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Topic: What are sets? again
Replies: 21   Last Post: Dec 9, 2012 10:12 AM

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Posts: 2,665
Registered: 6/29/07
What are sets? A correction
Posted: Dec 5, 2012 3:44 PM
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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 & ~ 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



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