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Replies: 1   Last Post: Dec 10, 2012 4:58 AM

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Posts: 2,665
Registered: 6/29/07
Posted: Dec 10, 2012 4:58 AM
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On Dec 9, 11:18 am, Zuhair <> wrote:
> This topic comes as a continuation to thought presented in this Usenet
> in posts:
> what are sets? again
> Attn: See the corrected form of this theory at the discussing thread
> of it.
> Background Theory.
> Here I'll present a simple modification of background theory that
> widens its conceptual coverage.
> Language: FOL (=,P,Rp,0)
> P is the binary relation "is part of"
> Rp is the binary relation "represents"
> 0 is a constant symbol.
> Background Theory is the collection of all sentences entailed (via
> rules of FOL(=,P,Rp.0)) by the following non logical axioms:
> Define: x PP y <-> x P y & ~ y P x
> ID axioms +
> I.  Part-hood: [forall z. z PP x -> z P y] <-> x P y
> ll. Anti-symmetry: x P y & y P x -> x=y
> Def.) atom(x) <-> ~ Exist y. y PP x
> Def.) x atom of y <-> atom(x) & x P y.
> Def.) c is a collection of atoms iff forall y P c (Exist z. z atom of
> y).
> Def.) g is atomless <-> ~ Exist x. x atom of g
> lll. Atomistic parts:
> [x is a collection of atoms & forall z. z atom of x -> z P y] -> x P
> y
> lV. Representation: x Rp c & y Rp d -> (x=y<->c=d)
> V. 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 &
>  forall y. y atom of x <-> atom(y) & phi))
> is an axiom.
> Define: x=[y|phi] <->
> [x is a collection of atoms & (forall y. y atom of x <-> atom(y) &
> phi)]
> For convenience writable finite collections of atoms shall be simply
> denoted by a string of those atoms embraced within solid brackets [],
> so [a] is the collection of atoms, that has one atom which is a, of
> course [a]=a; similarily [a,b] is the collection of atoms a and b.
> Vl. Null. atom(0) & ~Exist x. 0 Rp x.
> Define: x is descriptive <-> x=0 OR Exist y. x Rp y
> Define: x is a descriptive atom <-> atom(x) & x is descriptive.
> Define: x is a collection of descriptive atoms <->
> [forall y. y P c -> Exist z. z atom of y & z is descriptive].
> A set is defined as an atom that uniquely represents a collection of
> descriptive atoms or otherwise signify non representation.
> "signify" in the above definition refers to "witnessing" of absence of
> representation and that witness is some fixed non representing atom
> denoted by the primitive constant symbol 0.
> Define: Set(x) <->
> x=0 or [atom(x) & Exist y. y is a collection of descriptive atoms & x
> Rp y]
> Set membership is defined as being an atom of a collection of
> descriptive atoms that is represented by an atom.
> Define: x member of y <->
> Exist z. z is a collection of descriptive atoms & y Rp z & atom(y) & x
> atom of z
> The curly brackets shall be used to denote SETs, so {a,b,c,...} stands
> for the Set representing [a,b,c,...], also {x|phi} will stand for the
> Set representing [x|phi], { } stands for 0.
> /
> So for example the set {miami} is an atom that represent the atom
> "miami" which represent the real city MIAMI.
> The real city MIAMI can be viewed as some concrete collection of real
> atoms [m_1, m_2,...,m_n]; now miami is the atom representing MIAMI, so
> miami is a descriptive atom but it is not a set since what it is
> describing is not a collection of descriptive atoms, since the atoms
> that MIAMI is composed from are non descriptive objects. So miami is
> an Ur-element. However {miami} is a set since it is a descriptive atom
> representing the descriptive atom miami.
> So here with this approach Ur-elements can represent collections of
> physical atoms.
> Zuhair

A nice issue is the representation of parts.

Lets take an orange call it G. of course G is a collection of some
concrete physical atoms.

Now lets say that g is an atom that represents G.

So g is a descriptive atom. And since G is not a collection of
descriptive atoms, then g is an Ur-element.

Now take the orange G and cut it into two pieces G1,G2

Now take the collection of atoms of G1 and G2, this would be exactly G

But G1 and G2 is a different status from G, how can we describe this

This can be done if G1 and G2 have representative atoms g1, g2.

In this way the atom representing the collection [g1,g2] is a set of
course that reflects this splitting.

This gives some natural basis for the POWER set axiom.


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