But I remember learning about the Axiom of Choice in one of my philosophy classes. I just can't for the life of me remember what it is. Anyone want to venture a layman's explanation?
The Axiom of Choice tells you that when you have a set of sets, that there exists another set which contains exactly one element of each of these sets. So say you have {{0,1,2},{4,2,1},{5,3,1}}, there exists a set which has exactly one elementof {0,1,2}, one of {4,2,1} and one of {5,3,1}.
With these example this seems quite obvious, and indeed it is. But if you have more and larger sets, like infintely many infinite large sets, then this is not so clear anymore. Or better, it maybe looks like that should be true, but it doesn't follow from the other axioms one uses so far (Otherwise one wouldn't need another axiom).
The famousness of AC is that it is somehow on the edge of reasonablefullness. That is, while if you assume it is false, you get some unreasonable results (most known maybe that there exists Vectorspaces without a basis), it itself implies some unreasonable results (most known maybe the
Banach Tarksi Paradox , telling you that you can decompose a ball into subsets, than move and rotate these and end up with two balls).
Therefore, the AC is the most controversial of the usually used axioms, which means that everybody outside of maybe logic and set theory just uses it, when they need (Which is not that often at least explicitely, but as said, it comes in disguise sometimes like with the basis of Vectorspaces) But I think in these fields people care more, and also historically it was a bit disputed.
