When we say "complete", are we talking about completeness in the Godelian sense? According to Godel, and formal system (except for the possibility of the oddballs mentioned below--I hadn't heard of this possibility) is "incomplete" in that there will exist true statements that can not be proven given the axioms of that system. This does not have to be anything complex...a statement like 2+2=4 in some systems may be obviously true, but there's no way to "get there" given the other axioms in the system. This statement must then be added to the axiom list.
After that (old axioms +(the 2+2=4 axiom)), we now have a "new" formal system, and there will (not might!) exist another statement that is true but unprovable, and so on. In a sense, then, no system is ever "complete".
As for the nature of the system in which Godelian Incompleteness applies, I'm not enough of a number theorist to remember. BUT...in Turing terms I know that any system that is equivalent to a Turing Machine will have the Incompleteness property. (In addition, Godelian Provability is equivalent, I think, to the Turing Halting Problem. Statements that are true but unprovable will never halt...correct?)In other words, any system that can be used for useful computation will suffer from incompletenenss, so I would assume "para-consistent logic" would fall under that category (is that similar to fuzzy logic?).
I have not, however, heretofore considered that there could exist systems that had some form of completeness built in. My intuition (which is easily wrong) tells me that no such system could ever be useful in the real world, but who the heck knows?
From: Jim Choate <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Subject: Re: The End of the Golden Age of Crypto
Date: Wed, 13 Nov 2002 07:27:44 -0600 (CST)
On Wed, 13 Nov 2002, Peter Fairbrother wrote:
> Jim Choate wrote:
>
> >
> > What I'd like to know is does Godel's apply to all forms of
> > para-consistent logic as well....
>
> And I replied:
>
> No. There are consistent systems, and complete systems, that do not admit
> Godel's theorem, but apparently not a system that is both (although even the
> last is subject to dispute, and problems of definition).
Which is the point of Godel's, you're arguing in circles here son. Perhaps
you've been out in the sun too long ;)
[SSZ: text deleted]
> However you can have eg arithmetics without Peano counting, and so on, and
> there are ("trivial" according to Godel, but even he acknowledged that they
> exist) systems that are both complete (all problems have answers) and
> consistent (no statement is both true and false).
[SSZ: text deleted]
> Can you do interesting things in such systems? Yes. But you tend to leave
> intuition behind.
What the hell does 'counting' have to do with para-consistent logic on
this? Extraordinary claims...
Para-consistent logic is logic where statements -can't by definition- be
given an absolute true/false, in fact para-consistent logic allows
axiomatic statements that are in direct conflict. The 'para' comes
from 'paradox'.
Considering the state of the real world I doubt you'd leave very much
'intuition behind' by moving to a para-consistent model.
The answer of course is "Yes, Godel's applies to Para-Consistent Logic".
Irrespective of whatever logic you wish to use, it will be sensitive to
Godel's because Godel's is a sort of halting theorem that says that with
respect to decidability you can't devise an algorithm in any language or
representation that will -guarentee- and answer to the question of whether
a particular question has an answer. Godel's applies irrespective of the
contents of any given system, paradoxical or consistent be damned.
What really matters is the 'complete', not the 'consistent'. Godel's
doesn't apply to incomplete systems because by definition there are
statements which can be made which can't be expressed, otherwise it would
be complete. You can't prove something if you can't express it since there
is no way to get the machine to 'hold' it to work on it.
--
____________________________________________________________________
We don't see things as they are, [EMAIL PROTECTED]
we see them as we are. www.ssz.com
[EMAIL PROTECTED]
Anais Nin www.open-forge.org
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