In Jorge Luis Borges’ The Library of Babel, the narrator describes a never-ending world of hexagonal galleries, all filled with books. Each book is 410 pages, 40 lines per page, 80 characters per line, 25 characters in the script. Doing some quick calculations, that number of books is approximately a 1 with 1.8 million zeros behind it. For reference, the number of atoms in the known universe is a 1 with 80 zeros behind it, which is essentially nothing compared to the scale of the library.
The library is vast, but its volumes are not infinite. The narrator attempts to circumvent this by concluding that the library likely repeats itself. Infinity can be scary, but if the library were looking for new librarians, I have a suggestion for how to make the library truly infinite. If you still want all the books to fit in an infinite library, why limit yourself to 410 page books? For the sake of mathematical clarity, let’s assume that every book is the same thickness, no matter how many pages it has (pretend the pages get super thin). Then, if you allow books to be any finite length, then you will never need to repeat the library because there will always be enough books to fill the shelves without repeating. Assuming the library’s hexagonal galleries do continue forever, then every possible book of finite length will fit perfectly in the library. Finite length books are the way to go.
Infinite length books, however, pose a huge issue. Let’s again assume that books of infinite length still have the same thickness as all the other books. If you allow every book of infinite length to enter the library, then you will run out of space. It’s hard to believe that an infinite library could run out of space, but it’s true. Cantor’s famous diagonal argument proves this isn’t possible, though his proof was about infinite, binary sequences. Applied here, no matter how you arrange the books, you can prove that at least one book of infinite length has been left out. Surely, the library wouldn’t want to leave any books out of its vast volumes, so I advise caution when tasking a librarian with constructing an infinite library.
Regardless of how infinite you decide to make the library, the unfortunate problem remains that the Library of Babel is entirely useless as a library. Libraries function best as curators and disseminators of resources, working to best serve their community. Typically, this means the knowledge in the library should be relevant to the readers. At the very least, it should be true. Unfortunately, there is no curatorial process in the Library of Babel. Sure, one book might just so happen to contain the works of Shakespeare, but there will be countless others that look like works of Shakespeare that all get the endings wrong. Most comprehensible books will be wrong, and most books will be incomprehensible. Given how finite human knowledge is, the chances of finding something useful are essentially zero. Even if you do find something potentially useful, you would have to independently verify the source is credible, because no one is curating it for you. The Library of Babel is an interesting thought experiment, but I pity anyone that wastes more than a second in that building trying to find meaning in the empty volumes.