Re: Cantor Confusion

In article <1174654681.038185.299380@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 22 Mrz., 16:29, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1174562837.696384.225...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> On 21 Mrz., 16:09, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > In article <1174481309.873623.92...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx
> > > > What portion of the first node
> > > > is assigned to your path?
> > >
> > > How much does the last term of the geometric series contribute to
> > > the value o the series? This is the value which the first node
> > > contributes to he path.
> >
> > As there is no last term of the geometric series this makes no sense.
>
> Nevertheless, it makes sense to calculate the sum of the series.

You can calculate the sum of the series. But what the relation is to the
contribution of the nodes to the paths is extremely unclear. From your
statement I derive that the first node contributes nothing to the path.

the first node contributes exactly as much as the last term of the
geometric series contributes to the sum 2 of the series. As this term
does not exist, it does not contribute anything. The next one before
the last one also does not exist and not contribute. But somehow some
nodes manage to contribute enough to obtain the result 2 after all.

How do you obtain that result? You have not shown a proof at all for it.

If we exchange infinitely many terms of the geometric series, its sum
remains 2, because it is absolutely converging.

As the geometric series contains only the smallest possible infinity
of terms, it should not cause problems to read it from behind.

If there were a last one. As there is not a last one I have some
difficulty with it, because I do not know where to start.

> > At every node in the tree uncountably many paths go to the left and
> > uncountably many paths go to the right. I do now know what you are
> > meaning here.
>
> At no node in the tree there exists a single path. This means: There
> are no single paths.

Still unclear. Through each node go uncountably many paths. But what you
mean with "there are no single paths" is unclear. Every two individual
paths diverge at some node from each other.

Consider all of them simultaneously. You like to do so when Cantor's
diagonal proof is concerned. Every exchanged digit is followed by
infinitely many digits which have to be exchanged. Nevertheless you
say, it is possible to exchange all of them at one time. Consider the
paths o he tree - all at the same time.

Yes, I do. What now? Every two paths separate from each other at some
specific node, through all nodes go uncountably any paths, there is no
level where all paths do separate.

This is the big mistake of set theory. The unreasonable allowance or
prohibition of instantaneously possible actions. Cantor: yes. Tree:
no. Why?

Why not? I do it and come to the same conclusion.

> > > In the whole tree there can be no more separated paths than nodes.
> >
> > Pray give a *proof*, not just a statement.
>
> The number of separated paths up to level n is given by the number of
> nodes of level n. No level of the tree has an uncountable number of
> nodes.

Makes no sense at all and is no proof. At each level n there are 2^(n-1)
separated groups of paths, where each group contains uncountably many
paths.

That is wrong, because none of these paths ever gets isolated. Is it
similar to the quarks? Path-confinement?

How do you define "isolated"? By the common meaning of the word I would
say that indeed no two paths are isolated from each other because they
have a common node. So you apparently do mean something completely
different.

We can safely state that at no level there are uncountably many
separated paths.

What do you *mean* with uncountably many separated paths.

As every node is in the tree ad no node is outside, there are at no
place in the world uncountably many numbers.

Yes, that is what has been stated all the time. The number of *terminating*
paths is countable. What this has to say about non-terminating paths
escapes me.

> > > No. It means that the number of paths which consist only of nodes
> > > with natural indexes is countable.
> >
> > Not at all. Pray provide a *proof*.
>
> The number of separated paths up to level n is given by the number of
> nodes of level n. No level of the tree has an uncountable number of
> nodes.

Yes, so what? That is no proof of your statement. When we talk about the
set of paths, we are *not* talking about any finite level. What happens at
finite levels is irrelevant to the total result.

Wrong. Every number has digits only at finite distance from the
decimal point.
In the tree there is no level "infinite", but all the infinitely many
levels are there - each one in a finite distance from the root node.

Yes, so what? That is still no proof of your statement. What happens at
finite levels is irrelevant to the number of non-terminating paths.

> > > The "distance" is measured by the index of the due node, i.e., by
> > > the number n of the corresponding level. "At no finite distance
> > > from the root" means at no node which can be enumerated by a
> > > natural number.
> >
> > Right. There is *no* node where all non-terminating paths are
> > separated from each other.
>
> There is no point in the tree where more than countably many paths are
> separated, although the tree is infinite. Outside of the tree there is
> no mathematics of real numbers.

As at each point in the tree there are uncountably many paths going
through it, I wonder what you mean with "countably many paths are
separated".

At no point of the tree (and of the universe) more than countably many
path can be distinguished.

I would state that as "at no node in the tree more than countably many
groups of paths can be distinguished", or, equivalently "at no node in
the tree terminate more than countably many terminating paths". This
still does not say anything about the number of non-terminating paths.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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