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BACKGROUND THEORY
Posted:
Dec 9, 2012 3:18 AM


This topic comes as a continuation to thought presented in this Usenet in posts:
what are sets? again https://groups.google.com/group/sci.math/browse_thread/thread/a78c4e246a5c8a58/242e7fda3946ebe4?hl=en&lnk=gst&q=what+are+sets#242e7fda3946ebe4
Attn: See the corrected form of this theory at the discussing thread of it.
Background Theory. https://groups.google.com/group/sci.logic/browse_thread/thread/a79f12dff0095b3f?hl=en
Here I'll present a simple modification of background theory that widens its conceptual coverage.
EXPOSTION OF BACKGROUND THEORY:
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. Parthood: [forall z. z PP x > z P y] <> x P y ll. Antisymmetry: 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=[yphi] <> [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 {xphi} will stand for the Set representing [xphi], { } 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 Urelement. However {miami} is a set since it is a descriptive atom representing the descriptive atom miami.
So here with this approach Urelements can represent collections of physical atoms.
Zuhair



