Contents

# Contents

## Idea

What is known as the Banach-Tarski paradox is the theorem (Banach-Tarski 24) that the axiom of choice implies that any two bounded subsets in Euclidean space of dimension $d \geq 3$ may be partitioned by a finite number of pairwise congruent subsets.

This is perceived as a paradox due to its counter-intuitive interpretation, which becomes particularly vivid if one takes one of the two bounded subsets to be the disjoint union of two copied of the other: In this case the theorem says, intuitively, that it is possible to break up any shape in 3d Euclidean space into a finite number of pieces, such that re-assembling these pieces suitably yields not just the original shape, but that and an entire other copy of it.

It has been pointed out that it is not just the use of the axiom of choice that is responsible for this perceived paradox, but also the point-based concept of topological spaces as such, see the discussion In point-free topology below.

## In point-free topology

It is argued in (Simpson 12) that the Banach-Tarski paradox disappears if one works in point-free topology, hence with locales instead of just topological spaces:

We view spaces of interest as locales, and the notion of “part” is given by the standard notion of sublocale, $[\cdots]$. Every topological space determines a locale $[\cdots]$. However, when a space is viewed as a locale, the notion of sublocale gives rise to new “parts” of spaces that are not merely subsets, and need not be determined by their points.

The usual contradictions are avoided $[$this way$]$. The different pieces in the partitions defined by Vitali and by Banach and Tarski are deeply intertangled with each other. According to our notion of “part”, two such intertangled pieces are not disjoint from each other, so additivity does not apply. An intuitive explanation for the failure of disjointness is that, although two such pieces share no point in common, they nevertheless overlap on the topological “glue” that bonds neighbouring points together.

## References

The original article:

• Stefan Banach, Alfred Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes Fundamenta Mathematicae (in French). 6: 244–277, 1924 (pdf)

Textbook account: