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).
--------
Sorry, that was a little terse, and just a restatement of Godel. Slightly
drunk.
One way of looking at it is that Godel's theorem only applies in systems
that allow counting according to Peano arithmetic*.
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).
Or, to put it in another and possibly simpler way, if you limit the axioms
in a system in such a way that statements about statements are impossible to
formulate, then Godel doesn't apply.
Can you do interesting things in such systems? Yes. But you tend to leave
intuition behind.
-- Peter Fairbrother
*{axioms: assume Natural numbers, no 0 [can be stated in other ways, that's
original Peano], an add-one function exists, such that if x-add-one =
y-add-one then x=y, and an induction axiom showing there are infinite
numbers: applying Dedekind logic gives ax + ay = a(x+y) and so on, known as
Peano Arithmetic, which is basically ordinary arithmetic in Natural numbers
only, ie no subtraction, division etc}
