tree (or third degree if you want to adop the empty path) IS
> larger than the total number of nodes. And what I mean by paths those
> that can start with 1 or with 0, but with the condition that it must
> be unidirectional. And showed it clearly and I've illustrated each
> path. You have 9 paths (inclusive of the empty path) and only 7 nodes.

Are you are too dishonest, to confess your error? Or do you really not
understand, that your pieces of paths are irrelevant?

> > No. I proved that the number of infinite paths is countable by
> > constructing all nodes of the Binbary Tree by a countable set of
> > infinite paths.

> This only means that you can have a bijective function from a
> countable subset of infinite paths of the binary tree to the set of
> all nodes, which everyone already know that this is possible, because
> we all agree that the total number of nodes of the infinite binary
> tree is countable.

There is a bijective function between N and all finite words.
All distinct paths are defined by finite words.
Infinite words and infinite paths cannot be distinguished as I proved
by this complete infinite Binary Tree:

0 1

What kind of paths did I use to construct it?

> What would be a proof is if you manage to define an injection from the
> set of ALL infinite paths of the binary tree to the set of all nodes
> of the binary tree.
> If you managed to do that, the next question is:
>  where is that proof? please show us

I will it show it to you for all the paths that I used to construct
the above tree and, in addition, for all the paths that you can
identify as beeing missing there.

Regards, WM