I was in the process of writing this myself, I'll just add one more thing to clarify: As Andreas said, when dealing with a finite number of sets you can explicitly write the function in the form of a formula. However, in the infinite case you cannot write this sort of formula (it would require infinitely many quantifiers to mention which element is chosen) and therefore you're in need of a stronger tool that would guarantee the existence of such function.

@Donu: There can't be a countable $\mathbb{R}$-vector space, as any real vector space would have a surjective map onto $\mathbb{R}$, therefore would be of greater-or-equal cardinality, which is $2^{\aleph_0}$ which in turn is strictly greater than $\aleph_0$ and therefore uncountable.

In addition to Robin's suggestion, you can consider the polynomial ring in $\kappa$ variables (for an infinite cardinal $\kappa$) over a given commutative ring.

@Charles: If I recall correctly, there is a logarithm for ordinal numbers, it might be a fixed-point (i.e. $\log_\omega\epsilon_0=\epsilon_0$ but there's still a logarithm. Ergo, given $\epsilon_0$ you have that the CNF for it would be $\omega^{\epsilon_0}$. To clarify my point, there's no condition given for $\alpha_i$ (assuming the notation from my answer) except that it would be a decreasing sequence of ordinals. Joel just remarked something that I skipped, that there could be a situation in which $n=0,\ \alpha_0=\gamma$ and $\beta0=1$.

You are indeed correct. I'll do better not to dismiss the trivial case the next time.