Given that this wonderful result has been posted again[0] I thought I would again provide a link to this explanation not of the result itself, but of why it's an important result in a wider context and not just a gimmick.
Here's the basic idea ...
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[1] cannot have all four obviously desirable characteristics.
That's the basic idea ... if you want more details, click through to the post. It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[1] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
If I'd read this when I was first learning measure theory, I would have had a much easier time. In fact, it took me an embarrassingly long time to realize that sigma algebras were just the "nice sets and subsets" of things that we can extend measures from finite additivity to countable additivity.
I used to think mathematical objects were somehow "inherent". I was always amazed that people had discovered and proved so many interesting things about them. Once I realized they were often just defined to be the thing that has the property we want to prove something about, it got a lot less mysterious.
Note, I'm not saying we just stop there, or that this is somehow bad. The next obvious step taken by mathematicians is to start removing bits of the objects they study, and try to figure out what's still provable until we get to categories, logic, and start arguing about things like the axiom of choice.
> Once I realized they were often just defined to be (...)
Right, that's why I often flippantly say that maths is the study of all ideas which are interesting. We could also think about other things, but they haven't been considered interesting for one reason or another.
Yea, the more I learn about things, the more I realize that everything is defined relative to something else. Even measurements are defined relative to some standard (that beautifully manicured kilogram ball, the speed of light in a vacuum, your ruler's hand width or foot length, etc.) That's both very satisfying and extremely frustrating.
Programming languages can't escape it either; see the tautologies at the top of your favorite programming language (metaclasses, metaobject protocols, etc).
Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make. It's not a "wider context", it's a different thing altogether.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.
> Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make.
Let me try to make the analogy more explicit.
The interesting thing about the fifth postulate is that we show we can't prove it from the other axioms because there are models where the first four hold and the fifth doesn't.
The interesting thing about the Banach-Tarski Theorem is that it shows we can't have all four desirable properties of a metric because there are constructions that show they they can't all hold at once.
> ... you're completely glossing over the theory of measurable sets.
I'm not glossing over it, I'm showing why it is necessary and important.
I think there's an important difference between the situations.
With (non-)Euclidean geometry, you have a bunch of axioms and it turns out you don't have to accept the parallel postulate even if you accept all the others.
With measure theory, you have a bunch of things you'd like to be true and it turns out you can't accept all of them at once.
Those are quite different.
On the other hand, the analogy between geometry and, say, set theory is closer. There are a bunch of axioms for Euclidean geometry, the parallel postulate seems a bit dicey, and it turns out that you can accept the others and lose that one. There are a bunch of axioms for set theory, the axiom of choice seems a bit dicey, and it turns out that you can accept the others and lose that one.
From this perspective, the role of the Banach-Tarski paradox is to help show why the axiom of choice seems a bit dicey, maybe more so than the other commonly-adopted set-theoretic axioms.
(In set theory, too, there are situations where we can write down a bunch of axioms we would like to be true but that actually can't all be true at once, just like with measure theory. Russell's paradox is the best-known example, and it led set theorists to abandon the otherwise very attractive axiom of "unrestricted comprehension".)
The difference is that it's very easy to construct a non-Euclidean geometry in the physical world, at home, on your table, but it's impossible to construct an object anywhere in the real Universe where the Uncountable Axiom of Choice applies. It's purely a mathematical game of pretend.
They are as far apart from each other as possible, as similar as polar opposites.
Banach-Tarski is resolved by deciding that "the real numbers" aren't real, and nothing is lost except for dubious overly-simple proofs.
If one doesn’t accept the Axiom of Choice but uses instead Dependent Choice the paradox no longer holds. Is it the case in this situation that the 4 desirable properties hold?
You don't get new theorems if you remove assumptions. Rather, you get the ability to add different assumptions.
The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability.
There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets.
I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
Assuming the usual consistency caveats, the paradox is no longer a theorem of ZF+DC, but its complement isn't either. So in that case the analogue to the fifth postulate is even stronger, as there are both models in which you get the counterintuitive results of unmeasurable sets and those in which you don't, and the axioms are not strong enough to distinguish the two.
In ZF+DC is it true that measures satisfy the desirable properties mentioned by Colin? I think the sticking point is isometry invariance. Are there measures in ZF+DC of R^3 that are finitely (countably?) additive and isometry invariant?
I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bnach-Tarski follows from the Axiom of Choice which is independent of the rest of set theory.
Here's the basic idea ...
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[1] cannot have all four obviously desirable characteristics.
That's the basic idea ... if you want more details, click through to the post. It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[0] https://news.ycombinator.com/item?id=40797598
[1] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.