This early "science fiction" story explores the notion of a library containing every possible five hundred page book and an English translation appears in the classic mathematical fiction collection Fantasia Mathematica. It may have been an inspiration for the similar but more fantastical work The Library of Babel written by Borges.
I did not originally include an entry for this story in my database because it is more of an essay than a work of fiction. Moreover, I considered the mathematics to be too elementary. (In one sense, the mathematical content is just the computation of the number of works in the library, simple combinatorics.) However, prompted by an email from author and anthologist Charles Waugh, I reread it, learned more about its history and have reconsidered.
In addition to considering entertaining nonmathematical consequences of the library (such as the fact that the library contains a volume which is its own catalog, but also many volumes which are frustratingly incorrect catalogs and no obvious way to distinguish them) the story rather explicitly explores the boundary between the finite and infinite.
(quoted from The Universal Library [Die Universalbibliothek])
"I thought all along that it was infinite," said Burkel.
"No, that's just the point. The figure is not infinite, it is a finite figure. The mathematics of it are flawless. What is surprising is that we can write down on a very small piece of paper the number of volumes comprising all possible literature, something which at first glance seems to be infinite. But if we then try to visualize it–for example, try to find a specific volume — we realize that we cannot grasp what is otherwise a very clear and logical thought that we evolved ourselves."

So, let me thank Dr. Waugh for reminding me of it and encourage others to read it. Especially since it can now be "borrowed" for free from the Internet Archive (another interesting form of library that would years ago have seemed like science fiction), you have no excuse not to! 