Re: Relative Cardinality

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. They have changed a bit in the course of time. The
cardinals are defined for *all* sets. It is the ordinals that are defined
for well-ordered sets only. The cardinals are the "sizes" of sets in the
equivalence classes induced by the bijection relation. The ordinals are
the "sizes + order" of sets in the equivalence classes induced by the
order-preserving bijection relation.

> Although cardinality has nothing to do with
> order?

Indeed, the current definition does not regard order for cardinalities.

> > > Of course order is not fundamental, but if an order can be established,
> > > then my definition is a sharp criterion to determine whether other
> > > criteria are meaningful.
> >
> > Except that it is a false criterion.
>
> It is a very fine criterion! Would Kronecker already have seen it,
> nobody knew set theory today.

Ok, let's see how your criterion helps you. Consider two sets. Set A
contains the natural numbers , set B the negative even numbers. Consider
the following relative ordering of the elements of sets A and B in their
union:
b (in B) < -b/2 (in A) < b - 2 (in B)
by your definition WMCard(B) <= WMCard(A). Next:
-b/2 (in A) < b (in B) < -b/2 + 1 (in A)
by your definition WMCard(A) <= WMCard(B).

So the conclusion is that WMCard(A) = WMCard(B). Something you
rejected earlier (but is nevertheless true with standard cardinals).
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