Re: Relative Cardinality

In article <1120589916.067271.35290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> *** T. Winter wrote:
>
> > > 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:
> >
> Thanks for your construction, but is not necessary. If there is a
> bijection possible between two well-ordered sets A and B, then the
> elements of the two sets can be put in a single well-order such that
> there is always one a e A between two b e B and vice versa (+/-1). Then
> we find that Card(A) =< Card(B) and Card(B) =< Card(A). A conclusion
> can be drawn if the actual existence of A and B is assumed, namely
> Card(A) = Card(B). As this would lead to strange results like Card(N) =
> Card({Primes}), I doubt the actual existence of such sets. This solves
> the problem. Something not completely existing cannot be equal to
> some other existing or not existing object. The property of being not
> larger, however, remains meaningful even for potentially infinite and
> necessarily incomplete sets.

GIGO
.

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