Banach-Tarski Paradox

One of the implications of this paradox is that in an infinitely divisible world, it's possible to cut up a 3 dimensional ball into 5 pieces. Then rearrange them using rigid motion, i.e no strechting/scaling etc. so that you  will get two balls each of whom are equal in volume of the original ball. Pity the real world isn't infinitely divisible.

Theoretically it is infinitely divisible (mechanics works with real numbers and vector spaces/matrix groups over them, which are dense AND uncountable).

The problem is that there's no infinitely thin knife.