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.

• Ah, right, so it has measure zero. – Xodarap Aug 21 '17 at 21:30
• Not only that, there are computably countably infinitely many r.e. sets. – Andrej Bauer Aug 22 '17 at 16:29
• @AndrejBauer thanks for feedback, could you give me a reference to the definition of computably countably (infinite) set? – fade2black Aug 22 '17 at 16:46