[FOM] About Paradox Theory

[Bill Greenberg]

"The point of philosophy is to start with something so simple as not
to seem worth stating, and to end with something so paradoxical that
no one will believe it."
--Bertrand Russell

On Wed, Sep 14, 2011 at 1:03 PM, charlie <silver_1 at mindspring.com> wrote:
>
>        I'm sure your project has merit, but I can never overcome "Russell's Paradox" because of the following theorem of  first-order logic.
>
>           ~EyAx[F(xy) <--> ~F(xx)]
>
>               As a consequence, I tend to dismiss R's Paradox as having nothing to do with sets or anything else in particular.   I'm sure I must be wrong in this, as his paradox regarding classes or sets enjoys such wide popularity, perhaps due to the Frege connection, and was perhaps a guiding light for improved set theories.   So, perhaps someone will please give me a good reason not to trivialize it so.  Perhaps I'm making some significant mistake. I'd be glad to be corrected.
>
> Charlie
>
>
>
>
>
>
> On Sep 13, 2011, at 3:56 PM, Zuhair Abdul Ghafoor Al-Johar wrote:
>
> > Dear FOMers,
> >
> > I would like to know what would be the foundational
> > initial stand point on the following suggestion
> > of mine about establishing a comprehensive study
> > of logical paradoxes on their own.
> > A study that characterize those paradoxes in such
> > a manner that each category of paradoxes would bear
> > specific implications especially as regards machinery
> > of producing further paradoxes within each category.
> > A study that also depicts various inter-category issues,
> > i.e. the relation between each category to the other.
> > A study that also aids us in building a strategy for
> > avoiding such paradoxes when constructing theories
> > of interest.
> >
> > To illustrate an example of the above,
> > lets take Russell paradox, it seems that we
> > can build up a set of those paradoxes where all
> > can be described by a single simple rule that is:
> >
> > The set of all sets that are not e_i members
> > of themselves cannot exist, because otherwise
> > the i_singleton set of it would be paradoxical
> > i.e. both e_i of itself and not e_i of itself.
> > e_i is defined recursively as
> >
> > x e_0 y iff x e y
> >
> > x e_i y iff Exist z. z e_i-1 y and x e z
> >
> > for i=1,2,3,......
> >
> > 0_singleton (x) = y iff y=x
> >
> > i_singleton (x) = y iff Exist! z. z e y and z = (i-1)_singleton (x)
> >
> > for i=1,2,3,....
> >
> > Let's call the above Russell paradox category.
> >
> > Now this shows this paradox in more depth than the usual
> > presentation which is only the top tier of the above.
> >
> > Now we need to study further how can we produce other paradoxes
> > by working within the above category. For example, lets take
> > the second tier of Russell paradox category, i.e. that concerned
> > with e_1 membership, now we can have a sub-paradox of this
> > which is already known as Lesniewski's paradox which is
> > the set of all singletons that are not in their members
> > cannot exist, because the singleton of that set is paradoxical
> > i.e. it is in its sole member and not in its sole member. This is
> > exactly the same argument behind the second tier of Russell's paradox
> > category shown above. Seeing this connection one can go further
> > and define further Lesniewski's paradox category in an exactly
> > similar manner as to how it is defined above for Russell's.
> >
> > Another example is Russell's paradox of second order logic,
> > i.e. on predicates, this is also can be viewed as extending
> > the same argument but to a higher language, and i think
> > this can also be extended into a category like the above,
> > and possibly has sub-categories of it similar to Lesniewski's
> > paradox category, that is besides many other possible
> > sub-categories.
> >
> >
> > I personally think that a comprehensive study of paradoxes
> > would be fruitful in the sense of increasing the awareness
> > about them and thus facilitating constructing theories
> > of general interest that can avoids them.
> >
> > So what FOM would say about that?
> >
> > Regards
> >
> > Zuhair
> >
> >
> >
> >
> >
> >
> >
> >
> > _______________________________________________
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> > FOM at cs.nyu.edu
> > http://www.cs.nyu.edu/mailman/listinfo/fom
>
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