> I think the elements of the base need to be enumerable (proof needed but it feels natural)

Proof of what? Needed for what?

The elements of the number system are the base raised to non-negative integer powers, which of course is an enumerable set.

> transcendental numbers are not enumerable

Category mistake ... sets can be enumerable or not; numbers are not the sort of thing that can be enumerable or not. (The set of transcendental numbers is of course not enumerable [per Georg Cantor], but that doesn't seem to be what you're talking about.)