The Banach-Tarski Paradox

[MegaMayorca]

the hell?

[patu8010]

@panadevulpe It is OBVIOUS that it only works in theoretic geometry. It's kinda complicated, but you know there are infinite amount of real numbers between any two numbers, so "in [one] sense there are exactly the same number of numbers between 0 and 2 as there are between 0 and 1" (Irregular Webcomic). And, well... you should read the explanation.

[TheDeathtacle]

@patu8010 I'm not convinced by your analogy; by this analogy we can make a ball of infinite radius from a ball of unit radius, since [0,1] has the same cardinality as R. Ah I just realised why; R is clopen.

It all depends on whether you accept or reject AC, so you could argue that the Banach-Tarski paradox is invalid, just as 'strongly' as you can argue that it is valid.

[srph25]

it's not possible to do such a break-up. the key of the paradox is to detach unmeasurable parts from the ball, but in real life, we can only cut parts that have measure.