Today I learned about connections between musical rhythm and knot theory, abstract algebra and dance, a Navy ship from the mid 1800s called the U.S.S. Constellation, and some great undergraduate real analysis pedagogy. In today’s post, I’ll share with you some of the most interesting mathematical ideas I heard today.
Because I will be teaching first semester real analysis this semester, I spent a lot of time in the MAA Session on Topics and Techniques for Teaching Real Analysis. I loved Judit Kardos’s talk “Constructing continuous functions” in which she describes how she teaches her analysis students to construct continuous functions with pathological properties. Can you construct a continuous function on $[0,1]$ for which every point in its range has an infinite preimage? Or a monotonic nowhere differentiable continuous function with finite -finite length? Kardos’s technique is to use the fact that a function on $[0,1]$ is continuous if and only if its graph is compact. She then intersects a sequence of compact sets with fractal-like properties to obtain graphs of continuous functions with bizarre attributes.
My favorite talk from that session is Cesar E. Silva’s talk “One Proof Is Not Enough,” in which he reminded us that it can be beneficial for undergraduate real analysis students to see several different proofs of the same result. Silva presented several charming proofs that the real numbers are uncountable (which he collects in his paper on the arXiv), some of which are not well known. We all know Cantor’s diagonal argument, but it was not the first proof Cantor published of the uncountability of the reals. The first proof he published goes as follows. Let be a sequence of real numbers in $[0,1]$. Let be the smallest $k$ such that is in $(0,1)$. Now let be the smallest $k$ with such that is in $(0,1)$. Set and . Now choose to be the next elements in the sequence that are in . Iterate this process, generating nested intervals . The intersection of compact intervals is nonempty, but contains no points in . Thus cannot contain every point in $[0, 1]$, proving that $[0,1]$ is uncountable.
Cantor’s second proof also uses the fact that the intersection of compact sets is nonempty and is just as elementary. Call a set perfect if the set is equal to its set of accumulation points. So in particular, the interval $[0,1]$ is perfect. Now suppose (seeking a contradiction) that $P$ is both a countable set and that $P$ is perfect. Let . Since $P$ is perfect, contains infinitely many points of $P$. Choose to be an open interval about one of the infinitely many points in such that and . Iterate this process, choosing such that . Now consider the intersection of compact sets . This intersection is nonempty, contains accumulation points of $P$, but does not contain any of the . This is a contradiction. Thus perfect sets cannot be countable. In particular, $[0,1]$ is not countable.
Silva gives more proofs in his arXiv paper, including a game theoretic proof due to Matthew Baker, which are worth reading.
It’s already late, and I haven’t worked out my schedule for tomorrow. That’s the thing about the Joint Meetings: you never have enough time to do it all!