[gjm11]

Yup! (I think rathereasy chose an unfortunate example.) The point is that AC implies that there is a way to put an ordering on all of R such that every nonempty subset has a smallest element.

It's easy to find such an ordering for any countable subset -- i.e., one that set-theoretically is no bigger than the integers. For instance, we can do it for all the rational numbers by saying that we order numbers p/q (p,q integers, no common factor, q positive) by converting p/q to 2^signbit(p) 3^|p| 5^|q| or something of the kind, where signbit(x)=0 is what in C you write as (x<0), and comparing the positive integers that result.

It's much harder to see how to do it for all the real numbers. In fact, you provably can't see how, in the sense that there actually is known to be no construction that does it -- no way to say explicitly which numbers to put before which.

(This is a common pattern; cases where you need AC to do something are always ones where there is no explicit construction that does it. That kinda has to be true, because there are models of set theory in which AC is false.)