Language: FOL
Primitives: 'e' for 'is an element of'; '=' for 'is identical to'; 'Rp' for 'represents'
Axioms:
ID axioms+
Define: x=[x] <-> for all y (y e x <-> y=x)
1.Elements: Exist y (x e y) <-> x=[x]
2.Extensionality: for all z (z e x <-> z e y) -> x=y
3.Extensions: if phi is a formula in which x do not occur free, then all closures of "Exist x for all y (y e x <-> y=[y] ^ phi)" are axioms.
Define: x=[y|phi] <-> for all y (y e x <-> y=[y] ^ phi)
4.Representatives: Exist y (x Rp y) <-> x=[x]
5.Representationality: x Rp a ^ y Rp b -> [x=y <-> a=b]
Define (membership 'E'): x E y <-> Exist z (y Rp z ^ x e z)
Define (set): x is a set <-> Exist y (x Rp y)
Define (represented): x is represented <-> Exist y (y Rp x)
Define (x*): x Rp y <-> y=x*
Define ({,}): x= {y|phi} <-> x Rp [y|phi]
6. Representation of pure extensions: if phi is a formula in L(e,=) in which only symbols y,w1,..,wn occur free, then:
for all w1,..,wn
[w1,..,wn are sets or are represented ->
for all x (x=[y|phi] -> x is represented)]
is an axiom.
Define (inclusive): x is inclusive <-> x is a set ^ for all y (y C x* -> y is represented) where C is the subextension relation defined as:
A C B <-> for all z (z e A -> z e B)
Define (Hereditarily inclusive): x is inclusive ^ for all y (y E TC(x) -> y is inclusive)
Where TC stands for "transitive closure of .." defined in terms of E,=.
7. Reflective Set Representation: Any definable extension of Hereditarily inclusive sets not using other than set parameters, is represented by a set.
8. Reflective Inclusive Representation: Any L(=,E) definable set of Hereditarily
inclusive sets not using other than hereditarily inclusive parameters, is inclusive.
/Theory definition finished.
Informal account:
I think the idea of this theory is clear. We have the relation e which is 'element-hood' that builds extensions, the elements of those extensions are the individuals which are formalized here as self elemented singletons (with respect to relation e), however extensions can only be elements of others if they are e-singletons, in this sense extensions can be reasoned as FLAT collections, an empty extension is also formalized here although this clearly defies intuition yet it is a nice technical fix. Now the primitive relation "Represents" is added, symbolized as "Rp", with the intention to represent extensions. Construction of extensions is Liberal from ANY defining formula, i.e. for any phi whatsoever we have the extension of all elements that satisfy phi. However Representation is limited to certain extensions, i.e. not all extensions have representatives. A representative is fixed to be a self elemented singleton and it is called as a "SET", while membership "E" is defined as being an element of an extension represented by the set. By the membership relation we can pull up extensions to be members of other extensions and therefore enabling having a hierarchy of sets of sets of sets,etc.. And this is why representation is actually defined in the first place, i.e. in order to overcome the shortage of element-hod relation which cannot define a hierarchy of extensions being elements of extensions. The first and simplest rule of representation is that of representing pure extensions, and this is done in the pure language of extensions (i.e. L(=,e)) so all extensions definable in the language of extensions from parameters substituted by represent-able extensions or by sets, are represent-able. This enacts a Boolean structure over sets of this theory and also proves E-singletons. The next rule of representation is that of representing definable extensions of hereditarily inclusive sets, and the later mean sets that represent extensions having every sub-extension of them being represent-able, and hereditarily is used in the usual sense to mean that the set is inclusive and every element of its transitive closure (defined after membership E) is also inclusive. Now those kinds of sets are considered as somewhat safe from being involved in paradoxes, so careful representation rules of extensions of them is thought here to be safe. So the first rule states that ANY extension of hereditarily inclusive sets if it is definable without using non set parameters then it is represent-able. The second rule states that any set of hereditarily inclusive sets if it is definable in the language of sets (i.e.; L(E,=)) without using non hereditarily inclusive set parameters then it is inclusive. I tend to think that this theory is consistent and at least a little bit stronger than Ackermann's set theory. I think it affords some explanation to the primitives of Ackemrann's. However it is stronger than Ackermann's since it defines a model of it. Here the Ontology is richer we have proper classes and sets, and the later we have inclusive sets and those that are not inclusive, and from the inclusive sets we have those that are hereditarily so. The Big sets, like the universal set, are non inclusive sets. Hereditarily inclusive sets can be seen to be the sets of ZF and those of Ackermann's, while inclusive sets can be seen as the Classes of Ackermann's, however here we have higher than those classes which are the big sets like the universal set and it’s alike as well as we have proper classes, i.e. extensions that do not have representatives, we can easily modify the theory as to have Ur-elements, and also we already have objects that are neither elements nor are sets nor classes like non singleton extensions of members of sets, those are neither elements nor are set nor are classes. So Ontologically speaking this theory is richer than most set theories, it appears to encounter all possible Ontologies.
Zuhair