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.