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Re: My final formal answer as to what classes are and what class membership is!
Posted:
Apr 2, 2013 6:25 PM


On Mar 28, 3:02 pm, Zuhair <zaljo...@gmail.com> wrote: > See:http://zaljohar.tripod.com/sets.txt > > Below is the full quote from the above link. > > ********************************************************************************* > What Are Classes!
I swear to Allah I was going to ask the same question. Not because I figured out the answer or wondered, but because the answer hit me in the face in my refutations of faulty proofs that NST is inconsistent.
The umpteen axioms etc. below are shit  worthless. That misses the whole point.
1. If you throw out a bunch of axioms, then you are making arbitrary decisions and there is no way you can say that a class means that and only that. 2. The idea is to move AWAY from a bunch of arbitrary decisions and ask what is really wanted/needed. 3. Class is a primitive and so anything complex  having millions of smaller subsets  would define millions of things even more primitive. That is counterintuitive and contrary to the intent.
Why are there classes?
Because people are inconsistent.
1. Everything is a set. 2. x ~e x is not a set.
So what is x ~e x? A formula.
So you have the set of values that make a formula true, including something that is not a set . . .
???
Alls I know is that in Frege Logic we have concepts that are total Boolean functions, and that x ~e X is not total so we have a partial function.
Sets are total functions from everything to {TRUE,FALSE}.
Classes are partial functions from everything to {TRUE,FALSE}.
CB
> 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



