Re: Relative Cardinality



*** T. Winter wrote:
> In article <1120485719.564066.165390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> > Of course, you are right. But order can help to count the elements. Do
> > you know that Cantor's alephs (which I do not get by my method, perhaps
> > you intermingled these notions with cardinality) can only be definied
> > for well-ordered sets?
>
> Perhaps in Cantor's time. You are still working with Cantor's definitions
> and notions.

You are doing so by insisting upon bijection. But as far as I am
infored, Zermelo's assertion is accepted also today: "da bei Cantor der
Nachweis noch fehlt, daß jede Menge einer Wohlordnung fähig und daher
jede Mächtigkeit ein Alef ist."

> They have changed a bit in the course of time. The
> cardinals are defined for *all* sets.

Only because all sets are believed to be capable of being well-ordered,
erroneously.

> It is the ordinals that are defined
> for well-ordered sets only.

A cardinal is but an ordinal (OZ). Cantor: Eine OZ, die zu keiner
kleineren OZ äquivalent ist, heißt eine Kardinalzahl.
>
> Indeed, the current definition does not regard order for cardinalities.

You are explicitly wrong. A cardinal is but an ordinal (OZ). Cantor:
Eine OZ, die zu keiner kleineren OZ äquivalent ist, heißt eine
Kardinalzahl. No ordinal ==> no cardinal.

Regards, WM

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