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What are sets? A correction
Posted:
Dec 5, 2012 3:44 PM


Language: FOL + P, Rp
P stands for "is part of" Rp stands for "represents"
Axioms: Identity theory axioms +
I. Parthood: 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
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Zuhair



