```Date: Dec 5, 2012 3:44 PM
Author: Zaljohar@gmail.com
Subject: What are sets? A correction

Language: FOL + P, RpP 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. vP x  & v P z.Def.) atom(x) <-> for all y. y P x -> x P yDef.) 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. zatom of y].Def.) c is atomless <-> ~ Exist x. x atom of clll. 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 orabsence of atoms.Def.) Set(x) <-> Exist c. (c is a collection of atoms or c isatomless) & 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 isuniquely represented by an atom.Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atomof 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 ory=d/Zuhair
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