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 <2f2588c2ca27458695e395d29b7b5e9f@b8g2000yqh.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.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. > > > > Let's start with "distinguishability" 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 distinquishability is present at > > all levels below the root node level, so we have two distinguishable > > paths at level 1 that are 00 and 01. While at level 2 we have four > > distinguishable paths that are 000, 001, 010, 011. However the > > reason why we had increased distinguishability 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: 00 , 01. Only Two paths. > > Paths at level 2 are : 000, 010. Only Two paths. > > > > Similarly take the tree: > > > > 0 > > / \ > > 0 1 > > / \  \ > > 1 1 1 1 > > > > Paths at level 1 are: 00 , 01. Only Two paths. > > Paths at level 2 are : 001, 011. 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 selfimposed limitations on his mind. 

