After watching the Vsauce video many times I still can not seem to fully grasp it.
In: Mathematics
So the paradox has you split sphere into 5 parts, rotate those parts and reassemble the original sphere plus a copy.
The paradox lies in the fact that all the involved actions, splitting into finite number of pieces, moving them about and rotating them, are supposed to conserve volume. But start, you have 1 sphere worth of volume. At the end, two spheres worth of volume.
And if you then try to see where the magic happened, you notice that between first and last step, tthe intermediate pieces had either non-defined volume, or 0 volume. This “non-defined volume” is pretty wonky, since mathematicians actually have quite brutal ways to assign volume(or to speak of lengths, areas and other such measures, we often use “measure” as catchall) to anything pretty much.
So how does the non-measurable thing come to be? It turns out, existence of those things is a bit controversial, since the measure theory we have is so comprehensive, we cannot ever construct a thing, even in mathematics, that fails to have a measure. What allows this is axiom of choice, a non-constructive statement, about existence of certain sets. To put it simply, if you have possibly infinite pairs of shoes, you can choose one from each pair, just select the left shoe. You don’t need axiom of choice. But if you have infinite pairs of socks, you cannot construct the set of one-sock-from each pair explicitly, mathematician can for finite number of socks make a choice which sock to use, but not for infinite onts. Axiom of choice claims that however it is possible to do this, the same as with shoes, even though the choice function, selecting one sock from the two for each pair. It seems super logical and common-sensical hopefully, but the important bit is, the axiom just claims “you could do it, just pick whatever”, but it doesn’t do it. We don’t have this choice function defined.
So with the same logic, using axiom of choice, we can postulate certain type of partitioning of the sphere should exist, but it requires infinite number of kinda-arbitrary choices. We cannot actually have a real example of this partitioning, but if it exists, as axiom of choice claims it does, it would fail to have a measure. Concepts of volume or measure simply fail to apply to it. And as a kind of “proof” of this “your laws don’t apply to me”, Banach and Tarski showed that doubling volume is possible by just moving and rotating these non-measurable partitions.
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