the number of possible distinct books is finite but it's unknown if the library itself is finite. iirc it ended with the narrator speculating that, although the books themselves are devoid of meaning, perhaps the overall structure of the library is ordered, i.e. the same finite pattern of books repeats endlessly in infinite space.

The Library of Babel is finite if the books are unique. Interestingly, Borges does mention that one of the books in the library must be an index of the other books. This is similar to the notion of a universal computably enumerable language.

However, I doubt that Borges' claim is accurate. If the set of programs is finite, then I think there cannot be a comprehensive index of all programs. A finite set is a regular language, and there is no universal regular language in the set of regular languages.