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Is set of the indexsindices of c.e.sets that cover a productive set also productive one?

Given a productive set, there is a collection of c.e. sets union of which is the productive set, as we know that every c.e. set is with a c.e. function with a index. My question: is the set of the indexsindices of c.e.sets that cover a productive set also productive one?

Is set of the indexs of c.e.sets that cover a productive set also productive one?

Given a productive set, there is a collection of c.e. sets union of which is the productive set, as we know that every c.e. set is with a c.e. function with a index. My question: is set of the indexs of c.e.sets that cover a productive set also productive one?

Is set of the indices of c.e.sets that cover a productive set also productive one?

Given a productive set, there is a collection of c.e. sets union of which is the productive set, as we know that every c.e. set is with a c.e. function with a index. My question: is the set of the indices of c.e.sets that cover a productive set also productive one?

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Is set of the indexs of c.e.sets that cover a productive set also productive one?

Given a productive set, there is a collection of c.e. sets union of which is the productive set, as we know that every c.e. set is with a c.e. function with a index. My question: is set of the indexs of c.e.sets that cover a productive set also productive one?