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Date: 20-1-2022
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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.
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.
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