```Date: Dec 9, 2012 3:18 AM
Author: Zaljohar@gmail.com
Subject: BACKGROUND THEORY

This topic comes as a continuation to thought presented in this Usenetin posts:what are sets? againhttps://groups.google.com/group/sci.math/browse_thread/thread/a78c4e246a5c8a58/242e7fda3946ebe4?hl=en&lnk=gst&q=what+are+sets#242e7fda3946ebe4Attn: See the corrected form of this theory at the discussing threadof it.Background Theory.https://groups.google.com/group/sci.logic/browse_thread/thread/a79f12dff0095b3f?hl=enHere I'll present a simple modification of background theory thatwidens 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 (viarules of FOL(=,P,Rp.0)) by the following non logical axioms:Define: x PP y <-> x P y & ~ y P xID axioms +I.  Part-hood: [forall z. z PP x -> z P y] <-> x P yll. Anti-symmetry: x P y & y P x -> x=yDef.) atom(x) <-> ~ Exist y. y PP xDef.) 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 ofy).Def.) g is atomless <-> ~ Exist x. x atom of glll. Atomistic parts:[x is a collection of atoms & forall z. z atom of x -> z P y] -> x PylV. 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 simplydenoted by a string of those atoms embraced within solid brackets [],so [a] is the collection of atoms, that has one atom which is a, ofcourse [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 yDefine: 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 ofdescriptive atoms or otherwise signify non representation."signify" in the above definition refers to "witnessing" of absence ofrepresentation and that witness is some fixed non representing atomdenoted by the primitive constant symbol 0.Define: Set(x) <->x=0 or [atom(x) & Exist y. y is a collection of descriptive atoms & xRp y]Set membership is defined as being an atom of a collection ofdescriptive 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) & xatom of zThe curly brackets shall be used to denote SETs, so {a,b,c,...} standsfor the Set representing [a,b,c,...], also {x|phi} will stand for theSet 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 realatoms [m_1, m_2,...,m_n]; now miami is the atom representing MIAMI, somiami is a descriptive atom but it is not a set since what it isdescribing is not a collection of descriptive atoms, since the atomsthat MIAMI is composed from are non descriptive objects. So miami isan Ur-element. However {miami} is a set since it is a descriptive atomrepresenting the descriptive atom miami.So here with this approach Ur-elements can represent collections ofphysical atoms.Zuhair
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