Re: Heap-Set Theory H-S

On Jan 25, 5:42 pm, Zaljo...@xxxxxxxxx wrote:
Hi all,

The following is a link to a Theory of mine that blends Mereology with
set theory.

http://zaljohar.tripod.com/index.html#heap

Zuhair

It would be interesting to define the cardinality of a heap, which
informally speaking can be understood as the number of atoms in a
heap.

But here we would have a problem, that is we cannot use the heap of
all bijective heaps to define the cardinality of that heap, because
simply we will arrive at one cardinality for all heaps, since the heap
of all binary heaps is exactly the same as the heap of all atoms, and
exactly the same as the heap of n atoms. So we cannot use this method.

However we can add 'Cardinality' as a primitive one place function
symbole and define the formulas Cardinality(x)=cardinality(y)
cardinaltiy(x)<cardinality(y), and cardinality(y)<cardinality(x)
in a Cantorian manner.

Of course the functions here are not sets, they are actually heaps,
but it is not difficult to define relations, functions, and bijection,
injections
as being subheaps of the cartesian product heap between two heaps.

Actually what we call cardinality of a non empty set x is actually the
cardinality of the heap of all members of this set x. However we
cannot define cardinality of the empty set in this manner,that's why I
define cardinality of a set as the heap of all sets bijective to that
set, and in this way the cardinality of the empty set will be the
empty set itself.

Also I was trying to find a symbole to represent heaps , something
that is comparative to the brackets { } of sets.

I think the will define that symbole as ' '

So z= 'xy' <-> Ak ( k is atomic part of z <-> ( k=x or k=y ) ).

You see there is no need for the coma between the members of a heap
and I used the smallest symbole to surround these members of a heap.

of course from the above axioms it is understood that

'x' =x
'y'= y
x is atomic part of 'xy'
x is atomic part of x
y is atomic part of 'xy'
y is atomic part of y
'xy' is a part of 'xy'
x neq y -> 'xy' neq x
x neq y -> 'xy' neq y

Zuhair
.

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