# Reading elements of a countable set with Turing machine

I have a basic question about the behavior of a potential Turing machine.

Suppose that $$S$$ is a countable set of binary strings, so that we can enumerate $$S$$ as $$(s_i)_{n\in \mathbb{N}}$$.

Suppose that we want to construct a Turing machine $$M$$ that, on any input, simply writes each element of $$S$$ on a tape in order $$s_1, s_2, \ldots$$. (I realize that $$M$$ will never halt.)

But we can construct such an $$M$$ if and only if our particular bijection $$\mathbb{N} \to S$$ is computable, correct? Or is knowing that $$S$$ is countable somehow enough?

By definition, a set $$S$$ is recursively enumerable if there is a Turing machine that enumerates it.