[FOM] About Paradox Theory

[marcel.crabbe at uclouvain.be]

The argument  doesn't even require equality; e.g. the axiom of singletons can be replaced by: 

AyEzAx(F(xz) <--> F(xy) & F(yx))

Marcel Crabbé


Le 19 sept. 2011 à 02:22, hdeutsch at ilstu.edu a écrit :

> 
> I agree that the nonexistence of the class of grounded classes is not a theorem of FOL.  But I hesitate to agree that it requires specifically set theoretic premises.  The only required premise is AyEzAx[F(xz) <--> x = y] and as Vaughan Pratt mentioned, the argument does not even assume extensionality.  I must confess, though, I'm not sure what is at stake here.  Perhaps one should just say that the most significant application of the relevant reasoning is in set theory.
> 
> Harry Deutsch
> 
> 
> 
> 
> Quoting Vaughan Pratt <pratt at cs.stanford.edu>:
> 
>> Think of "transitive closure" as a taboo term in the language of FOL. Just because a term is taboo doesn't mean you can't work with it.  This is true of various terms arising not just in logic but personal relationships, divinity, etc.
>> 
>> Vaughan Pratt
>> 
>> On 9/17/2011 10:38 AM, David Auerbach wrote:
>>> 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?
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