Date: Dec 12, 2012 4:41 PM
Author: Virgil
Subject: Re: On the infinite binary Tree
In article

<d9a4a4e9-e88f-41d4-8305-f4f7886bc045@fn10g2000vbb.googlegroups.com>,

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

> On 12 Dez., 11:26, Zuhair <zaljo...@gmail.com> wrote:

> > WM has presented the idea that the infinite binary tree must have

> > countably many paths. It seems that he thinks that the total number of

> > paths in a binary tree is always smaller than or equal to the total

> > number of nodes.

Which for finite trees is quite true.

But WM has, as usual, has considerable difficulty comprehending that

what works for finite cases does always work quite the same way for

infinite ones.

Cantor has a perfectly valid proof that there is no surjective mapping

from any set, finite or infinite, to its power set.

WM has no counterproof.

And it is easily shown that the set of paths in a COMPLETE INFINITE

BINARY TREE bijects with the power set of the set of node levels, where

the root is at level 1 and the child nodes of a node at level n are at

level n+1, thus its power set must also biject with the power se of the

se of node levels.

There are countably many nodes distributed into countably many node

levels in any and every COMPLETE INFINITE BINARY TREE, but the power set

of such a set of node levels is necessarily UNcountable, so that the set

of all paths, being bijectable with an uncountable set is equally

uncountable.

That WM does not allow such arguments in his Wolkenmuekenheim does not

invalidate them in standard mathematics.

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