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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 24, 2012 3:47 AM
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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.
>
> Let's start with "distinguish-ability" at finite binary trees, and
> 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.
Regards, WM
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