Re: Relative Cardinality



*** 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.

Regards, WM

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