Re: Cantor Confusion

On 29 Mrz., 04:14, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1175105032.482322.292...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> On 26 Mrz., 17:01, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > > > > The union of all *finite* paths gives the same infinite set of
> > > > > nodes as the union of all finite trees. But there is no
> > > > > infinite path involved / required.
> > > >
> > > > Not in the uniting, no. But the union *does* contain infinite
> > > > paths. Just like the union of all finite initial segments of the
> > > > natural numbers does have as a subset an infinite initial segment
> > > > of the natural numbers.
> > >
> > > Just like the set of all negative unit fractions does cover zero?
> >
> > What does *this* mean? The set of all negative unit fractions does
> > contain infinite subsets. *That* is what we are talking about.
>
> But these infinite subsets do not cover zero.

And what is the relevance? We are talking about infinite subsets, not
about what they "cover", whatever that may mean.

Infinite sets of paths need not contain or be infinite paths. To show
that, you can consider the negative unit fractions. The infinite set
of these fractions does not cover the interval [-1, 0] but only the
interval [-1, 0). Thi is a similar case which might instruct you.

> > > The union of all paths with only zeros 0., 0.0, 0.00, ... does not
> > > contain and not possess an infinite path. And it does not require an
> > > infinite path to pass every finite level.
> >
> > Again that word "contain". The union of all those paths *is* an infinite
> > path.
>
> The union of all finite paths is an infinite union of finite paths.

Again shifting position. You were talking about the union of a specific
set of finite paths. The union of *all* finite paths is equal to the
union of *all* finite trees, which is a set of nodes that is not even a
path.

> Why should it be an infinite path?

The union of the paths 0., 0.0, 0.00, ..., is an infinite path by your own
definition of infinite path, namely a path that does not terminate.

If the union of finite paths contains, or is, an infinite path, then
the union of finite natural numbers contains, or is, an infinite
natural number.


> The union of all finite natural numbers is an infinite union (with
> cardinality aleph_0) but it is not an infinite natural number.

Indeed. Because by definition natural numbers are finite.

But I think you fail to see the distinction. Natural numbers are finite.
That is their definition. Paths can be infinite. That is their definition.

But finite paths cannot be infinite. That is their definition.


Regards WM

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