Informal understanding of Aggregate-Container theory.

Aggregates are taken to be wholes of atoms in the Mereological sense. For simplicity all objects are taken to be fusions of atoms, in other words for every object x every object y that overlaps (have a shared part) with x would overlap with an atom that is a part of x. Atomic supplementation states that if x is not a part of y then there must be an atom that is a part of x that is not a part of y. Now the *unrestricted Atomic Sum principle* would allow formation of any collection of atoms satisfying a formula phi once there is at least one atom that satisfies phi. Of course an atom is an object that has itself as the only part. The primitive relation part partially orders the universe. bottom object doesn't exist, so there is no object that is a part of every object. Now all of that is usual Atomic Mereology.

Now the relation that an atom bears to the aggregate it is a part of, is what we call as 'aggregate-membership' relation.

What's new in this theory is its informal vision about 'containers' which are taken to paraphrase 'sets' as present in set theory.

Containers can be informally understood as closed boundaries, like a line that encircles an aggregate of objects. Now a container divides space into two main compartments: inside the container and what is not inside the container. Now the aggregate of all objects inside the container is said to be 'contained' in that container, so is every part of that aggregate. This is like for example three specific eggs in some specific basket, the basket is the container and the totality (i.e. aggregate) of the three eggs is said to be 'contained' in the basket, also every aggregate of two of those eggs is also said to be 'contained' in that basket, and similarly every single egg of those is also said to be 'contained' in the basket. Now the relation that a single egg has to the basket is an example of 'set-membership' relation. More generally the relation that an 'atom' of an aggregate contained in a container bears to that container is what is named here as 'set-membership' relation. Now every object that contains an aggregate is a CONTAINER, but yet there is a CONTAINER that do not contain any aggregate inside it! that is the empty container. 

The general philosophy here is that those containers try as much as possible to mimic properties of aggregates, although the copying is not exact and it does break down here and there yet it is a kind of paralleling of what occurs at the aggregate level. This is found both in Extensionality of containers which is nothing but copying of Extensionality of aggregates, and also in Comprehension over containment which copies comprehension over aggregates. So a pure formula from the aggregate world (i.e. that do not mention containers and containment) after which there exists an aggregate of atoms satisfying it, would be a formula that decides containment in a container after a copied formula, this copying is done by replacing each occurrence of 'aggregate-membership' symbol in that formula by the symbol given to 'set-membership' and then we stipulate that a container exists containing objects that satisfy that copied formula. A clear mimicking principle!

So 'sets' are viewed as: Containers that mimic aggregates!

It is essential for this method to spell out conditions when a container becomes an atom, and this is stipulated to be when a container only contains containers inside it, a more harsh restriction is only when a container contains hereditarily 'atom container' containers inside it.

At last it is more aesthetic to stipulate that distinct containers are non overlapping (i.e. disjoint, aka. discrete).

This method can define the set of all sets (the container of all atom containers), also can define the set of all self membered sets, the set of all singletons that are self membered, etc.... something that NF cannot. So it speaks about some large sets that NF cannot address!

This method is quite strong compared to NFU, it does interpret FULL second order arithmetic by working on aggregates of hereditarily atom container containers.

It is nice to see how the customary well known set theoretic paradoxes are avoided in this theory! it is a nice piece of knowledge and is trivial to speak about it.

Formal Exposition:

Name: Aggregate-Container Theory
Language: FOL
Primitives: 
P for 'is part of'; 
Monadic C for "is a container";
Dyadic C for "is contained in".

Axioms:
[1] x P y P z -> x P z
[2] x P x
Def.): x=y <-> x P y & y P x
Def.): atom(x)<-> Ay. y P x -> x P y
Def.): x e y <-> x P y & atom(x)
[3] Ax(Ey e x)
[4] Az(z e y -> z e x) -> y P x 
[5] Ey(atom(y)phi) ->ExAy(y e x <-> atom(y)phi)
Def.): k=[x|phi] <-> Ax(x e k <-> phi)

[6] y C x -> C(x)
[7] y C x & z P y -> z C x
Def.): y E x <-> Ez(z C x & y e z)
[8] C(x) & C(y) ->(Az(z E x <-> z E y) -> x=y)
[9] Ex. C(x) & ~Ey. y C x
Def.): k={x|phi} <-> Ax(x E k <-> phi)

[10] [x|phi] exists -> {x|phi*} exists. 
where phi do not have C occurring in it, and phi* is obtained
from phi by replacing each occurrence of e by C (dyadic).

[11] (Ay. y E x -> C(y)) -> atom(x)
[12] C(x) & C(y) & Ez(z P x & z P y) ->x=y

/Theory definition finished.

A nice observation is to see that if we replace 10 and 11 by the 
following axioms:

[10] [a,b]exists ->{a,b} exists.
[11] C(x) -> atom(x)

then this is quite enough to interpret second order arithemtic in which
most of ordinary mathematics is formalizable.

A pure class\set varient of this observation is the theory in first
order logic with identity, with e as the only extra-logical symbol,
with the following axioms:

[1] if phi is a formula in which x is not free, then
    (E!xAy(y e x <-> Ek(y e k) & phi)) is an axiom.

[2] Ay(y e x ->y=a or y=b) ->Ek(x e k).

Actually the pairing axiom can be further restricted to be only of empty
or singleton classes (or containers). So we can have:

[10] a,b are singleton or empty containers -> {a,b}exists.

[2] (a,b are singleton or empty classes & Ay(y e x ->y=a or y=b)) -> Ek(x e k)

This is 'semi-constructive' since it is imaginable to construct an encircling object
around a pair of containers that are built by successive finite singlton encirclings (hereditarily
singleton or empty containers) starting form the empty container. The first 9 axioms have clear
justification even on constructive grounds, so the result is that second order arithmetic have
strong justification for it being consistent, almost semi-constructive justification.

The main merit of the 12 axiom system outlined above is that it can address some big sets 
that are not definable in NF, like the set of all self membered sets, the set of all singletons
that are members of their elements, the set of all sets that are members of members of there elements
etc.... So it might assisst in understanding such constructions.

Zuhair Al-Johar
8/11/2013