All recursive languages are also recursively enumerable. The unrestricted grammars characterize the recursively enumerable languages. This grammar class can generate arbitrary recursively enumerable languages. Every complete recursively enumerable first-order theory is decidable. The set of halting turing machines is recursively enumerable but not recursive. The halting language is therefore recursively enumerable. It is evident that Diophantine sets are recursively enumerable. Recursively enumerable languages are not closed under set difference or complementation. In computational complexity theory, the complexity class containing all recursively enumerable sets is RE. Any "axiomatizable" fuzzy theory is recursively enumerable. Some pairs of recursively enumerable sets are effectively separable and some are not. This method is primarily used to construct recursively enumerable sets with particular properties. The recursively enumerable sets are exactly the sets at level formula 85. Later research dealt also with numberings of other classes like classes of recursively enumerable sets. The class of all recursively enumerable languages is called RE.