Explain xkcd : It's cause you're dumb
explainxkcd.com
explainxkcd.com
First, in case this wasn't clear from the explanation, you have to carve the pumpkin into some very specific pieces in order to be able to re-assemble them into two pumpkins. In particular, you have to carve them into some pieces that are non-measurable sets, which basically means that the carving is so exotic that it is not possible to define a consistent concept of volume for these pieces. This is sort of the heart of the paradox: you get volume-preserving transformations to disobey preservation of volume by using intermediate steps with undefined volume.
Non-measurable sets are not 100% legit in mathematics. Constructing them relies on the Axiom of Choice. I won't get into the debate about the Axiom of Choice, but suffice it to say that if you reject the Axiom of Choice, then the Banach-Tarski Paradox need not be true, and you can self-consistently reject the BTP if you are OK losing the AOC.
The biblical reference from King Solomon is probably referring to the parable of splitting the baby. Two women came to King Solomon, each claiming to be the true mother of the baby. King Solomon suggested cutting it in half, in order to observe the claimants' reactions. The fake mother was ok with this, and the real mother would rather give up her child than see it killed, . XKCD suggests that King Solomon was actually attempting to use the Banach-Tarski paradox to create two babies.
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.
If you have a finite number of sets, you don't need the Axiom of Choice. You can enumerate them and pick out of each.
If you have a rule to choose items, like picking the smallest element in a set of natural numbers, you don't need the Axiom of Choice. There is no choice to be made, you just follow the rule.
The problems come out when you have an infinite number of sets and you pick items arbitrarily. This is what the Axiom of Choice allows you to do. Infinity is dangerous.
(Not that I can call pedantery given the sheer number of who-cares-about-the-details glosses in my original comment ;)
And I don't know that I'd call infinity dangerous so much as I'd call it sanity-stretchingly counterintuitive.
Which makes sense, really. Where would we have developed intuitions for dealing with the infinite?
XKCD is a condescending comic whose audience revels in its own perceived superiority, while any self-respecting nerd with a real sense of humor knows that putting a science or technology reference in a cartoon is not a joke in itself.
These days it rarely achieves all three.
(Shame you're being down-voted, I suppose hell hath no fury like a nerd scorned ...)
Although I do enjoy frame 3 on its own merits.
I have to say, though, that I'm surprised by how many people know Banach-Tarski, but who never heard the story about King Solomon threatening to cut the baby in two in order to discern who cared about its life the most (in other words, to find the real mother).
Come to think of it, maybe recent xkcd panels would amuse me more if I did not understand them.
No?
It works because, if you assume the axiom of choice, you can postulate "pathological" sets that don't have a mathematically well-defined volume. You can use this to backdoor volume changes, by cutting something with well-defined volume into a buncha pieces without a well-defined volume, then reassembling this into something with a different volume than the original.
You can't get around this trick by redefining what it means to measure something: If you pick any reasonable definition, it must admit the Banach-Tarski paradox. https://secure.wikimedia.org/wikipedia/en/wiki/Non-measurabl...
These pathological sets are too weird to exist in physical reality.