How can you tell when your theory has overstepped the bounds of reasonableness? How about when you start telling people your “facts” and their faces register with incredulity and disbelief? That is the response of most reasonable people when they hear about the “Banach-Tarski paradox”.
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembly.
The “theorem” is commonly phrased in terms of two solid balls, one twice the radius of the other, in which case it asserts that we can subdivide the smaller ball into a small number (usually 5) of disjoint subsets, perform rigid motions (combinations of translations and rotations) to these sets, and obtain a partition of the larger ball. Or a couple of balls the same size as the original. It is to be emphasized that these are cut and paste congruences! This was first stated by S. Banach and A. Tarski in 1924, building on earlier work of Vitali and Hausdorff.
This “theorem” contradicts common sense. In real life we know that it is not easy to get something from nothing. We cannot take one dollar, subtly rearrange it in some clever fashion, and end up with two dollars. It doesn’t work.
That is why most ordinary people, when they hear about this kind of result, are at first disbelieving, and then, when told that the “proof” involves “free groups of rotations” and the “Axiom of Choice”, and that the resulting sets are in fact impossible to write down explicitly, just shake their heads. Those pure mathematicians: boy they are smart, but what arcane things they get up to!
This theorem is highly dubious. It really ought to be taken with a grain of salt, or at least generate some controversy. This kind of logical legerdemain probably should not go unchallenged for decades.
The logical flaws involved in the usual argument are actually quite numerous. First there are confusions about what “free groups” are and how we specify them. The definition of a finite group and the definition of an “infinite group” are vastly different kettles of fish. An underlying theory of infinite sets is assumed, but as usual a coherent theory of such infinite sets is missing.
Then there is a claim that free groups can be found inside the group of rotations of three dimensional space. This usually involves some discussion involving real numbers and irrational rotations. All the usual difficulties with real numbers that students of my YouTube series MathFoundations will be familiar with immediately bear down.
And then finally there is an appeal to the Axiom of Choice, from the ZFC axiomfest, which claims that one can make an infinite number of independent choices. But this contradicts the Law of (Logical) Honesty that I put forward several days ago. I remind you that this was the idea:
Don’t pretend that you can do something that you can’t.
You cannot make an infinite number of independent choices. Cannot. Impossible. Never could. Never will be able to. No amount of practice will help. Whistling while you do it won’t make it happen. You cannot make an infinite number of independent choices.
So we ought not to pretend that we can; that is what the Law of (Logical) Honesty asserts. We can’t just say: and now let’s suppose that we can make an infinite number of independent choices. That is just an empty phrase if we cannot support it in ways that people can observe and validate.
The actual “sets” involved in the case of transforming a ball of radius 1 to a ball of radius 2 are not sets that one can write down in any meaningful way. They exist only in a kind of no-mans land of speculative thinking, entirely dependent on these set-theoretic assumptions that pin them up. Ask for a concrete example, and explicit specifications, and you only get smiles and shrugs.
And so the Banach-Tarski nonsense has no practical application. There is no corresponding finite version that helps us do anything useful, at least none that I know of. It is something like a modern mathematical fairy tale.
Shouldn’t we be discussing this kind of thing more vigorously, here in pure mathematics?