This is a fabulous result, both positive and negative, for many reasons. But one of the things people don't realise is that there is a reason why it's interesting mathematically and not just a gimmick.<p>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.<p>These attempts all failed, and we can show that they <i>must</i> fail, because there are systems that satisfy the first four, but do <i>not</i> satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.<p>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"[0] cannot have all four obviously desirable characteristics.<p>For more information, here's a blog post[1] I wrote some time ago:<p><a href="https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?xf26hn" rel="nofollow">https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo...</a><p>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.<p>[0] 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.<p>[1] In case people want to discuss that separately I've submitted it as a separate post here: <a href="https://news.ycombinator.com/item?id=40798224">https://news.ycombinator.com/item?id=40798224</a>