> If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up? Reverse mathematics (http://en.wikipedia.org/wiki/

> If you do give up the axiom of choice, is there a lot of cool/useful math that you'd also have to give up? Reverse mathematics (http://en.wikipedia.org/wiki/Reverse_mathematics http://en.wikipedia.org/wiki/Reverse_mathematics) is a reasonably recent branch of mathematics aiming to investigate those kinds of questions.