Banach–Tarski Paradox - A Sketch of The Proof

A Sketch of The Proof

Here we sketch a proof which is similar but not identical to that given by Banach and Tarski. Essentially, the paradoxical decomposition of the ball is achieved in four steps:

1. Find a paradoxical decomposition of the free group in two generators.
2. Find a group of rotations in 3-d space isomorphic to the free group in two generators.
3. Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
4. Extend this decomposition of the sphere to a decomposition of the solid unit ball.

We now discuss each of these steps in more detail.

The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a−1, b and b−1 such that no a appears directly next to an a−1 and no b appears directly next to a b−1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: abab−1a−1 concatenated with abab−1a yields abab−1a−1abab−1a, which contains the substring a−1a, and so gets reduced to abaab−1a. One can check that the set of those strings with this operation forms a group with identity element the empty string e. We will call this group F₂.

The group can be "paradoxically decomposed" as follows: let S(a) be the set of all strings that start with a and define S(a−1), S(b) and S(b−1) similarly. Clearly,

but also

and

The notation aS(a−1) means take all the strings in S(a−1) and concatenate them on the left with a.

Make sure that you understand this last line, because it is at the core of the proof. For example, there may be a string in the set which, because of the rule that must not appear next to, reduces to the string . In this way, contains all the strings that start with . Similarly, it contains all the strings that start with (for example the string which reduces to ).

We have cut our group F₂ into four pieces (plus the singleton {e}), then "shifted" two of them by multiplying with a or b, then "reassembled" two pieces to make one copy of and the other two to make another copy of . That is exactly what we want to do to the ball.