Maybe it helps to circularize the problem a bit? Or at least, if not help, not make it worse? And then mix in some quadratic equations?

http://en.wikipedia.org/wiki/Non-measurable_set#Consistent_d...

If B-T is true, then you've got to select 1 of 4 known cheat codes to allow the definition of volume of "normal real world things" at least the way non-math people like to measure volumes. The least icky is the option that demands sets with non-measurable volumes exist, weird as that sounds. Then again, how weird is it really, given that no one freaks out about an infinite number of irrational numbers existing in between all possible fractions. Pi, after all, is no fraction, but its handy to keep around anyway. If you assume B-T is true and in choose-your-own-adventure fashion select that non-measurable sets exist, then, it turns out that having non-measurable sets make B-T "obvious-ish" or at least less obscure sounding, it all kind of works out in a circular manner.

Why in the name of Occams Razor would you want a pair of weird ideas instead of dust bin both and stick to grade school geometry? Well the axiom of choice wedges in sideways between B-T and the non-measurable sets above, kind of like two balls wedge into the space one ball takes up above (making kind of a joke or tongue in cheek). Its not just two weirdo ideas that work together but a couple of them. And the axiom of choice is just so useful in so many ways (see its wikipedia page) I'd have to think for a second about chucking out the axiom of choice. I think it would be ickier than keeping it around. Life is so much easier if you keep all three hanging around, and all their hangers on.

A really good analogy would be some real world quadratic equations for a land survey (well, made up example) only have one solution in the reals although everyone knows there's two mathematical solutions to any quadratic even if you don't like negative sq roots, and thats OK, and you kinda have to look sideways at the solutions involving negative square roots. Its not that real world geometry problems are full of negative square roots in practice or it means anything in the context of land survey problems, but its kind of a place holder in math.

Kind of a "conservation of weirdness" physics theory where they cancel out over a large enough collection of theorems or a collection of weird ideas is in sum less weird than any individual idea. So if you'd like this and that, and its really interesting and handy and seems to work quite well, sometimes you're going to have to not look overly closely at weird point sets that literally do not have a defined volume, at least not as you'd define volumes, and then screwing around with those volumeless objects can result in super weird stuff like two balls for the price of one. Which is OK in the physical world because we don't have abstract spheres that can have anything happen to them, we have vaguely round piles of atoms with really complicated rules about what you can do to that pile of atoms.

It all vaguely resembles the manufacture of sausage where you'll probably be happier if you don't look to closely at things that shouldn't exist, yet, its a tasty breakfast sausage if you don't think too hard about where any individual part came from. This is a highly heretical view, we're only supposed to think about math as some beautiful, pure, and virtuous thing, which I'm convinced is sociologically some repressed Victorian views about virgin brides or some nonsense so I don't feel too bad about being a heretic.