[FOM] Cardinality Beyond Regularity and Choice!

[Monroe Eskew]

I would like to see a proof or disproof that this assumption "Z" is
strictly weaker than choice.  I wonder whether known models of ZF+~AC
can be used.


On Sun, Dec 13, 2009 at 12:12 PM, Zuhair Abdul Ghafoor Al-Johar
<zaljohar at yahoo.com> wrote:
>
> Cardinality(A) is the class of all sets equinumerous to A that are
> hereditarily strictly subnumerous to A.
>
> In ZFC those cardinals are non empty sets. However those Cardinals unlike Von Neumann's do not necessarily require full choice, they actually require the following assumption.
>
> (Z)For every set x: Cardinality(x) is a non empty set
>
> or in ZF style
>
> (Z)For every x Exist y ( y=Cardinality(x) & ~y=0 ).
>
> Now that assumption if added to ZF as an axiom.
> then ZF+Z is weaker than ZFC, perhaps strictly weaker actually.
>