Uses for the Axiom of Choice

The Axiom of Choice is one of those things that causes a lot of trouble if you think about it for too long. It says that if you have a collection of non-empty sets then you can choose a member of each set in the collection. This sounds perfectly plausible, but it turns out that you can either assume that the Axiom of Choice holds or that it doesn't hold and without causing any logical inconsistencies. I've never really cared much about the Axiom of Choice myself. The only thing that it's done for me is create a really bizarre memory from graduate school.

When I was in graduate school, there was a fellow graduate student named Mike who was fascinated with the Axiom of Choice. It was definitely his favorite topic of discussion, although I never quite understood why. Mike told me once how he once was on a date that he could tell just wasn't going well. Apparently he and the woman that he had asked out ended up having absolutely nothing in common. After a while, he decided that he was going to stop trying to find common interests and instead decided to talk about things that interested him. In this case, it was the Axiom of Choice. According to Mike, this caused a sudden and dramatic change in his date, who couldn't keep her hands off him for the rest of the night.

I don't know if this story is true or not, but it certainly is one of the more unusual applications for the Axiom of Choice that I've heard of. Another interesting use may be in economics.

According to a paper by Christopher Ayres, all of the foundations of economic theory rely of the Axiom of Choice. Because the Axiom of Choice seems to be something that you can either accept or not accept without really affecting anything, I'm not sure how much Ayres' results will withstand a careful look, but I'm sure that I can think of at least one person who'd be interested in reading his paper. Of at least he would have been several years ago.