are quite different from those of finite collections of things. The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is . Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals. References in fiction BBC Learning Zone repeatedly screened a 1996 one-off educational docudrama Hotel Hilbert set in the hotel as seen through the eyes of a young female guest Fiona Knight, her name a pun on "finite". The programme was designed to educate viewers about the concept of infinity. The novel White Light by mathematician/science fiction writer Rudy Rucker includes a hotel based on Hilbert's paradox, and where the protagonist of the story meets Georg Cantor. Stephen Baxter's science fiction novel Transcendent has a brief discussion on the nature of infinity, with an explanation based on the paradox, modified to use soldiers rather than hotels. Geoffrey A. Landis' Nebula Award-winning short story "Ripples in the Dirac Sea" uses the Hilbert hotel as an explanation of why an infinitely-full Dirac sea can nevertheless still accept particles. In Peter Høeg's novel Miss Smilla's Feeling for Snow, the titular heroine reflects that it is admirable for the hotel's manager and guests to go to all that trouble so that the latecomer can have his own room and some privacy. In Ivar Ekeland's novel for children, The Cat in Numberland, a "Mr. Hilbert" and his wife