Maybe-Mathematical Musings
One of the easiest ways to start a (friendly) fight in a group of mathematicians is to bring up the axiom of choice. This axiom has a really interesting place in the foundations of mathematics, and I wanted to see if I can explain what it means and why it’s controversial. As a bonus, we’ll get some insight into what an axiom is and how to think about them, and about how we use math to think about the actual world.
Most people find this principle pretty inoffensive, or even obviously right, on first contact. But it’s extremely controversial and produces strong emotions; and unusually for a mathematical debate, there’s essentially no hope of a clear resolution. And I want to try to explain why.
One reason the axiom of choice can sound trivial is that there are a lot of superficially similar rules that are totally fine; the controversial bit is subtle. So here are a few things that don’t cause controversy:
But if we can pick an element from one set, can’t we pick an element from the first set, and then the second set, and then the third, etc.? Eventually we’ll pick an element from each set.
