Ars Mathematica

I have identified the most disturbing math theorem ever. What makes it the most disturbing is that it does not involve the Axiom of Choice in any way. I’ve seen the theorem many times before, but I never really noticed how disturbing it was until a couple of days ago.

The theorem is this: for any positive constant c, there is an open set U that contains every rational point, but has measure less than c. Think about what that means, for a minute. The rationals are dense in the reals. Here’s a set that contains an open interval around every rational. Naively I would have believed that the set would have to be the whole real line (except with maybe a finite or countable number of exceptions). At the worst, I would have at least expected the set itself to have infinite measure, and the set’s complement to be measure zero. Instead, not only can we construct such a U with finite measure, we can make that measure be arbitrarily close to zero.

The proof of this is pretty easy, and is a standard result in any real analysis book that covers Lebesgue measure. Not only have I seen it before, I’m pretty sure I’ve seen it pointed out before that the result is surprising. Somehow I never took in how strange it is until just this week.