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Posts: 2,665
Registered: 6/29/07
Posted: Dec 9, 2012 3:18 AM
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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
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

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,

(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) &

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.


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