Are there any countable sets that are not computably enumerable?
Are there any examples of countable sets that are not enumerable?


Yes.  All subsets of the natural numbers are countable but not all of them are enumerable.  (Proof: there are uncountably many different subsets of&nbsp;$\mathbb{N}$ but only countably many Turing machines that could act as enumerators.)  So any subset of&nbsp;$\mathbb{N}$ that you already know is not recursively enumerable is an example&nbsp;&ndash; such as the set of all numbers coding Turing machines that halt for every input.