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Topic: Distinguishability of paths of the Infinite Binary tree???
Replies: 69   Last Post: Jan 4, 2013 11:11 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Distinguishability of paths of the Infinite Binary tree???
Posted: Dec 24, 2012 2:18 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 23 Dez., 21:20, Zuhair <zaljo...@gmail.com> wrote:
> > This is just a minor conundrum that I want to discuss about WM's
> > argument about the complete Infinite binary tree (CIBT). It is really
> > about Cantor's argument. But the discussion here will be at intuitive
> > level rather than just formal level.
> >
> > then make some rough analogy with the infinite binary tree.
> >
> > Let's take the binary tree with two levels below the root node level
> > which is the following:
> >
> >    0
> >   /  \
> >  0   1
> > / \   | \
> > 0 1 0  1
> >
> > Now with this three one can say that distinquish-ability is present at
> > all levels below the root node level, so we have two distinguishable
> > paths at level 1 that are 0-0 and 0-1. While at level 2 we have four
> > distinguishable paths that are 0-0-0, 0-0-1, 0-1-0, 0-1-1. However the
> > reason why we had increased distinguish-ability at level 2 is because
> > we had differential labeling of nodes at that level!  Now if we remove
> > that differential labeling we'll see that we can only distinguish two
> > longer paths by the labeling of their nodes, like in the following
> > tree:
> >
> >     0
> >    /  \
> >   0   1
> >  / \   | \
> > 0 0  0  0
> >
> > Now clearly the only distinguishability present in that tree is at
> > level 1 because all nodes at level 2 are not distinguished by their
> > labeling. So the result is that there is no increase in the number of
> > distinguishable paths of the above tree when we move from level 1 to
> > level 2. See:
> >
> > Paths at level 1 are: 0-0 , 0-1.  Only Two paths.
> > Paths at level 2 are : 0-0-0, 0-1-0. Only Two paths.
> >
> > Similarly take the tree:
> >
> >     0
> >    /  \
> >   0   1
> >  / \   | \
> > 1 1  1  1
> >
> > Paths at level 1 are: 0-0 , 0-1.  Only Two paths.
> > Paths at level 2 are : 0-0-1, 0-1-1. Only Two paths.
> >
> > So the increment in number of paths in the original binary tree of
> > level 2 after the root node, is actually due to having distinct
> > labeling of nodes at level 2. If we don't have distinct labeling at a
> > further level the number of distinguished longer paths stops at the
> > last level where distinguished labeling is present.
> >
> > This is obviously the case for FINITE binary trees.

>
> Don't forget: Every distinction in every Cantor list and in every
> Binary Tree occurs at a finite level. There is no difference, whether
> the digits or nodes are continuing or not. Evereything in mathematics
> happens at a finite level. Therefore blathering about infinite paths
> is useless.

EVERY "level" of an infinite path is finite, so there is no problem with
finiteness of levels.

The only problem with WM are the possible self-imposed limitations on
his mind.
--

Date Subject Author
12/23/12 Zaljohar@gmail.com
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/25/12 mueckenh@rz.fh-augsburg.de
12/25/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Virgil
12/28/12 mueckenh@rz.fh-augsburg.de
12/28/12 Virgil
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/30/12 fom
12/30/12 mueckenh@rz.fh-augsburg.de
12/30/12 fom
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
1/4/13 ross.finlayson@gmail.com
12/30/12 forbisgaryg@gmail.com
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/26/12 Zaljohar@gmail.com
12/26/12 Virgil
12/26/12 Zaljohar@gmail.com
12/26/12 gus gassmann
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Zaljohar@gmail.com
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Zaljohar@gmail.com
12/28/12 mueckenh@rz.fh-augsburg.de
12/28/12 Zaljohar@gmail.com
12/28/12 Virgil
12/29/12 Zaljohar@gmail.com
12/29/12 Virgil
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/28/12 Zaljohar@gmail.com
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/27/12 Virgil
12/26/12 fom
12/26/12 Virgil
12/26/12 fom
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 forbisgaryg@gmail.com
12/26/12 Virgil
12/26/12 fom
12/27/12 gus gassmann
12/27/12 Tanu R.
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Tanu R.
12/27/12 Virgil
12/28/12 Zaljohar@gmail.com
12/28/12 Virgil
12/27/12 fom
12/27/12 Virgil
12/24/12 Ki Song