[FOM] Reply to Monroe Eskew message under subject: A new definition of Cardinality

[Zuhair Abdul Ghafoor Al-Johar]

Hello Monroe

My comments are inserted below:

Monroe wrote

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.)
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My reply 

 I agree totally with you regarding this particular aspect.

Monroe wrote

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?
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My reply 

 yes in ZFC this can be proved. But weather this can be proved in ZF is the real question.

Thomas Jech has a paper on proving that the class of all hereditarily countable sets is a set without choice, it appears as if this can be generalized someway for any arbitrary x. So it seems initially that these cardinals I defined can be proved to exist for any arbitrary x, with or without choice. However this issue remains to be settled, and it appears to be a very complicated issue.

Zuhair