Cutting the Ribbon of Ordinals;

Hi All:

for any well ordered set A, the ordinal number of it_ is the set of all
orders of its members in it, were an order of a member in a set_ is the
set of all orders of its predecessors in it.

Example: A={a,b,c}

were a P b P c , then

The order of a in A is the set of the orders of its predecessors in A.

since a has not predecessors in A then the order of a in A is { }

The order of b in A is the set of the orders of its predecessors in A,
ie the set of the order of a in A.

so the order of b in A = { {} }

Similarily the order of c in A = { {}, { {} } }.

The ordinal number of A = { order of a in A, order of b in A, order of
c in A }

= { {} , { {} } , { {} , { {} } } }

For any set to have an ordinal number it should be well ordered .

The cardinal number of a set A is the intersectional set of all ordinal
numbers bijectable to A.

Also an ordinal can be defined as the set of sets which subsets their
power sets, and which itself subsets its power set.

x is an ordinal <-> x C P(x) /\ A m in x -> m C P(m).

An equivalent definition to this is:

x is an ordinal <-> x: A m in x -> m C P(m) /\ m C x.

This will form the following set:

O={0,1,2,3,..........,w,w+1,w+2,........,2w,..,..3w,..................,w^2,..,,,..,w^3,..........,w^w
, (w^w)+1,......}

Now it can be proved that there is no member in this set that is
bijectable to O.

if we say for example that O is bijectable to w

since w is bijectble to itself.

then w U { x: |x|= w } = K -> |K| = w

Now K is the set of all ordinals injectable to w.

it is clear that K itself is an ordinal and since it is not in itself (
Axiom of regularity) then K is an ordinal not bijectable to w.

Similarily it can be proved that there cannot be a bijection between
any ordinal in O and O itself.

Let K=w1

w1= { x: |x| =w } is not bijective to w

w2= { x: |x| = w1 } is not bijective to w1.

There is unlimited wi.

Now since |w| < | P(w) |

it is obvious that P(w) = w1

P(P(w))= w2.

It is clear that P(w) = w^w.

This means that w1= { x: x is an ordinal /\ x < w^w }.

similarily w2= { x: x is an ordinal /\ x < ( w^w ) ^ (w^w) }.

w3 = { x: x is an ordinal /\ x < [( w^w ) ^ (w^w)]^[( w^w ) ^ (w^w)]}

..
..
..
..

The long term question of P(w)=w? is solved above.


All of this depends on the assumption that P(w) can be well ordered.

Otherwize. the following definition of cardinality is to be followed.

A cardinal number of a set_ is the set of all pre-cardinals bijectable
to a set.

a pre-cardinal is a set that subsets its power set.

This new definition of cardinality is required for those who believe
that |P(w)| > wi

ie the cardinality of the power set of w is larger than all Alephs.

(note: what is called Aleph-1, Aleph-2,...... is to be reduced to
w1,w2,....... ).

Zuhair

.

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