You can formalize this problem by talking about limits of sets, where the limsup of a sequence of sets consists of the elements that are in infinitely many of them, and liminf consists of the elements that are in cofinitely many of them, and the limit exists if these are equal. In this case, if we look at the sets of balls in the urn at each time, we see that the limit of this sequence is the empty set. Of course, as you point out, in the case where you keep putting in 10, the limit of the cardinalities is ℵ0, and in the case of taking 1 out, it doesn't exist. So I think what we get out of this really is the conclusion that taking cardinalities is a discontinous operation...
Edit: I just realized, it would probably help to clarify if you're unfamiliar - the reason the above are sensible notions of limsup and liminf is because they're $\bigcap_{n\in\mathbb{N}} \bigcup_{k=n}^\infty A_k$ and $\bigcup_{n\in\mathbb{N}} \bigcap_{k=n}^\infty A_k$, respectively.
