Von Neumann paradox

In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such that the result is two planar figures of the same size as the original.

This was proved in 1929 by John von Neumann, assuming the axiom of choice.

Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original.

But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators.

The following is an informal description of the method found by von Neumann.

Being a free group means that all its elements can be expressed uniquely in the form

We can divide this group into two parts: those that start on the left with σ to some non-zero power (we call this set A) and those that start with τ to some power (that is,

All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum.

If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin.

We then choose another figure totally inside it, such as a smaller square, centred at the origin.

This theorem tells us that if we have an injection from set D to set E (such as from the big figure to the A type points in it), and an injection from E to D (such as the identity mapping from the A type points in the figure to themselves), then there is a one-to-one correspondence between D and E. In other words, having a mapping from the big figure to a subset of the A points in it, we can make a mapping (a bijection) from the big figure to all the A points in it.

Likewise we can make a mapping from the big figure to all the B points in it.

This sketch glosses over some things, such as how to handle fixed points.

The paradox for the square can be strengthened as follows: This has consequences concerning the problem of measure.

As von Neumann notes, To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain transformations, depends on what transformations are allowed.

As explained above, the points of the plane (other than the origin) can be divided into two dense sets which we may call A and B.

Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.

Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4).

[2] More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2,R) contains a punctured neighbourhood of the origin.