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.


1 Answer 1


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

  • $\begingroup$ Ah, right, so it has measure zero. $\endgroup$
    – Xodarap
    Aug 21, 2017 at 21:30
  • $\begingroup$ Not only that, there are computably countably infinitely many r.e. sets. $\endgroup$ Aug 22, 2017 at 16:29
  • $\begingroup$ @AndrejBauer thanks for feedback, could you give me a reference to the definition of computably countably (infinite) set? $\endgroup$
    – fade2black
    Aug 22, 2017 at 16:46

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