```Date: Dec 10, 2012 4:58 AM
Author: Zaljohar@gmail.com
Subject: Re: BACKGROUND THEORY

On Dec 9, 11:18 am, Zuhair <zaljo...@gmail.com> wrote:> This topic comes as a continuation to thought presented in this Usenet> in posts:>> what are sets? againhttps://groups.google.com/group/sci.math/browse_thread/thread/a78c4e2...>> 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/a79f12...>> 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.  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.>> ZuhairA nice issue is the representation of parts.Lets take an orange call it G. of course G is a collection of someconcrete 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 ofdescriptive atoms, then g is an Ur-element.Now take the orange G and cut it into two pieces G1,G2Now take the collection of atoms of G1 and G2, this would be exactly GBut G1 and G2 is a different status from G, how can we describe thisdifference.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 ofcourse that reflects this splitting.This gives some natural basis for the POWER set axiom.Zuhair
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