> The following is an account about what sets are, first I'll write the > exposition of this base theory in brief, then I'll discuss some > related issues. > > Language: FOL + P, Rp > > P stands for "is part of" > > Rp stands for "represents" > > Axioms: Identity theory axioms + > > I. Part-hood: P partially orders the universe. > > ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.
x subset y, y not subset x -> some z subset y with x not subset z. x proper subset y -> some z subset y with x not subset z x proper subset y -> y\x subset y, x not subset y\x
Oh my, no empty set.
> Def.) atom(x) <-> for all y. y P x -> x P y > > Def.) x atom of y <-> atom(x) & x P y. > > Def.) c is a collection of atoms iff for all y. y P c -> Exist z. z > atom of y. > > Def.) c is atomless <-> ~ Exist x. x atom of c > > lll. Representation: x Rp c & y Rp d -> (x=y<->c=d) > > lV. Representatives: x Rp c -> atom(x) > > V. Null: Exist! x. (Exist c. x Rp c & c is atomless). > > A Set is an atom that uniquely represents a collection of atoms or > absence of atoms. > > Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is > atomless) & x Rp c & atom(x) > > Here in this theory because of lV there is no need to mention atom(x) > in the above definition. > > Set membership is being an atom of a collection of atoms that is > uniquely represented by an atom. > > Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom > of c & atom(y) > > Here in this theory because of lV there is no need to mention atom(y) > in the above definition. > > Vl. Composition: if phi is a formula in which y is free but x not, > then > [Exist y. atom(y) & phi] -> [Exist x. x is a collection of atoms & > (for all y. y atom of x <-> atom(y) & phi)] is an axiom. > > Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or > y=d > / > > This theory can interpret second order arithmetic. And I like to think > of it as a base theory on top of which any stronger set theory can > have its axioms added to it relativized to sets and with set > membership defined as above, so for example one can add all ZFC axioms > in this manner, and the result would be a theory that defines a model > of ZFC, and thus proves the consistency of ZFC. Anyhow this would only > be a representation of those theories in terms of different > primitives, and it is justified if one think of those primitives as a > more natural than membership, or if one think that it is useful to > explicate the later. Moreover this method makes one see the Whole > Ontology involved with set\class theories, thus the bigger picture > revealed! This is not usually seen with set theories or even class > theories as usually presented, here one can see the interplay between > sets and classes (collections of atoms), and also one can easily add > Ur-elements to this theory and still be able to discriminate it from > the empty set at the same time, a simple approach is to stipulate the > existence of atoms that do not represent any object. It is also very > easy to explicate non well founded scenarios here in almost flawless > manner. Even gross violation of Extensionality can be easily > contemplated here. So most of different contexts involved with various > maneuvering with set\class theories can be easily > paralleled here and understood in almost naive manner. > > In simple words the above approach speaks about sets as being atomic > representatives of collections (or absence) of atoms, the advantage is > clearly of obtaining a hierarchy of objects. Of course an atom here > refers to indivisible objects with respect to relation P here, and > this is just a descriptive atom-hood that depends on discourse of this > theory, it doesn't mean true atoms that physically have no parts, it > only means that in the discourse of this theory there > is no description of proper parts of them, so for example one can add > new primitive to this theory like for example the primitive "physical" > and stipulate that any physical object is an atom, so a city for > example would be an atom, it means it is descriptively an atom as far > as the discourse of this theory is concerned, so atom-hood is a > descriptive modality here. From this one can understand that a set is > a way to look at a collection of atoms from atomic perspective, so the > set is the atomic representative of that collection, i.e. it is what > one perceives when handling a collection of atoms as one descriptive > \discursive whole, this one descriptive\discursive whole is actually > the atom that uniquely represents that collection of atoms, and the > current methodology is meant to capture this concept. > > Now from all of that it is clear that Set and Set membership are not > pure mathematical concepts, they are actually reflecting a > hierarchical interplay of the singular and the plural, which is at a > more basic level than mathematics, it is down at the level of Logic > actually, so it can be viewed as a powerful form of logic, even the > added axioms to the base theory above like those of ZFC are really > more general than being mathematical and even when mathematical > concepts are interpreted in it still the interpretation is not > completely faithful to those concepts. However this powerful logical > background does provide the necessary Ontology required for > mathematical objects to be secured and for > their rules to be checked for consistency. > > But what constitutes mathematics? Which concepts if interpreted in the > above powerful kind of logic would be considered as mathematical? This > proves to be a very difficult question. I'm tending to think that > mathematics is nothing but "Discourse about abstract structure", where > abstract structure is a kind of free standing structural universal. > Anyhow I'm not sure of the later. I don't think anybody really > succeeded with carrying along such concepts. > > Zuhair >