This post is the sum of the thoughts I had watching this great video.
Although it seems very convincing, you cannot divide infinity. When you take an object as infinite set of points(and lines connecting them to the centre), you can always divide into two, or any finite number, and say each one has infinite points again and thus it is same as the original.
In short, Banach–Tarski Paradox is the physical version of the following argument. On a number line, number of real numbers between 0 and 1(uncountably infinite, just like the points on the sphere) is equal to real numbers between 0 and 2. But this doesn't prove that 1 = 2. Similarly just because you ended up with same number of, 'infinite', points you cannot make the conclusion that you have the whole object.
But are we sure that 1 is indeed not equal to 2? Actually you can say, mathematically, it is plausible and we cannot prove it one way or the another. You can argue that 1 is clearly not equal to 2 because by definition they are different, but as stated before, one can argue that since there are equal number of rational/real numbers between 0 and 1 and between 0 to 2, 1 and 2 are actually equal. Well, at least the physical world seems to contradict this as we cannot make two objects from one object.
One explanation for this phenomenon can be that the world we live in is not continuous and if we were to draw xyz axes in the real world, we know the number next to zero to be planck's length. So there is an upper limit to the number of points on any plane of a finite area and thus there are no infinite number of points on the surface of the sphere(or any physical object) and you cannot make the assumption, at 16:34, that the number of elements in the sets after rotating right would be equal to original number of points(they would be one fourth).
So, if there is a universe, which is continuous, it is plausible that we could make two objects from nothing but one. To sum it up, the proof that Banach-Tarski Paradox doesn't hold true is that Banach-Tarski Paradox doesn't hold true. Quite a paradox!
