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From: zuhair on 2 Jan 2010 18:51
Hi all,

in previous discussions about the subject of Cardinality the following
question raised:

is the following a theorem of ZF,

For all x Exist y ( y is hereditary and y equinumerous to x )

y is hereditary is defined as y having every member of its transitive
closure strictly subnumerous to y.

Define(hereditary): y is hereditary <->
for all z ( z e TC(y) -> z strictly subnumerous to y )

I personally don't have a proof of that but it looks as if it can be
proved.

My account on that is the following:

For every set x, there must exist a set y such that
y is equinumerous to x of the lowest rank.

Lets put things in this way:

We have the Cumulative Hierarchy with stages V_0, V_1,.....,V_i
for every ordinal i.

Now for any set x there must exist a minimal ordinal i
such that there exist a set y that is equinumerous to x and y subset
of V_i ,so for all j < i there do not exist a set y that is
equinumerous to x and a subset of V_j.

Now, either i is a limit ordinal, or is a successor ordinal,

Lets say that i is a successor ordinal, so there is i-1,

Now y would be a set of subsets of V_(i-1)

V_(i-1) itself is not supernumerous to y, i.e. it is either
incomparable to y or strictly subnumerous to y.

However V_(i-1) is transitive!

So there must exist a subset y of V_i that is equinumerous to x, and
that have all members of V_(i-1) as members of it, i.e. V_(i-1) is a
subset of y, the reason is because all sets equinumerous to x that are
subsets of V_i are nothing but different combinations of subsets of V_
(i-1), because V_i is actually Power(V_(i-1)),
and V_(i-1) is transitive , so all its members are also members of
V_i, so there must exist a subset y of V_i that is equinumerous to x
and at the same time having V_(i-1) as a subset of it.
If this is correct then V_(i-1) would be strictly subnumerous y, and
since all members of y are subsets of V_(i-1) then
all are subnumerous to V_(i-1) and thus strictly subnumerous to this
y.

This means that one of the sets that are equinumerous to x
in V_i must be hereditary. Thus establishing the proof.

Although the above is not a full formal proof, because it gives strong
intuitive background for such an assumption, i.e. the assumption that
for every set x there exist a hereditary set that is equinumerous to
x, of course working in ZF.


Another issue that is also important, I am seeing that among well
Equinumerous orderable hereditary sets the one which has the maximal
rank would be the ordinal. This can give a motivation
of proving the existence of the set of all sets that are hereditarily
strictly subnumerous to a set ( i.e. the set of all sets that are
strictly subnumerous to a specific set x, were every member of their
transitive closures is strictly subnumerous to x).

If this turns to be the case then my definition of Cardinality of x as
the set of all sets equinumerous to x having every member of their
transitive closures strictly subnumerous to x. would be workable in
ZF.

Zuhair
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