Jim Choate wrote:
>
> 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....
>
>> 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...
Godel's (allegedly?) applies, as Ben pointed out, to "any sufficiently
complex system". The requirement of "sufficient complexity" is that the
system contains Peano counting.
Systems described by Presburger, by Skolem, and by Tarski are among those
which do not include Peano counting, and which are both consistent and
complete.
The relevance of non-Peano counting is simply that you can often do more
things in a system that includes some form of counting.
One way of stating Godel is "No system that includes Peano counting is both
consistent and complete".
> The answer of course is "Yes, Godel's applies to Para-Consistent Logic".
Trivially, to the extent that all paraconsistent systems are not consistent
by definition, you can say "yes".
You can also say "no"! Not all paraconsistent systems include Peano
counting. Depends what you mean by "apply".
Godel also has connotations of consequences _within_ the system, eg
regarding decideability. Let me introduce a term, "Godellike", to describe a
system that obeys those supposed consequences.
Are paraconsistent systems Godellike? Not necessarily, that's one of the
reasons for the development of paraconsistent systems.
> 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.
Ahh, those problems of definition again. "Complete" is normally* taken to
mean that every statement expressable within a system is provably true or is
provably false within the system. I don't know offhand of any paraconsistent
systems that have that property, but it's not impossible afaik.
IMO "complete" has nothing to do with "statements which can be made which
can't be expressed" - though I may be wrong, as I don't understand exactly
what that means.
-- Peter Fairbrother
*As in Godel's other famous theorem, the completeness theorem, which is
completely (ouch) different to his incompleteness theorem, the one we are
discussing.
