Is the set of all Turing machines whose language includes the set of all even length strings recursively enumerable? My intuition tells me the answer should be no, but I can't prove it.
I know that it should be non-recursive because the property is nontrivial (i.e there are some TM-recognizable languages which have this property and some which do not), by Rice's theorem.
Also by Rice's theorem, if P is a non-monotone property then Lp (the set of all TMs whose language satisfies P) is not r.e.
However, the property "includes the set of all even length strings" appears to be monotone, since if a set includes the set of all even length strings, then all supersets of that set must also satisfy that property.
On the other hand, the opposite property, let's call it Q, "does not include the set of all even length strings" does not seem to be monotone, hence Lq should be non-recursively-enumerable.
This doesn't prove anything though, since knowing that the complement of a set is non-r.e doesn't tell you anything about the cardinality of the set itself. That's where I'm stuck.