Problem: Our class takes a field-trip to Archimedes’ Library. Before entering the library, your tour guide makes you notice the sign on the main doors which reads: “Observe the Rule of Archimedes’ Library: Every book has another book on top of it.”
(a) For x and y books in Archimedes’ Library, define q(x, y): Book x is on top of book y. Using logical symbolism, rewrite the “Rule of Archimedes’ Library” in terms of q(x, y).
(b) All of the books in Archimedes’ Library are lying flat. Without opening the doors and counting, how many books are in the library? Justify your answer as logically as possible.
So for 12a) I have figured out that this is ∀y∃x q(x,y). Not sure how to solve 12b. There cannot logically be an infinite number of books in the library as a library has finite space. It is possible that books are placed in a circular format so that one is on top of one another, but that means the books are not lying flat. Is it possible that there are no books in the library?
(a): Correct.
(b): Sketch of proof: If there are finitely many books, but at least one book, then each book has a finite number of books on top of it. Let $b$ be the book with the least number of books on top of it (this exists by the well-ordering principle, which is equivalent to induction). Then $b$ cannot have zero books on top of it, otherwise it would contradict the library rule. But the book on top of $b$ has less books on top of it than $b$, contradicting the choice of $b$. Therefore the assumption that there is at least one book is false, otherwise there must be infinitely many books.