Re: Godel's Theorem under Fuzzy Logic?

prof. Giangiacomo Gerla (gerla@matna2.dma.unina.it)
Mon, 20 Apr 1998 14:39:40 +0200 (MET DST)

In some paper we define and study the notion of recursive enumerability for
fuzzy subsets and the one of productive or creative fuzzy subset.
Also, we prove that if a fuzzy logic is "universal" (i.e. able to represent
any recursive enumerable fuzzy set) then in such a logic Goedel limitative
theorems holds. Nevertheless we was not able to give a good example of
"universal fuzzy logic". In any case I suggest to read

L. Biacino and G. Gerla, Decidability, recursive enumerability and Kleene
Hierarchy for L-subsets, Z. Math. Logik Grund. Math. 35 1989.
where some limitative theorem is given.

See also
L. Biacino and G. Gerla, Fuzzy sets: a constructive approach. Fuzzy Sets and
Systems 45 (1992).
At 04.02 06/04/98 +0200, you wrote:
>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.
>
>