The Banach-Tarski Paradox

Study Notes

The Banach-Tarski paradox states that a ball can be cut up into pieces and rearranged to create two identical balls.
This seems impossible, but it can be done through rotations that create points seemingly out of nowhere.
A circle with a missing point can be filled in by picking points one radian clockwise and rotating them counterclockwise.
The key to the Banach-Tarski paradox is figuring out how to create points on a graph that can work on a ball.
Points on the graph represent series of rotations of a ball, and each point is associated with a point on the sphere.
Two different words, or series of rotations, must not represent the same set of points.
The ball is split up into pieces, and each set of points is associated with a word or set of rotations.
Sets of points are created by rotating the sphere in different directions, and undoing the last rotation.
The heart of the Banach-Tarski paradox is that two balls can be created by cutting up the sphere into a few pieces, rotating and moving them around.
Important details have been glossed over in the explanation.
The Banach-Tarski paradox demonstrates that a ball can be cut into pieces and reassembled into two identical spheres.
This process is impossible from a physical point of view and assumes matter is infinitely divisible.
The key problematic axiom in the proof is the Axiom of Choice.
The Axiom of Choice allows for the choice of infinitely many points.
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