[FOM] A new definition of Cardinality.

[Zuhair Abdul Ghafoor Al-Johar]

Dear Mr. Forster:

 You seem to be right regarding the concept of not every set is equinumerous to a well founded set. However the way how I see matters, is that every set MUST have a cardinality that is a set despite choice and regularity. We need to find a general definition of cardinality in ZF minus Regularity (and without choice).

However as a first step, Scott's trick is acceptable, for at least it defines cardinality for all well founded sets despite choice.

By the way, I have the following question in my mind, which might be related to Scott's trick?

We know from ZF that for every set x there exist a transitive closure 
TC(x) that is a set. 

Now does ZF(with Regularity of course)
prove or refute the following? 

For all x Exist y 
(y equinumerous to x  and
not Exist z (z equinumerous to x and TC(z) strictly subnumerous to TC(y))) 

were 

x subnumerous to y  <-> Exist f (f:x-->y, f is injective) 
x equinumerous to y <-> Exist f (f:x-->y, f is bijective) 


Zuhair