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The Point of the Banach Tarski Theorem (2015)
One of my favourite Limited Audience Jokes (although technically it's actually a riddle) goes: This is not intended to explain why the Banach-Tarski theorem is true, nor to give a proof, nor to talk about what the pieces look like, etc. No ...
So a "measure" is a function that takes a set and returns a number that somehow represents the length, area, volume, whatever. The Banach-Tarski theorem says that if $B$ is the unit ball in $\mathbb{R}^3$, there exist pair-wise disjoint sets $\{A_i\}_{i=1}^n$ and isometries $\{\tau_i\}_{i=1}^n$ and $\tau$ such that: You should follow me on twitter I've decided no longer to include comments directly via the Disqus (or any other) system.
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