[FOM] About Paradox Theory
David Auerbach
auerbach at ncsu.edu
Sat Sep 17 13:38:50 EDT 2011
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
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