Date: Nov 30, 2012 1:17 PM Author: Zaljohar@gmail.com Subject: What are sets? again The following is an account about what sets are, first I'll write the

exposition of this base theory in brief, then I'll discuss some

related issues.

Language: FOL + P, Rp

P stands for "is part of"

Rp stands for "represents"

Axioms: Identity theory axioms +

I. Part-hood: P partially orders the universe.

ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.

Def.) atom(x) <-> for all y. y P x -> x P y

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.) c is atomless <-> ~ Exist x. x atom of c

lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)

lV. Representatives: x Rp c -> atom(x)

V. Null: Exist! x. (Exist c. x Rp c & c is atomless).

A Set is an atom that uniquely represents a collection of atoms or

absence of atoms.

Def.) Set(x) <-> Exist c. (c is a collection of atoms or c is

atomless) & x Rp c & atom(x)

Here in this theory because of lV there is no need to mention atom(x)

in the above definition.

Set membership is being an atom of a collection of atoms that is

uniquely represented by an atom.

Def.) x e y iff Exist c. c is a collection of atoms & y Rp c & x atom

of c & atom(y)

Here in this theory because of lV there is no need to mention atom(y)

in the above definition.

Vl. 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.

Vll. Pairing: for all atoms c,d Exist x for all y. y e x <-> y=c or

y=d

/

This theory can interpret second order arithmetic. And I like to think

of it as a base theory on top of which any stronger set theory can

have its axioms added to it relativized to sets and with set

membership defined as above, so for example one can add all ZFC axioms

in this manner, and the result would be a theory that defines a model

of ZFC, and thus proves the consistency of ZFC. Anyhow this would only

be a representation of those theories in terms of different

primitives, and it is justified if one think of those primitives as a

more natural than membership, or if one think that it is useful to

explicate the later. Moreover this method makes one see the Whole

Ontology involved with set\class theories, thus the bigger picture

revealed! This is not usually seen with set theories or even class

theories as usually presented, here one can see the interplay between

sets and classes (collections of atoms), and also one can easily add

Ur-elements to this theory and still be able to discriminate it from

the empty set at the same time, a simple approach is to stipulate the

existence of atoms that do not represent any object. It is also very

easy to explicate non well founded scenarios here in almost flawless

manner. Even gross violation of Extensionality can be easily

contemplated here. So most of different contexts involved with various

maneuvering with set\class theories can be easily

paralleled here and understood in almost naive manner.

In simple words the above approach speaks about sets as being atomic

representatives of collections (or absence) of atoms, the advantage is

clearly of obtaining a hierarchy of objects. Of course an atom here

refers to indivisible objects with respect to relation P here, and

this is just a descriptive atom-hood that depends on discourse of this

theory, it doesn't mean true atoms that physically have no parts, it

only means that in the discourse of this theory there

is no description of proper parts of them, so for example one can add

new primitive to this theory like for example the primitive "physical"

and stipulate that any physical object is an atom, so a city for

example would be an atom, it means it is descriptively an atom as far

as the discourse of this theory is concerned, so atom-hood is a

descriptive modality here. From this one can understand that a set is

a way to look at a collection of atoms from atomic perspective, so the

set is the atomic representative of that collection, i.e. it is what

one perceives when handling a collection of atoms as one descriptive

\discursive whole, this one descriptive\discursive whole is actually

the atom that uniquely represents that collection of atoms, and the

current methodology is meant to capture this concept.

Now from all of that it is clear that Set and Set membership are not

pure mathematical concepts, they are actually reflecting a

hierarchical interplay of the singular and the plural, which is at a

more basic level than mathematics, it is down at the level of Logic

actually, so it can be viewed as a powerful form of logic, even the

added axioms to the base theory above like those of ZFC are really

more general than being mathematical and even when mathematical

concepts are interpreted in it still the interpretation is not

completely faithful to those concepts. However this powerful logical

background does provide the necessary Ontology required for

mathematical objects to be secured and for

their rules to be checked for consistency.

But what constitutes mathematics? Which concepts if interpreted in the

above powerful kind of logic would be considered as mathematical? This

proves to be a very difficult question. I'm tending to think that

mathematics is nothing but "Discourse about abstract structure", where

abstract structure is a kind of free standing structural universal.

Anyhow I'm not sure of the later. I don't think anybody really

succeeded with carrying along such concepts.

Zuhair