[michaelochurch]

I have to commend Michael Stevens (VSauce) for the courage to take on Banach-Tarski at all. I was surprised that someone could make it that accessible inside of 20 minutes.

I do agree that a "there is mathematical dispute [++] over whether this operation is valid" disclaimer could have accompanied the part where AoC was used.

[++] Side note: "mathematical dispute" doesn't mean what many non-mathematicians think it does. AoC isn't as "controversial" as some make it out to be. There is no disagreement over whether it's "true" or "false" in any Platonic sense. Rather, there is nearly universal acknowledgement that a valid mathematical system exists with AoC and that a valid mathematical system exists without it... and that both have useful properties. Mathematical dispute over an axiom means that a valid mathematics exists with or without the axiom-- not necessarily that mathematicians hold strong and divergent opinions on whether to include it.

As for the physical interpretations, he's right in that nothing that we know about physics rules out a Banach-Tarski-like behavior at the subatomic level. That said, it's obviously impossible to Banach-Tarski an orange or a ball of gold, since we'd have to literally split every atom. A macroscopic Banach-Tarski event is almost certainly impossible and would be, even with some unforeseen physical capability that made it possible, extremely expensive in terms of energy due to mass-energy equivalence.