```Date: Dec 9, 2012 12:40 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: [for all 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 for all y. y P c -> Exist z. zatom of y.Def.) g is atomless <-> ~ Exist x. x atom of glll. Atomistic parts: [x is a collection of atoms & for all z. z atomof x -> z P y] -> x P ylV. 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 ofatoms& (for all y. y atom of x <-> atom(y) & phi)]) is an axiom.Define: x=[y|phi] <-> [x is a collection of atoms & (for all y. yatomof 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 <->[x is a collection of atoms & for all y. y atom of x -> y isdescriptive].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 descriptiveatoms & y Rp z & atom(y) & x atom 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 represents the atom"miami" that represents the collection of all atoms of the real cityMIAMI.So MIAMI is the collection of real atoms (i.e. physical atoms)m_1,m_2,...,m_nMIAMI = [m_1, m_2,...,m_n]and miami is the atom representing MIAMI, so miami is a descriptiveatom but it is not a set since what it is describing is not acollection of descriptive atoms, since atoms MIAMI is composed fromare not descriptive objects. So miami is an Ur-element.However {miami} is a set since it is a descriptive atom representingthe descriptive atom miami.So here with this approach Ur-elements can represent collections ofphysical atoms.Zuhair
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