There's a bit more to the subtext than that. The Axiom of Choice sounds quite innocuous: If you have a bunch of sets, you can construct a new set by picking on

There's a bit more to the subtext than that. The Axiom of Choice sounds quite innocuous: If you have a bunch of sets, you can construct a new set by picking one element out of each of them. Turns out you can reason your way from that to "I can make any solid 3d object into any other solid 3d object just by cutting and rearranging the parts". (Banach-Tarski is often explained as the ability to turn one sphere into two, which is true, but the full implication is more along the lines of being able to rearrange a pea into the earth.) This is just one of many scary dragons that lurk in set theory. Historically speaking, Banach-Tarski is why the Axiom of Choice is controversial. The other (I forget, nine?) axioms in ZFC, garner a tepid response--along the lines of, "Hooray, someone axiomatized set theory." But AC remained controversial for a long, long time. I cannot find the quote, but there was a mathematician who refused to accept the Axiom of Choice, and when asked about it, commented, "It is because I do not believe one and one make three." He was referencing Banach-Tarski. That is the dilemma that has faced mathematicians historically: either reject something so innocuous as the Axiom of Choice or embrace something so nonsensical as the Banach-Tarski paradox. "Controversial" is putting it lightly; that's a hard decision. One still sees textbooks that flag all theorems depending on the axiom of choice, so you can reject them if you want to. I would say the field as a whole has come to embrace AC and accept Banach-Tarski; one of my professors one commented, "If we need the axiom of choice in this proof, we'll use it; why do mathematics with one arm tied behind your back?" I think this reflects the common attitude. But it wasn't an easy choice. And that's the joke. If the guy didn't want one pumpkin to equal two, he should have gone the other way and rejected AC. It is not unfair to characterize Banach-Tarski as the sorry consequence of something mathematicians really wanted to do -- collateral damage, the lesser evil. One could regret it.