BT exists just before we make that last transition and abuses the fact that the constructions we've made so far are not required to have any reasonable sense of "volume" whatsoever. All of the "infinity" bits are sort of a red herring as there's no reasonable way to think about taking physical things and cutting them into aphysical things. Instead, this all arises from taking incredibly aphysical things (raw, theoretical sets) and building them up until they feel physical.
If you lay out the constraint that "cuts" must divide measurable things into measurable things, which is a reasonable expectation we have of physical items, then BT will vanish.
> immerse it in water, in a bathtub ready
> to overflow, and start manipulating the
> pieces.
> At what point does the water level rise?
The only "realistic" answer I can give is that each of the four pieces would "take up" the full volume of the sphere as you moved them apart. Each of the four pieces is "shaped" like the whole sphere, but with infinitesimally small holes where the other four go. So not a single "water droplet" would ever go inside the four pieces, and while you were moving them around they'd take up the space of four of the original sphere.
I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.
I've sent you an email, as requested.
From the sound of it, though, the answer to my puzzlement, is that it doesn't apply to this universe. In which case, I don't think it's a problem at all. I'm quite happy to imagine mathematicians doing work on geometries that don't map to the real world, for example.
Mostly, then, is this just a question of presentation? I mean, if you said "this does not apply to the real world, but to some fictitious mathematical assumption that assumes things can be divided up infinitely," I don't think anyone would have an issue with it. It sounds like it's a problem to most people because it's presented as if it's a real-world result.
Or, at least, that seems to be the impression I get, given the other comments.
What I mean is, is there's a question of misdirection here? I.e. the theorem is presented as this non-intuitive thing that can't possibly be the case in the real world, then when someone asks what would happen if you tried it in the real world, the answer is "it's assuming some things that aren't true for the real world." Because if that's the case, I'm not sure I see the paradox. Assuming a weird set of ideas, I would expect you can come up with weird answers.
Here's a question, though, because something is nagging at me. And I'm going to assume this is a universe of infinite points and no atoms (as I understand it, at least).
Let's say we have a sphere of volume 4/3 pi r^3 = 100
Now you do your cuts, but don't reassemble yet. The sphere is still the original sphere, with all the shapes it has been cut into still in virtually their original spaces.
The total volume still has to be 100, right? I mean they all still fit into the original space.
So now, you immerse it in water, in a bathtub ready to overflow, and start manipulating the pieces.
At what point does the water level rise?