TITLE: Is there such a thing as a countable set with an uncountable subset?
QUESTION [16 upvotes]: Is there such a thing as a countable set with an uncountable subset?

Actually I know the answer. Well, I believe I know the answer, which is NO.
Unfortunately, the professor in a Theory of Computation class said that yes, there is such a subset.
This is to settle a discussion with fellow students. A discussion that is going nowhere so we go to the internets for a verdict.
Thanks in advance for weighing in on this question.

REPLY [2 votes]: If your course was a TCS-Course, then maybe your professor mixed up countable sets with enumerable sets. (This happens sometimes, especially in german language areas, where it is "abzählbar" vs. "aufzählbar).
Every subset of a countable set is countable or finite (or empty). However, not every subset of an enumerable set is enumerable.
For example, $\mathbb{N}$ is trivially enumerable. But the codomain of the busy bever function, which is a subset of $\mathbb{N}$, is not.