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From: zuhair on 15 Jan 2010 01:36
Hi all,

I am still seeking a definition of Cardinality using the methodology
of hereditarily sets that is supposed to work in ZF a lone (i.e
without Choice).

First we have the following as a theorem of ZF

For every ordinal d Exist x For all y
( y e x <-> x is hereditarily strictly subnumerous to d ).

x is hereditarily strictly subnumerous to d iff
x is strictly subnumerous to d &
for all z ( z e TC(x) -> z is strictly subnumerous to d ).

We define H_d in the following manner:

x=H_d iff For all y
( y e x <-> x is hereditarily strictly subnumerous to d ).

So we have: for all d Exist H_d as a theorem of ZF.

Now I come to define the notion of a "nearest Hereditarily_ordinal
set".

First I shall define the "iterative powering of H_d" , in a recursive
manner,

P0 (H_d) = H_d
P1 (H_d) = P (P0 (H_d)) = P(H_d)
P2 (H_d) = P (P1( H_d)
..
..
..
Pi (H_d) = P ( Pi-1 (H_d)) , or any successor ordinal i.

For any limit ordinal j

Pj (H_d) = Union(i<j) Pi (H_d)

Now it is a theorem of ZF that for any set x , there exist an ordinal
d and an ordinal j such that x subnumerous to Pj (H_d).

(Note: x subnumerous to y iff Exist f (f:x-->y, f is injective)).


This is easy to prove when x is finite, and also when x is infinite
then every x is actually a subset of an iterative power of
H_Omega. Since every x is a subset of the Cumulative
Hierarchy, and every iterative power of H_Omega is a stage
of the Cumulative Hierarchy.

Now I come to define the minimal iterative power of H_d having x
subnumerous to it.

For any set x and for any set H_d were x is subnumerous
to some iterative power of H_d, we call Pj(H_d) the minimal
iterative power of H_d having x subnumerous to it, iff Pj(H_d)
is a subset of every set Pi (H_d) that has x subnumerous to it.

Note that for every ordinal i , Pi(H_d) is transitive.

Now I come to define " nearest Hereditarily_ordinal" sets.

For any set x, H_d is a nearest Hereditarily_ordinal set to x
iff
there exist an ordinal j such that x subnumerous to Pj (H_d) &
For every every ordinal k were there exist an ordinal i
such that x subnumerous to Pi(H_k), then if Pj (H_d)
and Pi(H_k) are the minimal iterative powers of H_d
and H_k respectively having x subnumerous to them, then
j is ordinally smaller than or equal to i.

Now for any set x, if H_d is a nearest Hereditarily_ordinal set to x,
then for every ordinal k>d , H_k is a nearest Hereditarily_ordinal set
to x also.

Now obviously we can have the minimal of all nearest
Hereditarily_ ordinal sets to x.

Now we can define cardinality of any set as:

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

Now this definition have two requirements:

(1) For every ordinal d, the set H_d exists.

(2) For every x, there exist ordinals d,i such that
x subnumerous to Pi(H_d)

Both of which are theorems of ZF, but not of ZF-
so these should be added to ZF- for those cardinals to work.

If we modify the above definition to:


Card(x) is the set of all pure well founded sets Equinumerous to x,
that are subsets of the minimal iterative power of the minimal nearest
Hereditarily_ordinal set to x (Coret's)

then all what we need is the assumption that every set x is
equinumerous to some well founded set, for that definition to work.


Scott cardinals do not work with (ZF-)+(1)+(2).
Modified Scott cardinals work with Coret's assumption in ZF-
but also the same modification makes the above definition work
with Coret's assumption in ZF-.

So this definition seems to be a little bit stronger than Scott's.

Zuhair





















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