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 & ~ Exist v. v P x & v P z.
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