The Importance of Choice in Mathematics

The axiom of choice tells you that given a collection of sets that are non-empty, you can select something from each one of them. But it doesn't tell you how you're going to make the selection. This was an axiom that allowed us to construct all kinds of weird paradoxical things. It's really very fruitful for developing different branches of mathematics and I think most mathematicians are not comfortable working without this axiom.