True, that is the subtle bit -- but I think most people misunderstand (I'm not saying you are!), and don't realise you can always split an infinity into two -- this is just about splitting a sphere into some 'point clouds' such that you can cleverly stitch them back together into the same original space. In particular, the 'cutting' really makes no real-world sense (at least, as far as I can understand).
I agree that the "cutting" makes no real world sense. In a way, that's one of the points of the exercise.
And I agree that most people don't (initially) understand that an infinite set can be divided into two infinite sets that kinda "look the same", such as dividing Z (or N) into the evens and odds.
But BT is more than that. What follows isn't really for you, but is for anyone following the conversation.
Let's take a set A. It's a subset of the unit sphere, and it's a carefully chosen, special set, not just any random set. It's complicated to define, and requires the Axiom of Choice to do so, but that's what the BT theorem does ... it shows us how to define the set A.
One of the properties of A is that we can rotate it into a new position, r(A), where none of the points of r(A) are in the same position as any points of the original position, A. So the sets r(A) and A have a zero intersection. For the set A there are lots of possible choices of r ... we pick a specific one that has some special properties. Again, the BT theorem is all about showing us how to do this.
Now we take the union: B = A u r(A)
The bizarre thing is this. If we've chosen A and r (and therefore by implication, B) carefully enough, it ends up that there's another rotation, call it s, such that s(B)=A, the set we started with.
So whatever the volume of A, the volume of A u r(A) must be twice that, but that's B, and B can be rotated to give A back to us. So B must have the same volume as A. So 2 times V(A) must equal V(A), so A must have zero volume.
Well, we can kinda cope with that.
But if we've chosen A carefully enough, we find that a small, finite number of them, carefully chosen and rotated appropriately, together make up effectively the entire sphere (we miss out countably many points, but they have zero total volume, and we can fix that up later). So if finitely many copies of A make up a solid sphere, they can't have zero volume.
And that's the "paradox".
The conclusion is that we can't assign a concept of "volume" to the set A, and this is explained a little more in a blog post I've submitted here before:
There's a lot more going on than just the "I can split infinite sets into multiple pieces that kinda look the same as the original", although that is certainly part of it, and lots of people already find that hard to take.
To any who has got this far, I hope that's useful.
