[FOM] On the nature of Sets

[Zuhair Abdul Ghafoor Al-Johar]

Dear FoMers

The following is a further refinement of my previous approach about
defining the notions of set\class\ur-elements (Links to which are
are provided at the end of this account). One can regard *every* first
order set\class theory as a suitably defined extension of this system.


EXPOSITION:

language first order logic with primitive binary relations of:
"part of" and " is a Label of".

Axiom I:
x part of y iff  [(for all z. z part of x -> z part of y) and
Exist u. u part of y and not u part of x].

Define (atom): x is an atom iff not Exist y. y part of x.

Define (atom of): x atom of y iff ( x part of y & x is an atom).

Axiom II: For all x. x is an atom or Exist y. y atom of x.

Any object in this theory is to be called an *aggregate* weather
it is an atom or not, so a singleton aggregate is an atom while
a non singleton aggregate is clearly an aggregate of atoms.

Define (=): x=y iff for every non atom z. y part of z iff x part of z

Axiom III: for all x. for all y. y is a Label of x -> y is an atom

Axiom IV: for all x. (Exist y. x is a label of y) ->
Exist! y. x is a Label of y.

Axiom scheme V: if phi is a formula in which x is not free, then
[(not Exist! atom y. phi) -> Exist x (for all y. y atom of x iff (y is an
atom & phi))] is an axiom.


A class is defined as:
x is a class iff for all y,z. y atom of x and z atom of x and not y=z
-> Exist u,w. y is a Label of u and z is a Label of w and not u=w.

Epsilon membership e is defined as:
y e x iff [x is a class & Exist z. z atom of x & z is a label of y].

A set is defined as:
x is a set iff [x is a class and Exist y. y is a label of x].

So a set is a labeled class.

An Ur-element is defined as an aggregate that is not a class.

Or if one demands membership then it would be a labeled aggregate
that is not a class (thus not a set).

All of the above is an exact formal system
specifying exactly what those terms are. Then
if one want to add say axioms of ZF then she\he
can extend the above system with those 
axioms restricted to *sets*. One can extend
this system with any set\class theory provided
she\he makes the correct restrictions. Also this
opens the door for studying many types of
Ur-elements like those that are non class
aggregates of labels of classes, those 
that are aggregates of many non labeling atoms
or a mixture of both kinds, etc...

Zuhair

PS:Older versions posted to FOM:
http://www.cs.nyu.edu/pipermail/fom/2011-May/015375.html
http://www.cs.nyu.edu/pipermail/fom/2011-May/015456.html