[FOM] A new definition of Cardinality.

[Monroe Eskew]

Hello Zuhair,

It is worth pointing out that your definition still has a disadvantage
if you don't assume choice.  Without choice, not all cardinalities are
comparable.  If they were then all cardinalities in your sense would
be comparable to a cardinality that contains a Von Neumann ordinal,
but from this you could derive choice.  (It is nice to have a linear
order on set sizes.)

I'm not sure if you can prove in ZF that every set has a cardinality
in your sense.  For every set X is there a set Y and a function f such
that Y is hereditary and f is a bijection between X and Y?

Monroe

On Sun, Nov 22, 2009 at 2:16 PM, Zuhair Abdul Ghafoor Al-Johar
<zaljohar at yahoo.com> wrote:
>
> 1) Von Neumann's cardinals has the limitation of being dependent on
> choice, without choice one cannot know what is the cardinality of
> Power(omega) for example.Accordingly in any theory which do not have
> the axiom of choice among its axioms most of its sets would be of
> indeterminable cardinality, which is a big draw back.
>....
> 4) The cardinality of any set x is: The class of all sets
> that are equinumerous to x were every member of their transitive
> closure is strictly subnumerous to x.
>
> So for any set x, any y is a member of the cardinality of x,
>  if and only if, y is equinumerous to x and every member of the
> transitive closure of y is strictly subnumerous to x.