In a paper published in 1924, Stefan Banach and Alfred Tarski proved a theorem which has sparked a great deal of controversy since. As improved over the years, the theorem roughly states that a solid ball in 3-dimensional space (actually, bodies having any dimension larger than 2) can be split into as few as five non-overlapping pieces, such that these pieces can be moved around and reassembled to regain the original ball and a second copy of the original ball.
These two product balls are indistinguishable from each other and from the original. The reassembly process involves only moving the pieces around and rotating them, without changing their shape by bending, stretching, or any other transformation. In addition, the pieces can be chosen so that they can be moved continuously into their new positions without running into one another.
Naturally enough, this unintuitive result generated considerable skepticism among mathematicians, leading to calling this the Banach-Tarski paradox. It is well known that neither rotation nor translation can change the volume of an object.
How, they asked, could it be possible that a ball having, say, a volume of one cubic inch be so transformed into two balls each having a volume of one cubic inch?
Part of the answer is that the pieces themselves are complicated. They are not continuous solids, such as cubes or pyramids. Rather they are uncountably infinite scatterings of points.
Dealing with infinities is always a tricky business, and we are not going to try to prove the Banach-Tarski theorem here. However, it is important to get some idea from whence the theorem emerges, and to appreciate that, although the result is unintuitive, it is not truly paradoxical.
As a reminder, a set is a collection of elements, which for our purposes will usually be points. The elements of two sets S and T can be placed into 1-to-1 correspondence if each element of S can be linked to a unique element of T, and vice versa. When S and T can be placed into 1-to-1 correspondence, it means that they have the same number of elements.
But what if S has an infinite number of elements? If the elements of S can be placed into 1-to-1 correspondence with the natural numbers, S is called a countable set. A countable set is said to have A0 elements (pronounced aleph-null).
A0 does not behave like regular numbers. For example, let’s take a set N which consists of the natural numbers [1,2,3,…]. Since the elements of N are the natural numbers, N has A0 elements.
Now take a set I consisting of the integers […,-2,-1,0,1,2,…]. The elements of I can be put into 1-to-1 correspondence with the natural numbers as follows:
n = 2 x abs(i) + sgn(i) [Abs(i) is the absolute value of I, while sgn(i) is equal to 1 unless i is negative. Sgn(0)=1.].
This relationship says that i=0 goes to n=1, i=1 goes to n=3, i=2 goes to n=5, etc, and i=-1 goes to n=2, i=-2 goes to n=4, and so on. The result is that all of the elements of I can be put into 1-to-1 correspondence with the elements of N. By the earlier definition, this means that both N and I have measure A0, and the same number of elements, even though N is a proper subset of I!
If the elements of an infinite set S cannot be placed into 1-to-1 correspondence with the natural numbers, S is called an uncountable set. We will be considering only one kind of uncountable set – a set having the same number of elements as appear in the real numbers between 0 and 1. This kind of uncountable set is said to have c elements, where c stands for the continuum.
Uncountable sets show the same sort of unintuitive behavior as do the countably infinite sets. If A is the set of real numbers in the unit interval [0,1], and B is the set of real numbers in the interval [0,2], the elements of A can be put into 1-to-1 correspondence with the elements of B by the mapping b = 2 x a. This means that A and B both contain c real numbers, even though B is twice as long as A when considered as line segments.
In mathematical geometry, the points of a 3-dimensional body are labeled (x,y,z), corresponding to the location of the point in length, width, and height. Such a point has no volume. Note that in taking this viewpoint, any body is assumed to be continuous, that is, there is no smallest ‘atom’ of structure having a finite volume.
Given that the set of points of the unit interval [0,1] is uncountable, can we decide whether or not the set of points in a unit cube has the same number of points? The answer is that we can and it does. Proving this is a bit complex, and doesn’t really aid the current mission. The simplest proof depends on the existence of space-filling curves, which are everywhere 1-dimensional but overlap every point of a unit cube.
Note that this results in another unintuitive behavior of infinite sets. The set of points of half the unit cube is the same size as the set of points of the unit cube itself, despite a difference of a factor of 2 in their volumes. Also, it would naively appear that an uncountable infinity times the zero volume of a point equals a finite volume – but different finite volumes depending on where the points came from.
This nave viewpoint is not spot on, but gives us a metaphor with which to envision the Banach-Tarski result. There is a 1-to-1 correspondence between the points comprising the unit ball and the points comprising 2 unit balls, and thus a unit ball has the same number of points as does a pair of unit balls.
The essence of the Banach-Tarski result is that the unit ball is broken up into a number of pieces, where each piece is comprised of an uncountable number of scattered points. Although the volume of the unit ball is unequivocally pi/6, these pieces do not have well-defined volumes. Such pieces are called nonmeasurable sets.
The existence of nonmeasurable sets, such as those in the Banach-Tarski paradox, has been used as an argument against the axiom of choice, which is an underlying requirement for their existence. Nevertheless, most mathematicians are willing to tolerate the existence of nonmeasurable sets, given that the axiom of choice has many other mathematically useful consequences.
We are almost there. The act of breaking the unit ball into unmeasurable pieces allows us to imagine that these pieces can be reassembled into two unit balls, each having the same volume as the original ball, since the pair has the same number of points as does the original.
The logical sequence is that the original ball is broken into pieces whose individual volume is not defined. These pieces are then moved about and reassembled into a pair of unit balls, having the same number of points but twice the original volume. This is far from a proof, but is perhaps enough of a plausibility argument to go on with.
What in the world does all this have to do with the nature of physical reality? A simple example is that the Banach-Tarski theorem, combined with the principle of conservation of mass, shows that matter cannot be continuous, but must have some sort of atomic structure in the sense used by the early Greek philosophers.
Otherwise it would be possible, at least in principle, to take a gold ball weighing 1 kilogram and convert it into two gold balls with a total weight of 2 kilograms. Although such a procedure was the topic of an April Fool’s article in Scientific American, the impossibility shows us that matter is composed of atoms. This is a significant result in physics founded nearly entirely on pure mathematics.
More generally, most of our pictures and models of physical nature are based on continuum approximations. The Banach-Tarski theorem shows us that reasoning based on such approximations can yield hopelessly nonphysical results.
It seems reasonable to suspect that results which are only mildly misleading might also result. This type of interaction between pure mathematics and physical reality is largely unexplored, and may offer a fruitful area for further research.
