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From: zuhair on 18 Jan 2010 23:45
Hi,

I believe that I managed to come with a definition of Cardinality
that works under grounds strictly weaker than that of Coret's
assumption of every set being equinumerous to some well founded set.

Scott's cardinals require Coret's assumption, and as far as I know
among all known defined cardinals, Scott's cardinals work under
weaker conditions than the others.

So here I shall define those cardinals that I think they work under
conditions strictly weaker than Coret's assumption.

Define(H(x)):

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

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

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

Define( minimal for x ):

For every set x ,For all ordinals i,d

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 ).

Pi(H(d)) is said to be the "minimal for x iterative power of H(d)".

Define (near to x):

For every set x, For every ordinal d.

H(d) is near 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)).

Define (the nearest to x):

For every set x, For every ordinal d

H(d) is the nearest to x iff
H(d) is near to x &
For every ordinal k (H(k) is near to x -> d subset of k))

The Cardinality of any 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 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 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))".

Lets denote this assumption by the letter "Z".

Now Z is a theorem of ZF, so these cardinals
work in ZF.

It turns that Z is strictly weaker than Coret's, since the later is
nothing but a special case of Z ! this can be seen clearly by simply
fixing "d" in Pi(H(d)) to be the empty set, so Z would be converted
to:

For every set x, there exist an ordinal i, such that
x subnumerous to Pi(H(0)).

Which is Coret's assumption itself.

We can have a model of (ZF-) + Z.
in which Coret's assumption fail,
but we cannot have a model of (ZF-)+Coret's
in which Z fails!

So these cardinals do work under strictly weaker conditions than those
of Scott's cardinals, thus defining cardinalities of a larger subclass
of sets than does Scott's cardinal define.

Weather this is by itself an interesting result, I actually don't know?
but I would tend to believe so!

Zuhair






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