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Re: My final formal answer as to what classes are and what class membership is!
Posted:
Apr 2, 2013 6:50 PM


On Apr 2, 6:25 pm, CharlieBoo <shymath...@gmail.com> wrote: > 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 Hide quoted text  > >  Show quoted text 
The real questions are why do you have classes? What are we trying to do?
The answer is: Avoid Russell's Paradox in set definitions while allowing everything else.
(And if we knew of other definitions that led to a contradiction we would avoid them, too.)
No?
I think that means any predicate calculus wff defines a set but x ~e x does not. Then is x ~e x a predicate calculus wff? And the higher order definitions of sets?
Mainly I say to parallel the Theory of Computation for a consistent definition. TOC defines r.e. sets ("sets") and nonr.e. sets ("classes".)
CB



