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
>
>
> _________________________________________________________________
> Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
> Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-
> forum/
> Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
> Shared Files: http://ontolog.cim3.net/file/
> Community Wiki: http://ontolog.cim3.net/wiki/
> To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
> (05)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (06)
|