For the Banach-Tarski paradox to work, you need to use the axiom of choice.

Similarly, the Banach-Tarski paradox fails because a volume cannot be taken apart into points.

This result is now called the Banach-Tarski paradox.

They proved the following more general statement, the strong form of the Banach-Tarski paradox:

The Banach-Tarski paradox can also be used to turn a small sphere into a huge sphere.

There is, however, an important difference between the two: the Banach-Tarski paradox relies on non-measurable sets.

This makes it plausible that the proof of Banach-Tarski paradox can be imitated in the plane.

The existence of Nikodym sets is sometimes compared with the Banach-Tarski paradox.

Using this terminology, the Banach-Tarski paradox can be reformulated as follows:

The proof of the much more famous Banach-Tarski paradox uses Hausdorff's ideas.

The well-ordering theorem has consequences that may seem paradoxical, such as the Banach-Tarski paradox.

This Cayley graph is a key ingredient in the proof of the Banach-Tarski paradox.

A free group on a two-element set S occurs in the proof of the Banach-Tarski paradox and is described there.

The axiom of choice must be wrong because it implies the Banach-Tarski paradox, meaning that geometry contradicts common sense.

Some examples of this are the Banach-Tarski paradox and the existence of Vitali sets.

The Banach-Tarski paradox.

Also read in the Mathematical Intelligencer an article on the Banach-Tarski Paradox.

Thus, he argues that certain paradoxical phenomena in elementary particle physics parallel the Banach-Tarski paradox in set theory.

The Banach-Tarski paradox shows that there is no way to define volume in three dimensions unless one of the following four concessions is made:

This theorem, known as the Banach-Tarski paradox, is one of many counterintuitive results of the axiom of choice.

Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach-Tarski paradox.

Moreover, the Hahn-Banach theorem implies the Banach-Tarski paradox.

Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach-Tarski paradox.

The Banach-Tarski Paradox by Vsauce gives an overview on the fundamental basics of the paradox.

Since only free subgroups are needed in the Banach-Tarski paradox, this led to the long-standing Von Neumann conjecture.