Re: Cantor and the binary tree



Gottfried Helms wrote:
> Am 24.05.05 14:58 schrieb mueckenh@xxxxxxxxxxxxxxxxx:
>
> > of paths always equals that of the nodes + 1. It is simply impossible
> > to assume that one of these numbers becomes uncountably infinite while
> > the other remains countably infinite.
>
> No number can have the property of "being uncountable".

The number of elements of a set can be finite or infinte. It can be
countable (finite or aleph_0) or uncountable. In set theory such
numbers are defined, although you are correct.

What does
> that mean? A mathematical concept may require "uncountable many
> numbers (or uncountable many of whatever)". So it is "simply
> impossible to assume, that one of these numbers becomes uncountably
> infinite", I think so; and as well it is impossible, that "the
> other remains countably infinite".

We can compare the number of nodes between level 0 and level n with the
number of all nodes. If we switch from one to the other, the display
switches from finite to infinitely-countable. In case of path we can
switch between the number of paths already separated on leven n to the
number of all paths. The display switches from finite to
infinitely-countable.

Even the property of being "countable"
> - is commonly used only as a property of an aggregate, not of a single
> number

This property of a set is expessed by or as a transfinite number. (In
German: transfinite Zahl)

Regards, WM

.

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