The definition of "set' and " membership "

Hi all,

Actually using Mereology we can DEFINE 'set' and 'set membership' in
the following manner.

We introduce a two place primitive relation symbole N
which stands for " is the name of " , to the primitives of identity
and part-hood that is used in Mereology, and then define epislon
membership in the following manner:

Define: x e y <-> Ez ( x N z & y atomic part of z )

were x N z is read as x is the name of z.

Define: x is a set <-> Ez ( x N z )

So sets are names of heaps.

It is not difficult to introduce axioms that will lead to all ZF
axioms.

For example extensionality will be

Axiom: x N y -> [~Ez(~z=x & z N y) & ~Ez(~z=y & x N z )]

i.e N is bijective from names to heaps.

Axiom: x N y -> x is atomic

To build the hierarchy of sets.

If we want to element the ur-elements we axiomatize the following:

Axiom: x is atomic -> x is a set

To produce ZFC we axiomatize the following

Size limitation: x subnumerous to V <-> x is a set.

were V is the heap of all atoms.

Power:x is a set -> Power(x) is a set

Infinity: the heap of all finite ordinals is a set.

This will produce all axioms of null,
pairing,union,separation,replacement and global choice.

We can also use ackermann's approach.


Note: the variable present above are heaps.
.

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