$S$ is a subset of the class of all recursively enumerable languages over some finite symbols then $S$ is recursively enumerable iff
- If $L$ is in $S$ and $L'$ is a language such that $L ⊆ L'$ and $L'$ is recursively enumerable, then $L'$ is in $S$
- If $L$ is an infinite language in $S$, then there exists at least one finite subset of $L$ that is in $S$
- The set of all finite languages in $S$ is enumerable, i.e. a Turing machine can list all the finite languages in $S$
Source of the statement: https://cs.stackexchange.com/q/2322 and some online notes
Isn't the 3rd one contradictory to the 1st one?