Date: Dec 13, 2012 3:23 PM
Author: Virgil
Subject: Re: On the infinite binary Tree

In article 
WM <> wrote:

> 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.

Provably not all of them.

> 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.

Zuhair's mistake was trivial, and even he ignored it by using the
correct relatoonship further on.

WM's error in his claimed proof is a killer, conflating the difference
in properties between injection and surjection
> >
> > 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?

Wm is certainly too dishnest to admit his own mistakes.
> > > 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.

There is that same error again, confusing the properties of surjetion
and injection.

A surjection from set A to set B proves Card(A) >= Card(B).

But what WM claims it proved is Card(A) <= Card(B), which is provably

> >
> > 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.

Claimed, but not proved, and disprovable anywhere outside

> Infinite words and infinite paths cannot be distinguished as I proved
> by this complete infinite Binary Tree:

If a word requires use of letters and a path can be identified as an
infinite string of digits then even WM ought to be able distinguish
between them. As usual, WM is being too sloppy in his terminology to
support clear thinking.
> 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.

Promises, promises!