Virgil
Posts:
4,486
Registered:
1/6/11
|
|
Re: On the infinite binary Tree
Posted:
Dec 12, 2012 4:41 PM
|
|
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.
--
|
|
