I didn’t read too deeply about the axiom of choice, but what I took from it was the notion that there is a theoretically infinite amount of pumpkin, and he carved so much that he ended up with enough material to make a second pumpkin. I think. I may be way off.

Basically, if you go off a certain set of assumptions involving infinity, you can conclude that you can take a sphere, slice it up into a near-infinite number of near-infinitely small pieces, and arrange them into two spheres. I think.

And while we’re at it, could some-one remind me what setting I need to change in Firefox to make the mouse-over feature work on that comic strip? Thanks.

I don’t think it has any associated? If you view it normally here, then it should show up for you. Otherwise, I don’t see any alt text on the properties.

I did try to investigate what the axiom choice and the Banach-Tarski theorem were before I posted. But I’m not seeing how you can divide up an object into a bunch of pieces and then reassemble those pieces into two objects that are identical to the original. Can somebody explain how this works in terms a layman can understand?

Well, I don’t think I can, anyway, though maybe I’ll try if nobody else does. But I can at least point out where your intuition is probably failing. You’re probably thinking, OK, I have a ball of volume 1. I cut this up into a number of pieces–surely these still have total volume 1. Now I reassemble them–surely I still have volume 1… but, wait! Now you say the total volume is 2? That can’t be right!

The problem is that the “pieces” you have to use are not the sort of pieces you’re used to thinking about. In particular, these pieces are so confusing that you can’t meaningfully assign them “volumes” of the sort you’re used to thinking about. (It’s not that they have volumes of zero, it’s that the notion of volume doesn’t work for these pieces. In mathematical terms, they are non-measurable sets. In particular, read about the implications of Banach-Tarski for measurability there. You might also want to follow the link to Vitali sets to find a simpler one-dimensional example of the sorts of sets you need to be thinking about in order to understand the decomposition.) Objects in the real world, made up of discrete atoms, can’t have this property–you could use “number of atoms” as a volume measure, preserved under all physically-realizable cuts–so it’s natural for physical intuition to fail here.

The real point of the Banach-Tarski theorem is that you can’t consistently define the volume of a subset of a three-dimensional space in such a general way that it would apply to any arbitrary subset. You’re fine if you just define it on pieces with some ordinary shape. You just can’t define what the volume of a piece that is merely a bunch of disconnected points should be. This doesn’t really apply then to cutting up a pumpkin.

First, you have to understand that this is not something that works in the real world. It’s a theoretical math thing. It would never work with real pumpkins (or real anything else). I’m pretty sure everyone here already gets that, but I said it just in case.

The reason it won’t work in the real world is that it involves cutting a finite number of infinitely complex pieces. Each of these pieces have an infinitely long border. Think of fractals if that helps.

Maybe some real mathematicians (as opposed to my CompSci BS) can help out here if I don’t have the next part exactly right. The infinitely long border implies, at least mathematically, an infinite (or at least indeterminate) volume. Since the volume is undefined and undefinable, you can pretty much reassemble the pieces into whatever volume you want, depending on how you shape them and how you rearrange them.

Mathematically you can use infinity to achieve all kinds of counter-intuitive results. Hilbert’s Grand Hotel is another example which is probably easier to understand and may even help you with Banach-Tarski.

It strikes me that the Banach–Tarski paradox makes for a good reductio ad absurdum proof that matter cannot be subdivided infinitely. At least not without losing it’s identity as matter.