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As far as I understand, in the Axiom of Choice book, by Thomas J. Jech, in page 3, there's an explanation on the why by cutting a ball, in $\mathbb R^3$, in finitely many parts, they then can be arranged again to try to form the original ball, and then there can be form 2 balls, in $\mathbb R^3$, and both exactly as the original first ball.

Looking at this answer https://math.stackexchange.com/a/1243483/ by Asaf Karagila

there's mentioned the Banach-Tarski paradox; on which then it's enough to divide the ball into 5 parts? and not finitely many parts?

and is it that the ball in $\mathbb R^3$, in this case, is not to be considered as an actual physical ball in 3D? if yes, how then is this ball considered?

and one more question, could you suggest the proof for these?

Thanks in advance for your help.

Weronika
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The Banach Tarski construction does not cut and rearrange "an actual physical ball in 3D". The parts of the ball that you rearrange are not created by "cuts" in any physical sense. They have weird "shapes" you cannot visualize.

Five parts is finitely many parts.

Ethan Bolker
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