How many recursively enumerable sets are there?

I'm trying to get some notion of how "many" recursively enumerable sets there are.

Here's one way of formalizing the question: consider $S =\{q: q\in Q\cap [0,1]\}$ and $R\subset S$ the set of recursively innumerable subsets of $S$.

What is the measure of $R$? My intuition is that it is 0 (i.e. a given set is almost surely not RE), but I am having difficulty proving it.

Countably infinite. Each RE set corresponds to a TM $M$ recognizing that RE set, where the set of all TMs is countably infinite.