This has been submitted a lot[0], and discussed a little. What's rarely mentioned is why this is a relevant and interesting result. A lot of people dismiss it

This has been submitted a lot[0], and discussed a little. What's rarely mentioned is why this is a relevant and interesting result. A lot of people dismiss it as unrealistic and pointless, but it tells us a lot about the structures that underpin calculus, and highlights the dangers of "plausible thinking". In particular, when you want to talk about area or volume, you want the following things to be true: (0) The "unit object" has measure 1; (1) Invariant under translation, reflection, and rotation; (2) Given lots of things[1], the measure of the union is the sum of the two measures; (3) Everything can be measured. The Vitalli set[2] shows that these things are mutually incompatible People investigated relaxing "Countable Additivity(2)" to "Finite Additivity" and found that definition of length and area can be found to satisfy these "plausible properties". The Banach-Tarski theorem shows that that's not enough in three dimensions and higher. That's why we keep countable additivity and accept the idea that some things are simply weird enough to defy measurement. If you think all of this is stupid and irrelevant, you're not alone, but it's all tied in to how calculus works, which turns out to be pretty useful. I wrote about this a while ago[3][4]. ================================ [0] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=wikipedia%20Banach%20Tarski%20Paradox&sort=byDate&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que... [1] Countably many with bounded union [2] https://en.wikipedia.org/wiki/Vitali_set https://en.wikipedia.org/wiki/Vitali_set [3] https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?uf09hn https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheo... [4] https://news.ycombinator.com/item?id=27446844 https://news.ycombinator.com/item?id=27446844