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Topic: My final formal answer as to what classes are and what class
membership is!

Replies: 7   Last Post: Apr 7, 2013 4:34 AM

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Charlie-Boo

Posts: 1,585
Registered: 2/27/06
Re: My final formal answer as to what classes are and what class
membership is!

Posted: Apr 2, 2013 6:50 PM
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On Apr 2, 6:25 pm, Charlie-Boo <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}.
>
> C-B
>
>
>

> > 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, Part-hood 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 phi-ers 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 atom-hood.

>
> >  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 Ur-element is defined as an element of a class that is not a
> > class.
> > Or alternatively a non-class 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 part-hood 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 Al-Johar
> > 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 non-r.e. sets
("classes".)

C-B



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