Re: Cantor Confusion

On 2 Apr., 14:43, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1175261756.864645.326...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> On 30 Mrz., 05:14, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > > > > > No, the sum is undefined. If you think it is defined, *prove* it and
> > > > > > *prove* that the sum is 2. What is the sum of the "sequence":
> > > > > > "..., 1/4, 1/2, 1"?
> > > > >
> > > > > Well, what is it? If any infinite series ever had a value, then the
> > > > > sum of this sequence is 2.
> > > >
> > > > But it is not a sequence according to mathematical definitions.
> > >
> > > It is a sequence according to mathematical definitions, if you read it
> > > from the left hand side.
>
> Oh, that should read "from the right hand side"! But you read it as it
> was meant.
> >
> > So there is a first element, being 1, a second element, being 1/2, the only
> > difference is that you apply right to left reading.
> >
> > > Further you can determine a unique limit value by lim{n-->oo} (1/2^n
> > > + ... + 1/8 + 1/4 + 1/2 + 1}.
> >
> > Yes, because you can revert finite sequences without consequence. You
> > do not even need convergence for that.
>
> And if the series is absolutely converging, then you can exchange all
> terms you like. The result is independent of the order.

Not arbitrarily. The result must also be a sequence. If I start with the
sequence (1, 1/2, 1/4, 1/8, ...) I have a convergent series. If I
apply in sequence the transpositions (1 2), (2 3), (3 4), ... I do not
know what I get because it is not defined in mathematics. On the other
hand, the only reasonable definition for a result I can come up with is
the sequence (1/2, 1/4, 1/8, ...), because there is *no* place for the
element "1". So the sum is changed.

You can mirror the series without change of value. You can decide to
write from right to left.

> > But in mathematics sequences
> > are defined as having a first element. On the other hand, I wonder how
> > you prove that the series of interchanges on the initial sequence lead
> > to your final "sequence".
>
> It is the same as Cantor's "proof" that he gets ready.

It is not.

It is.

> > > You know, this node cannot be determined. Therefore I use the limit
> > > (which does exist):
> > > lim{n-->oo} (1/2^n + ... + 1/8 + 1/4 + 1/2 + 1} = 2
> >
> > Yes, you use limits to show something which you can not determine. The
> > strange thing is that the first node contributes nothing, as does the
> > second nodes, as do all nodes in a finite distance from the root. And
> > as all nodes in an infinite path are a finite distance from the root.
> > Nevertheless you maintain that all nodes together contribute 2 because
> > there is no node that contributes one, there is no node that contributes
> > 1/2, etc. So there is a sequence of no nodes that contribute 2. And in
> > some mysterious way you conclude that that sequence of no nodes is the
> > same as the sequence of nodes.
>
> Take the geometric series. It contains exactly the same terms as my
> reverted series.

What is that in relevance to my remark?

To show tha its sum is 2.

> > > Do you know of a way how to finish Cantor's diagonal?
> >
> > Is there any need to finish it at all?
>
> Yes, if you want to conclude that it is different from any other list
> entry, then it must be finished. Otherwise you only know that it
> differs from some entries.

No. You can *prove* that whatever entry you take the diagonal is different.

But you cannot prove that for entries which you did not yet take. And
that is the majority and remains so.

And, this is independent of the entry you take. So the diagonal differs from
*all* entries.

Wrong. That holds only fnly for finite segments, not for infinite
segments of the diagonal.

> > Apparently you see a need to finish
> > it. For the proof it is only needed to show that there *is* a real number
> > that is not on the list. The algorithm that describes that real number is
> > sufficient.
>
> You cannot say: "there is a real number that is not in the complete
> list" unless you have searched the complete list.

You can. If you can prove there is a number that is different from each
member of the list, as is done.

Only for finite segments.

> > "Every finite part" means
> > just that: "every finite part", it does not apply to "infinite parts",
> > which is "all". On the other hand with Cantor we have "every finite element"
> > and that means "all elements", because there are no "infinite elements".

You have no remark to this, which *shows* you are wrong?

The same holds for paths. Every finite node of a path means the whole
path because there are no "infinite nodes".

> > > > Why should I? To perform the n-th exchange in Cantor's process you do
> > > > *not* have to do the first n-1 preceding exchanges first.
> > >
> > > How would you know which element the n-th element is.
> >
> > By putting n in the mapping given.
>
> The mapping given includes and requires the counting.

Why? How do you know? If the mapping is (for instance) f: N -> R,
f(n) = sqrt(n), I see no counting involved at all.

That is deplorable. (N is created by counting.)

> > > Therefore we know that only countably many are there.
> >
> > A proof, please, for once.
>
> Every separation takes place at a separation point. No separation
> takes place at any other point. There are only countably many
> separation points. And there is only one initial separate path.

Yes. So what?

Yes. That's it.

Regards, WM

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