Many years ago, when I was an undergrad, I stumbled upon the Banach-Tarski Paradox (BTP), aka “one can cut a sphere into two disjoint parts which both have the same volume than the initial sphere“, and was quite puzzled by it — all the more since none of the courses I attended back then discussed it.
This is typically the kind of statement that, if not explained carefully, can drive people away from math (and sadly it probably has done so many times already). And of course it provides the perfect occasion to point out differences between physics and mathematics, as well as modifications of the undergraduate curriculum, so I’d like to discuss this a little bit. Comments welcomed (especially on the issue whether or not the countable version of the Axiom of Choice (AC) is enough to prove the BTP).
(more…)