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 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. Part-hood: [for all 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 for all y. 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 & for all 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

& (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. 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 <->

[x is a collection of atoms & for all y. y atom of x -> y 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 represents the atom

"miami" that represents the collection of all atoms of the real city

MIAMI.

So MIAMI is the collection of real atoms (i.e. physical atoms)

m_1,m_2,...,m_n

MIAMI = [m_1, m_2,...,m_n]

and 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 atoms MIAMI is composed from

are not 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.

Zuhair