Effectiveness and Godel Theorems in Fuzzy

It is well known that the notions of a "decidable subset" and "recursively enumerable subset" are basic one for classical logic. In particular, they are basic tools for the proof of the famous limitative theorems about the undecidability and incompleteness of first order logic. Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E. S. Santos in an interesting series of papers. Indeed, Santos, starting from an idea of L. Zadeh (Zadeh [1968]), proposed the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program. Santos proved that all these definitions determine the same notion of computability for fuzzy maps (see Santos [1970] and Santos [1976]). As in the classical case, a corresponding definition of recursively enumerable fuzzy subset is obtained by calling recursively enumerable any fuzzy subset which is the domain of a computable fuzzy map. Successively, a notion of recursive enumerability was proposed in Harkleroad [1984] where a fuzzy subset s is said to be recursively enumerable if the restriction of s to its support is a partial recursive function. In a large series of papers L. Biacino and the author proposed a definition of recursive enumerability which is a proper extension of both definitions of Santos and Harkleroad. In this paper, we will refer to the resulting theory. Background in recursion theory is required for understanding the arguments in this paper (see, for example, Rogers [1976]). A more complete version of this technical note can be find in Chapter 11 of my book in Fuzzy Logic by Kluwer Editor.