M.Sipser’s Introduction to the Theory of Computation offers the following problem in its chapter on decidability:

*Let A be a Turing-recognizable language consisting of descriptions of Turing machines, {⟨M1⟩,⟨M2⟩, …}, where every Mi is a decider. Prove that some decidable language D is not decided by any decider Mi whose description appears in A. (Hint: You may find it helpful to consider an enumerator for A.)*

My qualm about this is that the question seems to imply finding a decidable language, the decider for which is not in the set of all deciders, which goes against the definition of decidability of languages.

Could you explain whether my doubts are justified? And if not, could you provide a proof (or a sketch of a proof) for the given problem (with or without the enumerator that is mentioned in the hint)?