Those are not mathematical entities in the same way as are the sets W (whole numbers) or N (natural numbers, indices). It is reasonable to specify your domain closely, otherwise you end up with sqlite.
"Why not a type for datetimes or complex numbers?" would be a better question, as those are different kinds of things but not quite so frequently used in programming.
It seems like you're trying to use SQLite as an example of something you don't want to end up with. This seems backwards so it weakens your argument. SQLite is one of the most widely deployed, reliable, broadly applicable and all round useful pieces of software ever written.
Same for other sets like "not-primes", or the odd numbers.
Sets can be nicely approximated with types. (Some people say even that types correspond directly to sets).
Sets as such are mathematical entities. (Isn't there even a category of sets where sets are objects)?
But apparently not everybody agree as this got down votes; or someone don't see the parallels between sets and types? Strange.