This video is somewhat misleading. I appreciate the attempt at making Banach-Tarski accessible to a general audience, but it dwells on the wrong aspects of what makes Banach-Tarski interesting, making the construction look more like a magic trick with sleight-of-hand. I wish the video had at least mentioned the Axiom of Choice somewhere, as that is fundamentally what Banach-Tarski is about.
The sleight-of-hand comes in around 14:30 into the video, where we are told to create the sequence for an "uncountably infinite number of starting points." That's exactly the point where the construction is non-constructive, and the infamous Axiom of Choice is used. There is no construction - in the sense of constructive mathematics - that can achieve what is described at this point.
Banach-Tarski is not generally regarded as some deep fact about mathematics, a point the video mistakenly belabors. Rather, it is a consequence about particular axiomatizations of set theory which admit the Axiom of Choice. Banach-Tarski is only valid with the Axiom of Choice, and in fact that is the main interest in the paradox.
In my personal opinion, the Banach-Tarski paradox isn't much more enlightening than the simpler construction of the Vitali set (assuming the Axiom of Choice), which is a non-measurable set of real numbers (with Lebesgue measure, i.e., length).
Another part of the video I find misleading has to do with the hyper-dictionary, where he describes the hyper-dictionary by putting some parts of the dictionary "after" other parts which are infinitely long.
The putative applications of Banach-Tarski to physics are ridiculous. Uncountable sets are fundamentally unphysical. The Axiom of Choice serves mainly as a convenience to mathematicians when either a proof avoiding the Axiom of Choice would be more complicated, or so that mathematicians can state properties of objects which are set-theoretically larger than anything that can be relevant to physics anyways.
I think he did: you don't need real numbers to formulate classical physics. Everything measurable has finite precision so you can always get away without postulating that your limits actually converge to something.
Of course, the reality is that then you would wind up with awkward limit-taking machinery in your answers. Real numbers encapsulate that complexity so you might as well use them to simplify both the notation and manipulation of limits. But you don't need to.
Yep, this is what I meant to allude to, and you've worded it much better than I could have.
Perhaps a nice way to say it is that the mathematical objects necessary for physics that I can think of are separable (such as the real numbers). Basically, whenever you have uncountable sets, they come along with some topological structure which must be handled continuously.
There's even an argument that separability is sufficient for most of mathematics let alone physics (http://arxiv.org/pdf/math/0509245.pdf) but you're going to have to work very hard to persuade me that if we're going to describe any sets as physical then uncountable ones are less physical than countable ones.
The set of real numbers is uncountable. There are infinitely many members.
The set of natural numbers is countable. There are infinitely many members.
It would be better to say that infinite sets are fundamentally unphysical, since there are no actual infinities. Talk about uncountable versus countable is tangential.
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.
The sleight-of-hand comes in around 14:30 into the video, where we are told to create the sequence for an "uncountably infinite number of starting points." That's exactly the point where the construction is non-constructive, and the infamous Axiom of Choice is used. There is no construction - in the sense of constructive mathematics - that can achieve what is described at this point.
Banach-Tarski is not generally regarded as some deep fact about mathematics, a point the video mistakenly belabors. Rather, it is a consequence about particular axiomatizations of set theory which admit the Axiom of Choice. Banach-Tarski is only valid with the Axiom of Choice, and in fact that is the main interest in the paradox.
In my personal opinion, the Banach-Tarski paradox isn't much more enlightening than the simpler construction of the Vitali set (assuming the Axiom of Choice), which is a non-measurable set of real numbers (with Lebesgue measure, i.e., length).
Another part of the video I find misleading has to do with the hyper-dictionary, where he describes the hyper-dictionary by putting some parts of the dictionary "after" other parts which are infinitely long.
The putative applications of Banach-Tarski to physics are ridiculous. Uncountable sets are fundamentally unphysical. The Axiom of Choice serves mainly as a convenience to mathematicians when either a proof avoiding the Axiom of Choice would be more complicated, or so that mathematicians can state properties of objects which are set-theoretically larger than anything that can be relevant to physics anyways.