The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to fuzzy set theory is a crucial one. A first proposal in such a direction was made by E.S. Santos by the notions of "fuzzy Turing machine", "Markov normal fuzzy algorithm" and "fuzzy program" (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable. For example, in one shows that the fuzzy Turing machines are not adequate for fuzzy language theory since there are natural fuzzy languages intuitively computable that cannot be recognized by a fuzzy Turing Machine. Then, they proposed the following definitions. Denote by "Ü" the set of rational numbers in Then a fuzzy subset "s" : "S" formula_4[0,1 of a set "S" is recursively enumerable if a recursive map "h" : "S"×"N" formula_4"Ü" exists such that, for every "x" in "S", the function "h"("x","n") is increasing with respect to "n" and "s"("x") = lim "h"("x","n").