This comic is a reference to the American custom of carving pumpkins to set out on porches and front steps for the American holiday of Halloween, which occurs on October 31st. The pumpkin has the inside emptied out and a face or design carved in the side. Then a light in placed inside (usually a candle). These are called "Jack-O'-Lanterns". The Jack-O'-Lantern in the 3rd frame is the typical and standard design for a carved pumpkin.
Beret Guy, naturally, stays oddly on-topic by carving a pumpkin in his pumpkin. 
In the 2nd frame, Black Hat is putting Nitroglycerin (an explosive) into his carved pumpkin in the hopes that someone will attempt to smash it and it will explode. Black Hat references chest pains because Nitroglycerin is used to open blood vessels to quickly improve blood flow when someone has chest pains.
In the 3rd frame, Megan is our typical emotional xkcd comic character. She is projecting herself onto the jack-o'-lantern as she tries to distract herself with holiday traditions that won't work to distract her.
In the 4th frame, is the first to reference the Banach-Tarski theorem, which states that if you carve up a 3-dimensional ball, in this case a pumpkin, to a finite number of pieces, you can then reassemble the pieces into two different balls - identical to the original. The Banach-Tarski theorem is also called a paradox for obvious reasons. The person off-screen in that frame references the Axiom of choice which is a mathematical axiom that says that given a set of buckets or bins (each that contain one or more object(s)) it is possible to select exactly one object from each bucket. The Banach-Tarski rests on several axioms which are fairly well respected, but also requires the Axiom of Choice to work correctly. So a person who does not believe in the Axiom of Choice would not have been able to do what Cueball managed to do.
The title text says that King Solomon developed the Banach-Tarski theorem first. This is a reference to the story of two women being brought before him. Both were arguing that a particular child was their own. Solomon said that the solution was to cut the child in half and give each woman one of the halves. One of the two women said that the other should have the baby whole. Solomon then knew she was the true mother, and gave her the child. The joke is that Solomon, may not have intended to kill the child, but knowing that two whole children could be made from the one, intended give a baby to each woman, and the Banach-Tarski paradox states that it should be possible.
[Beret guy stands next to a pumpkin with a picture of a pumpkin carved into it]
Interlocutor: So what did you-
Beret Guy: I carved a pumpkin!
[Black hat guy stands next to a pumpkin and a box labeled "Nitro-glycerin. Do not shake."]
Interlocutor: Taking on teen vandals, I see.
Hat Guy: Heavens, No. My pumpkin simply has chest pains. In fact, I'll leave a note warning them not to smash it.
[Long-haired stick figure stands next to a jack-o' lantern]
Hair Figure: My pumpkin's name is Harold. He just realized that all the time he used to spend daydreaming, he now spends worrying. He'll try to distract himself later with holiday traditions, but it won't work.
[Generic stick figure stands next to two pumpkins and a knife]
Stick Figure: I carved and carved, and the next thing I knew I had two pumpkins.
Interlocutor: I told you not to take the axiom of choice.
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I'm bringing over a comment from the blog, because it helps in understanding the Banach-Tarski theorem and Axiom of Choice.
- I realize that you probably aren’t checking these comments anymore, but I’m math grad student currently wasting time I shouldn’t be, and can’t resist answering this question.
- So the short answer is pretty much exactly what Jonathan said–the Axiom of Choice is in someways the “extra” axiom of ZFC set theory (in fact ZFC stands for “Zermelo–Fraenkel set theory *with the axiom of choice*”) and for a long period of time mathematicians were attempting to either
- (1) prove the axiom of choice as a consequence of the other axioms of set theory
- (2) discover a compelling reason why we should not accept the axiom of choice.
- The Banach-Tarski theorem was published at the height of the debate/research, and is still frequently the first thing cited by someone who doesn’t accept the axiom of choice (although most working mathematicians I know do accept the axiom of choice, in part because it just seems silly to handicap yourself unnecessarily).
- What makes the Banach-Tarski theorem seem so paradoxical is simply the fact that they show it is possible to cut a ball into a finite number of pieces (5, to be specific) and reassemble these pieces only using rotations and translations (ie, only by movements you can make with your own hands) to produce two balls, each identical in volume to the first–ie, in someways "1 [ball] = 2 [balls]", which certainly feels a bit shady.
- All of the other theorems and axioms on which the Banach-Tarski theorem relies are standard and relatively accepted areas of mathematics, while the axiom of choice (which is used at a key point of the proof to “choose” elements not fixed in place by particular nice rotations of the sphere) was not, so many people consider this a reason not to accept the axiom of choice.
- If we don’t accept the axiom of choice, it becomes impossible to pick those points, and the whole proof breaks down. In fact, it can also be shown that if we don’t accept the axiom of choice, there don’t exist any “unmeasurable” sets–sets such as those created in the cutting of the ball whose volume we can’t really talk about, as it would need to be add up to 1 (after all those 5 sets together form a ball of volume 1!) and add up to 2 (after all, those 5 sets together form two balls of volume 2!), which makes some people very happy with the idea of rejecting the axiom of choice.
- For myself, I actually *don’t* find the theorem very paradoxical at all–these 5 “pieces” into which the ball is decomposed are incredibly crazy, and nothing you could ever cut with a scalpel, or even a laser, no matter how good you are. Additionally, the theorem really follows relatively easily from a theorem everyone accepts, which is that you can similarly split up the group of rotations to four disjoint pieces and “reassemble” them via rotations into two copies of the group of rotations. It seems a little odd at first, but the thing to keep in mind that any time infinite things get involved, things are going to get a little odd (after all–what’s infinity + infinity? What about infinity/2?). Honestly, at this point, the thing which I find the most “paradoxical” about the Banach-Tarski paradox is that it can’t be done by dividing the ball into only 4 pieces, but this could be a sign I’ve been drinking the math-koolaid for too long. — Ashley
- To clarify something on the point above, the 5 "pieces" are described as sets of points rather than actual objects with areas, and thus cannot be created in physical space. I edited the page to accentuate this, and to remove what I believed to be a contradictory statement. The original statement "This paradox has been proven for just about anything... except objects made of atoms, which our universe is comprised of." implies that a) objects made of atoms are not considered divisible and b) that most things are considered divisible. "Just about anything" could mean the physical universe, in which case the truth is that nothing is divisible and "just about anything" is misguiding, or both things that are within the physical universe and hypothetical things, in which case it deserves further explanation. Thus I edited to explain slightly further, being a safe move to improve the article in the case of either intention.22.214.171.124 05:17, 18 May 2014 (UTC)
- "In fact, it can also be shown that if we don’t accept the axiom of choice, there don’t exist any “unmeasurable” sets"
- That's not right. If we don't accept the axiom of choice, we can neither prove nor disprove the existence of unmeasurable sets. If we could disprove the existence of unmeasurable sets without the axiom of choice, we could disprove it with the axiom of choice by just not bothering to use it. 126.96.36.199 03:59, 25 October 2014 (UTC)
There you have it. lcarsos (talk) 17:42, 4 September 2012 (UTC)
"sited" changed to "cited"--DrMath 08:52, 4 June 2014 (UTC)
- There are so many *citation needed* options here, but I don't know how to put them in :( 188.8.131.52 23:33, 1 May 2017 (UTC)
- http://www.explainxkcd.com/wiki/index.php?title=285:_Wikipedian_Protester&action=edit§ion=3 Two opening-braces, "Citation needed", two closing-braces. 184.108.40.206 22:51, 2 July 2017 (UTC)