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My final formal answer as to what classes are and what class membership is!
Posted:
Mar 28, 2013 3:02 PM


See: http://zaljohar.tripod.com/sets.txt
Below is the full quote from the above link.
********************************************************************************* What Are Classes!
This account supplies THE final answer as to what classes are, and what is class membership relation, those are defined in a rigorous system with highly appealing well understood primitive notions that are fairly natural and easy to grasp. It is aimed to be the most convincing answer to this question. The formulations are carried out in first order logic with Identity, Parthood and Naming binary relations. Identity theory axioms are assumed and they are part of the background logical language of this theory. The mereological axioms are those of GEM (Generalized Extensional Mereology), they are the standard ones. The two axioms of naming are very trivial. The definitions of classes and their membership are coined with the utmost care to require the least possible assumptions so they don't require grounds of Atomic Mereology or unique naming or the alike.., so they can work under more general situations. Also utmost care was taken to ensure that those definitions are nearer to the reality of the issue and not just a technical fix. I simply think that what is given here do supply the TRUE and FINAL answer to what classes are and to what is their membership!
The General approach is due to David Lewis. Slight modifications are adopted here to assure more general and nearer to truth grounds.
Language: FOL(=,P,name)
Axioms: ID axioms +
1.Reflexive: x P x
2.Transitive: x P y & y P z > x P z
3.Antisymmetric: x P y & y P x > x=y
Define: x O y iff Exist z. z P y & z P x
4.Supplementation: ~y P x > Exist z. z P y & ~ z O x
5.Composition: if phi is a formula, then ((Exist k. phi) > Exist x (for all y. y O x iff Exist z. phi(z) & y O z)) is an axiom.
Definition: x is a collection of phiers iff for all y. y O x iff Exist z. phi(z) & y O z
6.Naming: n name of y & n name of x > y=x
Definition: n is a name iff Exist x. n name of x
7.Discreteness: n,m are names & ~n=m > ~n O m /
Definitions of "Class" and "Class membership":
Define: x E y iff Class(y) & Exist n. n P y & n name of x.
1. Class(x) iff x is a collection of names.
2. Class(x) iff x is a collection of names Or x never overlap with a name.
when x never overlaps with a name then it is to be called an inert object.
Definition: x is inert iff ~Exist n. n is a name & x O n
3. Class(x) iff x is a sum of an inert object and (an inert object or a collection of names)
Sum defined as:
Sum(x,y) = z iff for all q. q O z iff q O x Or q O y.
1 is incompatible with the empty class. 2 is incompatible with the subclass principle that is : "Every subclass of x is a part of x".
3 does the job but it encourages gross violation of Extensionality over classes since having multiple names for an object is the natural expectation!
If we assume the subclass principle and use definition 3 then full Extensionality over classes is in place and it follows that the empty Class is an atom. Although attractive on the face of it (since the empty set is just a technical fix), however it is not that convincing since there is no real justification for such atomhood.
If we strengthen the subclass principle into the principle that: "For all classes X,Y (Y subclass of X iff Y P X)", then only definition 1 can survive such a harsh condition, and this would force all names to be atoms and shuns the existence empty classes altogether! such a demanding commitment that despite the clear aesthetic gain of having internally pure classes in the sense that all classes are only composed of parts that are classes, yet still this is a very demanding commitment that do not seem to agree with basic natural expectations about naming.
So a definition of classes that proves Extensionality over them without restricting multiple naming per object is what is demanded.
Define: x is an equivalence collection of names iff there exist y such that x is the collection of all names of y.
Define: y is a fusion of equivalence collections of names iff y is a collection of names & for all a,b,c (a P y & a name of b & c name of b > c P y)
Define V' as the collection of ALL inert objects.
4. Definition: Class(x) iff x is a sum of V' and (V' or a fusion of equivalence collections of names)
As far as the concept of class is concerned Extensionality is at the core of it, so 4. is the right definition of classes.
It is nice to see that the *Empty Class* is just the collection of all inert objects.
For the sake of completion of this approach, we may say that Definition 4. is an Equivalence rendering of Definition 3. Similarly we can introduce two further definitions that are Equivalence renderings of Definition 1 and Definition 2. But those are rarely applicable in class\set theories.
Now one can easily define a set as a class that is an element of a class. An Urelement is defined as an element of a class that is not a class. Or alternatively a nonclass object. All kinds of circular membership can be explained; paradoxes can be easily understood. Also non definability of some classes can be understood.
This account explains membership and classes in a rigorous manner. And actually supplies the FINAL answer!
Somehow those definitions might be helpful in orienting thought about some philosophical questions about mathematics founded in set theory. For example identity and parthood are expected natural relations and they can be reasoned about as being human independent, but Naming might present some challenge, definitely it favors human dependency but still it can be human independent! Philosophical debate about the nature of sets would become a debate about the nature of naming procedures.
Zuhair AlJohar March 21 2013 ****************************************************************** Zuhair



