Productive Set

A set  of integers is productive if there exists a partial recursive function  such that, for any , the following holds: If the domain of  is a subset of , then  is convergent,  belongs to , and  does not belong to the domain of , where  denotes a recursive function whose Gödel number is .

Productive sets are not recursively enumerable.

REFERENCES

Davis, M. Computability and Unsolvability. New York: Dover, 1982.Kleene, S. C. Mathematical Logic. New York: Dover, 2002.Rogers, H. Theory of Recursive Functions and Effective Computability. Cambridge, MA: MIT Press, 1987.