Alex Simpson: Three Resolutions of the Banach-Tarski Paradox

Alex Simpson
University of Edinburgh 

Abstract: The Banach-Tarski Theorem states that a 3-dimensional ball can be partitioned into a finite number of pieces which can then be reassembled to form two balls each the same size as the original. This result is so startling that it is often called a paradox.

The usual resolution of the paradox is to accept the theorem, and to live with its main consequence: it is impossible to define a finitely additive measure on all subsets of Euclidean space that is invariant under isometries. This shows the necessity for the familiar technology of measure theory, in which the notion of measure is applied just to special subsets, called measurable.

A second resolution of the paradox is to reject the Axiom of Choice, which is used in an essential way in the proof of the Banach-TarskiTheorem. A famous theorem, due to Solovay, states that, in the absenceof the Axiom of Choice, it is consistent to assume that every subset of Euclidean space is Lebesgue measurable. Hence the Banach-Tarski Theorem does not hold.

In the talk, I will outline the proof of the Banach-Tarski Theorem, and discuss the two standard resolutions above. I shall then propose a third possible resolution, which is new. There is a natural notion of "part" of space which is more liberal than the notion of subset. Surprisingly, one can prove that there exists an isometry-invariant measure that assigns a value to every "part" of space. Hence, in particular, every subset has a measure. Moreover, this result is consistent with the Axiom of Choice (though does not require choice in its proof).

Vabljeni!