Re: An uncountable countable set


*** T. Winter schrieb:

In article <1152979000.961294.290560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
> > In article <1152884145.033413.104250@xxxxxxxxxxxxxxxxxxxxxxxxxxx> muecken=
> h@xxxxxxxxxxxxxxxxx writes:
...
Let's start with an English translation, makes it quite a bit easier for
me (I have not to do two translations with three different languages):
Works, p. 214, translated:
The question, through which transformations the ordinal number of a
well-ordered set is changed, and through which it is not changed, is
easily answered, those, and only those transformations leave the
ordinal number unchanged that can be rewritten as a finite or infinite
set of transpositions, that is, of interchanges of two elements.
Agree that this is a reasonable translation?

OK

> > No, but still, not *any* set of transpositions is permitted. Only those
> > sets of transpositions that are the result of rewriting a transformation
> > ("Umformungen"). So it remains interesting what Cantor is meaning when
> > he writes "Umformungen". And when he is meaning with that word arbitrary
> > re-orderings, he is wrong (because *all* re-orderings can be rewritten as
> > an infinite set of transpositions).
>
> You do completely misunderstand what is defined and what is the
> defintion.

I think not.

> Defined is Umformungen: "diejenigen und nur diejenigen Umformungen =
> those and only those transformations"

Yes, so that is not a definition of transformations, it restricts the
transformations to a subset of them, namely only those transformations
that have a particular property.

Of course. "Umformungen" includes many transformations like those of
iron to hammers.

Apparently there are transformations
that do *not* have that property. So the question remains, what is the
*definition* of "transformations"? (I can easily enough give a
transformation that does not have that property at all: transform {1, 2}
to {1} by mapping both elements of the first to 1. It can not be
re-written as a sequence or set of transpositions. So while it is a
transformation, it does not satisfy the requirements to leave the
ordinal number unchanged.)

So again, what is the definition of "transformation"?

The definition of *allowed transformations* was given by Cantor in
above text. There is *no* further question except that you are
unwilling to confess your error and your insult that I was unable to
understand the German text.

If it is
unrestricted (and so any re-ordering is allowed), the statement is
trivially false.

It is not the question whether Cantor was wrong, but only what was the
meaning of his quote.

If there are restrictions (and so not any re-ordering
is allowed), the statement might be true. But whatever, when the
first is true (any re-ordering is allowed as a transformation), the
statement is false and you can not use it, because it is not proven.
On the other hand, if there are restrictions, you have to show that
your set of transpositions form a transformation before you can use
the statement.

> they are defined by: "welche sich zur=FCckf=FChren lassen auf eine
> endliche oder unendliche Menge von Transpositionen, d. h. von
> Vertauschungen je zweier Elemente =3D which can be traced back to a
> finite or infinite set of transpositions, i.e., interchanges of two
> elements."

Yes, that is the definition of the property of the transformations that
leave the ordinal number unchanged. But the question remains, given
any sequence (or set) of transpositions. Is the result a "transformation"?
I do not know, because I do not know the definition of "transformation".

Then read again the text in English which you quoted above. There it is
stated explicitly.

> > > Of course it is false, whether there is a sequence or not. My
> > > transpositions form a sequence. One cannot do better.
> >
> > Yes, so what? Is yuor set of transpositions a rewriting of some
> > "Umformung" in the sense Cantor implies? So there is no "of course"
> > here. We need to know what Cantor means when he writes "Umformungen".
> >
> He says it clearly: those and only those transformations which are
> realized by a finite or infinite set of transpositions, i.e.,
> interchanges of two elements, do not change the Anzahl =3D ordinal
> number.

Yes, that is a subset of the Umformungen. What is his definition of that
term? What does he mean when he writes "Umformungen"?

I you are you unable to read English as well as German, try to get a
copy in Dutch.

> No. Cantor obviously was correct, if infinite sets would exist.

Again you state "obviously" when I ask for a proof. So you are apparently
not able to provide a proof but think that what Cantor writes is always
true. Who coined the term matheology?

There is no proof necessary. If infinite sets exist and can be
exhausted, then the set of transpositions can be finished. As no single
transposition changes the ordinal number, it remains unchanged during
the whole proess..

See above where I have written my rebuttal. You apparently have no idea
what a good definition entails. He does not define the term "Umformungen",
but defines a subset of "Umformungen" that have some particular property.
Until we know that the larger term "Umformungen" means, we have no idea
what that subset contains.

The full meaning is uninteresting, as it covers even mechanical
transformations. For our case only those tranformations are
interesting, which Cantor defined precisiely.


> > > Therefore it never becomes 0 but always remains > 0 like 0.11... never
> > > becomes 1/9.
> >
> > But apparently you do not use the standard definition of 0.111... . Is it
> > a number or something else?
>
> It is a number with aleph_0 places, which are all occupied by 1's.

Eh, it is a number that changes value? You state "never becomes". Strange
to state something like that about a number.

You will have to become accustomed with that. As only potential
infinity exists, not infinite process can be regarded as finished and
no sequence of 1's can be regarded as static unless it is finite.

Regards, WM

.

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