[FOM] About Paradox Theory

[David Auerbach]

Might it be that it is full generalization of the paradox (to chains of any length) that isn't first-orderizable, even though there's a first-order version for each length? And that that's what T. Forster meant?  


David Auerbach                                                      auerbach at ncsu.edu
Department of Philosophy and Religious Studies
NCSU
Raleigh, NC 27695-8103

On Sep 17, 2011, at 11:05 AM, hdeutsch at ilstu.edu wrote:

> 
> Here is the argument concerning the "paradox of grounded classes" to save people from having to look it up:
> 
> The following argument is first-order valid:
> 
> AyEzAx(F(xz) <--> x=y).  Therefore,
> 
> -EwAx(F(xw) <--> Au([F(xu) --> Ey(F(yu) & -Ez{F(zu) & F(zy)])]).
> 
> The claim is that this is the paradox of grounded classes "as described in [Montague's paper mentioned in my last message]".
> 
> Harry Deutsch
> 
> 
> 
> 
> 
> Quoting T.Forster at dpmms.cam.ac.uk:
> 
>> Vaughan, Agreed, but how then *would* you characterise the difference between Russell's paradox and other indisputably set-theoretic paradoxes such as Mirimanoff? Charlie has pointed to *something*. what do you want to say about that something?
>> 
>> 
>> On Sep 16 2011, Vaughan Pratt wrote:
>> 
>>> 
>>> 
>>> On 9/14/2011 1:03 PM, charlie 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
>>> 
>>> This theorem holds in a Boolean topos, but I don't know how much further you can take it than that, those better grounded in category theory should be able to say.  The theorem is set-theoretic to the extent that the category Set is the canonical Boolean topos, so I don't think it's fair to say it has nothing to do with sets.
>>> 
>>> In less categorical language, the semantics with which you give this sentence meaning is set-theoretic.
>>> 
>>> Vaughan Pratt
>>> _______________________________________________
>>> FOM mailing list
>>> FOM at cs.nyu.edu
>>> http://www.cs.nyu.edu/mailman/listinfo/fom
>>> 
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
>> 
> 
> 
> 
> ----------------------------------------------------------------
> This message was sent using Illinois State University RedbirdMail
> 
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom