[FOM] About Paradox Theory
charlie
silver_1 at mindspring.com
Wed Sep 14 16:03:59 EDT 2011
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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