Re: Godel's Theorem under Fuzzy Logic?

Torkel Franzen (torkel@sm.luth.se)
Mon, 6 Apr 1998 04:01:34 +0200 (MET DST)

Christian Borgelt <borgelt@iws.cs.uni-magdeburg.de> writes:

>Completeness means that starting from the axioms and applying only
>the allowed inference rules you can, in principle, prove any true
>formula of the formal system.

No it doesn't. Completeness is a syntactic concept: for every A,
A or the negation of A is provable in T.

>Another difficulty, which is not limited to fuzzy logic, is whether
>G"odel's proof is actually a proof. To prove incompleteness, we have
>to interpret the formula and have to understand that what it says is
>true. That is, the result is not achieved by formal reasoning, but
>by some meta-reasoning done from outside the system.

This is a misunderstanding. Godel's theorem for a theory T is
an ordinary mathematical theorem, provable (for standard theories
T) in T itself.