Banach-Tarski

The Banach-Tarski Theorem, often called a paradox, is one of the most famous results in math that concerns the Axiom of Choice. This video does a really good job of explaining it. He does though skip over where the Axiom Of Choice comes into it (because it’s very technical). It is used when you perform some of the steps on the uncountable remaining points. Roughly speaking, you talk about countable things by creating finite sequences and taking their limit. But you can only do that countably often. So you cannot get at all the surface that way, it’s uncountable. You need to be able to operate uncountably. The key step is where you can “find” the uncountable number of new starting points that have not been colored.  Anyway, an excellent explanation that you don’t need to know set theory to follow.