
Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 24, 2012 3:47 AM


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.
Regards, WM

