Are there any countable sets that are not computably enumerable? A set is countable if it has a bijection with the natural numbers, and is computably enumerable (c.e.) if there exists an algorithm that enumerates its members.

Any non-finite computably enumerable set must be countable since we can construct a bijection from the enumeration.

Are there any examples of countable sets that are not computably enumerable? That is, a bijection between this set and the natural numbers exists, but there is no algorithm that can compute this bijection.
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.