Re: Cantor Confusion


Franziska Neugebauer schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:

The finite union of two or more finite trees is a finite tree. An
infinite union of finite trees is the infinite tree.

What is an infinite union? (Please give a definition)

An infinite union of trees is the union of all (or nearly all) finite
trees which reach from level 0 to level n where n eps N. (By the way
the result is the same when taking only the last levels of all finite
trees.)

It is easy to see that these unions result in the infinite tree, i.e.,
in the tree containing all levels n for n eps N. Therefore this tree
contains all paths which represent sequences of bits which can be
indexed by natural numbers. There is no such path or part of such path
outside of the union of all finite trees.

Since the countable union of finite sets is countable, the result is
that all paths in the infinite tree belong to a countable set. This is
already clear when considering the countablility of nodes and edges,
because there can be no more paths than edges. But now we can be
absolutely sure: Only persons with very limited logic capabilities can
claim that an infinite tree conaining all edges and all nodes and all
levels which can be enumerated by natural numbers, will and
simultaneously will not contain all possible paths (infinite sequences
of nodes which can be enumerated by natural numbers).

Regards, WM

.

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