[FOM] Cardinality beyond Scott's:

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs: 

The following definition of "Cardinality of a set"
require weaker conditions than those required for 
Scott's cardinals.


Define (H(x)): 

H(x) ={y| for all z ( z e TC({y}) -> z strictly subnumerous to x )} 

Define (Base-set):

x is a Base-set iff Exist d (d is ordinal & x=H(d))


Define(Pi(H(d))) by recursion:

For every ordinal d 

P0(H(d))=H(d) 
Pi(H(d))=Pi-1(H(d)) for any successor ordinal i 
Pi(H(d))=Union(j<i) Pj(H(d)) for any limit ordinal i

were "P" stands for "power set". 


Define( minimal for x ): 

For all x ,For all ordinals d,i

Pi(H(d)) is minimal for x
  iff 
x subnumerous to Pi(H(d)) &
For every ordinal j
(x subnumerous to Pj(H(d)) -> i subset of j). 

So when Pi(H(d)) is minimal for x, then Pi(H(d))
is said to be the 

"minimal for x iterative power of H(d)". 


Define (the nearest Base-set to x): 

For every set x, For every ordinal d.

H(d) is the nearest Base-set to x 
 iff 
Exist an ordinal i (Pi(H(d)) is minimal for x & 
For every ordinals k,j
(Pj(H(k)) is minimal for x ->(i subset of j & d subset of k))). 


The Cardinality of a set x can be defined as: 

Define(Card(x)): 

Card(x)=A 
 iff 
for all y ( y e A iff ( y equinumerous to x & 
Exist i,d ( H(d) is the nearest Base-set to x &
Pi(H(d)) is minimal for x & y subset of Pi(H(d)) ) ) ). 

In words: 

-------------------------------------------------------------------
Card(x) is the set of all sets Equinumerous to x, that are subsets
of the minimal for x iterative power of the nearest Base-set to x. 
-------------------------------------------------------------------

This definition require the following assumption to work: 

"For every set x, there exist ordinals d,i such that 
 x subnumerous to Pi(H(d))". 


Which is weaker( possibly strictly weaker) than the assumption of
"every set being equinumerous to some well founded set",
which is the assumption required for Scott's cardinals
to work under.

If these cardinals are denoted as "Z cardinals",
then the following is a theorem of (ZF-)

For all x

Exist y ( y is non empty Scott's cardinal of x )
 ->
Exist y ( y is non empty Z cardinal of x)

However the opposite is not necessarily true, I do 
think that we can have a model of (ZF-) in which

Exist x (Exist y (y is non empty Z cardinal of x) &
Exist y (y is empty Scott cardinal of x)).

is a theorem.

Best Regards.

Zuhair