Re: Review of Mueckenheims book.

On 5 Mrz., 02:33, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1173024358.088126.251...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck....@xxxxxxxxxxxxxxxxx writes:

> On 4 Mrz., 03:53, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > In article <1172913211.189020.243...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
> > > If you think so...
> >
> > The original sequence (9.2) remains unchanged. If you think that is
> > changes, you should have stated so. New elements are sought in that
> > original sequence, not in a sequence with transpositions applied.
>
> In order to find the new elements in the original sequence, this
> original sequence has to be searched from the beginning, i.e., from
> its first element till to that one fitting the requirements.

Indeed.

> This
> process of searching consists of transpositions: The pointer has to be
> transposed continuously with the elements.

What does *this* mean? I would think that the search is:
set pointer to 1

set pointer in front of q_1

, does q_1 fit the requirements?
no
set pointer to 2, does q_2 fit the requirements?
no/yes
etc. Where are the transpositions?

Exchanging the position between the pointer in front of 1 and q_1,
such that the pointer is in front of q_2

> Here are all the quotes containing "Transposition" and "Umformung" of
> his collected works:
>
> Durch Umformung einer wohlgeordneten Menge wird, wie ich dies in Nr. 5
> wegen seiner Wichtigkeit wiederholt hervorgehoben habe, nicht ihre
> Mächtigkeit geändert, wohl aber kann dadurch ihre Anzahl eine andere
> werden.

I agree with this. So any "Umformung" does *not* change the cardinality,

beause no element is taken away and no element is added.

but *can* change the ordinality.

Yes, Cantor says "die Anzahl".

I may note in passing that an "Umformung"
is actually an "Umformung" that leaves a well-ordered set, otherwise we
can not talk about the ordinality. So changing:
{1, 2, 3, ...}
to
{..., 3, 2, 1}
is *not* an Umformung in this sense. (There is no ordinality for the latter
ordered set.)

An Umformung is every change of order. By an infinite set of
transpositions we get the Umformung from {1,2,3,...} to {...,3,2,1}.
That is why Cantor was wrong.

This is dubiously worded, but probably right within his view. The
question is, can the Umformung
{1, 2, 3, ...} -> {2, 3, ..., 1}
be brought back to an infinite sequence of transpositions? The answer
is, no.

If an infinite sequence of transpositions exists, the answer is and
must be yes. Because every *finite* sequence of transpositions leaves
the 1 at a *finite* position {2, 3, ..., n, 1, n+1, ...}. So there
exists no infinite sequence of transpositions. Reason: There exists o
infinite set.

> Ich hebe noch folgendes hervor: wenn in einer wohlgeordneten Menge M
> irgend zwei Elemente m und m' ihre Plätze in der gesamten Rangordnung
> wechseln, so wird dadurch der Typus nicht verändert, also auch nicht
> die "Anzahl" oder "Ordnungszahl". Daraus folgt, dass solche
> Umformungen einer wohlgeordneten Menge die Anzahl derselben ungeändert
> lassen, welche sich auf eine endliche oder unendliche Folge von
> Transpositionen von je zwei Elementen zurückführen lassen, d. h. alle
> solche änderungen, welche durch Permutation der Elemente entstehen.

Seems right, indeed. But note that this requires that an Umformung
leaves a well-ordered set. So you first have to state your Umformung
and *next* how it can be represented as an infinite sequence of
transpositions. Your exposition fails here, because there is *no*
smallest rational number, there is *no* well-ordering of the rational
numbers with the smallest one in the first position.

That is because there is no infinite set.

Regards, WM

.

Quantcast