Date: Dec 13, 2012 5:42 AM
Subject: Re: On the infinite binary Tree

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

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.

> > that it must begin with 0 since it is representing a number in
> > the interval [0,1], that's OK, then according to that I can see that
> > you are correct since the number of paths do corresponds to the number
> > of ending node, and with my calculation it would be less than the
> > total number of nodes by one. I'll need to check this again, but let
> > me agree with you on that for the current moment.

> > But still you have a problem. What you really managed to prove is that
> > the total number of FINITE undirectional paths beginning from 0 in the
> > infinite binary tree is countable, since we can simply draw an
> > injective map from those paths to their ending nodes.

> 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.

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