[FOM] Cardinality Beyond Regularity and Choice!

Monroe Eskew meskew at math.uci.edu
Mon Dec 14 14:48:22 EST 2009

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.

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