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Re: [ontolog-forum] Current Semantic Web Layer Cake

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Francis McCabe <frankmccabe@xxxxxxx>
Date: Wed, 1 Aug 2007 11:26:50 -0700
Message-id: <F2391DCC-A4B7-4157-864B-7A7F8419F754@xxxxxxx>
Actually, I believe that there is something analogous to nearly- 
consistent:    (01)

So-called paraconsistent systems. If you know that the shortest proof  
of inconsistency is N, then in a paraconsistent system, any  
conclusion involving fewer than N steps is consistent.    (02)

Frank    (03)

On Aug 1, 2007, at 9:29 AM, Christopher Menzel wrote:    (04)

> On Jul 31, 2007, at 5:09 PM, Kathryn Blackmond Laskey wrote:
>> [Pat wrote:]
>>> Set theory is ..., usually in the form of
>>> Zermelo-Fraenkel (Z-F) set theory
>>
>> or the equally powerful, somewhat less intuitive, but finitely
>> axiomatizable von Neuman / Godel / Bernays (NGB) set theory
>>
>>>  ... still the widely
>>> accepted mathematical foundational language that
>>> is as near to consistent as anything can be.
>>
>> It either is or is not consistent.  You can't be "near to consistent"
>> any more than you can be "almost pregnant."
>
> Pat of course knows that (as I am sure you are yourself aware).
> Surely all he meant was that, in over 100 years of extensive
> theoretical examination and extremely heavy use, no set theoretic
> paradoxes have arisen in ZF (likewise, it follows, for the equi-
> consistent NGB).  While that of course is not a formal proof, this
> gives logicians great confidence that the theory is consistent.
> (There is also a very intuitive informal model of ZF -- the so-called
> cumulative hierarchy -- whose role is analogous to the natural number
> structure for Peano Arithmetic.  We can't prove PA consistent (in
> PA), but it is quite intuitively clear that it *is* consistent given
> our belief in the existence of the natural number structure.
> Similarly for the cumulative hierarchy and the consistency of ZF.)
>
> -chris
>
>
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