The Axiom of Choice really grinds my gears.

Given a set of nonempty sets, the set of Cartesian products of all of those sets is obviously also nonempty. That is, we can always choose an element from a nonempty set, even when we can’t explicitly define a “choosing function” for an arbitrary set.

To make this more clear, we use Bertrand Russell’s analogy. Given a bunch of pairs of shoes it is easy to define a function that can take in those pairs and output a particular shoe from each pair. Just let the function choose the left shoe of each pair. But what if we had a bunch of pairs of socks? Socks are indistinguishable so it is no longer clear how we can define a function that can take in a pair of socks and output one from that pair. But this doesn’t mean that we can’t take a pair of socks and output one from that pair. The Axiom of Choice asserts that despite the fact that we can’t explicitly define a function that takes in a pair of socks and spits out a single sock, such a function must still exist because it is obviously not impossible to take an element from a nonempty set.

Seriously, who sat down one day and thought about this?

Anyway, another topic in the back of my head is figuring out the eigenvector of a two-dimensional rotation. Such a vector must be nonreal since no real vector will yield a multiple of itself when subjected to a rotation that is not a multiple of \(\pi\). This sort of computation could be a blog post on its own.

I also wonder about the oscillation of a dipole in an electric field. That could make an interesting paper for the physics section. My old research question of what force fields satisfy the shell theorem also still stands, though I am a little stuck on that mathematically. See here for the result.

As far as the math section goes, I don’t really know. I plan on uploading all my Putnam stuff once the seminar ends, but beyond that, I have no clue what I should work on next, apart from my running solutions of 100 Geometry Problems. I also have a new number theory book, so once I start working on there, maybe I can make more elaborate number theory things.

I’m moving at a snail’s pace right now :/