Re: Relative Cardinality

In article <1120569593.600126.173770@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> *** T. Winter wrote:
>
> > > I suspect, though I am not sufficiently interested to try and prove it,
> > > that according to WM's relative cardinality definition, all infinite
> > > sets will turn out the have the same "size".
> >
> > Not really. If you can order the elements of P(N) (and I think this
> > requires
> > quite a bit more) you can show that WMCard(N) <= WMCard(P(N)), but not the
> > reverse. Because the reverse would mean that there is a bijection between
> > the two.
>
> Q is equivalent to N, R/Q = X is equivalent to P(N), both by bijection.
> I have shown: Card(Q) >= Card(X) by my method. If bijection is a
> correct tool, then Card(N) >= Card(P(N)). This would lead to the only
> conclusion that the assumed existence of infinite sets in ZFC is
> inconsistent. But then we will no longer need bijections.

Without injection/bijection (pairing off members from two sets to see
which, if either, runs out first) there is no counting at all.
.

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