Recently, a lot of blogs (incited by the Everything Seminar) have been talking about the Axiom of Choice and it’s more horrifying implications. Well, I’m going to jump in on the bandwagon and present deGroot’s problem. This is essentially a continuous version of the Banach-Tarski paradox, and is therefore somewhat more horrifying than the standard version. The solution is due to Trevor Wilson, and my understanding is that it came out of an REU. This post is mostly taken from having to present his paper to one of my professors as part of an independent study on Banach-Tarski.
Standard Banach-Tarski is a discrete paradox. That is, the pieces move instantaneously from the unit ball you begin with to the two unit balls you finish with. De Groot asked if there were a continuous version, that is, a way to actually pull the ball apart with a continuous family of isometries. To this end, we define are continuously -equidecomposable, written if there are finite partitions , of , respectively and a family of -paths so that for all we have and for all and all . We omit if it is the full isometry group. This is an equivalence relation on subsets of . We also define to be the bounded subsets of with and we require that , that is, that the group of isometries contains all of the translations in the first two coordinates.
We define , that is, continuous equidecomposability classes, and define with such that lies to the left of in the first coordinate. This summation is well-defined, associative and commutative, by fairly straightforward arguments. We’d like to be able to pick , so we call a pair of disjoint sets extricable if . For a finite family of pairwise disjoint sets we call it extricable if . This means that, intuitively, two sets are extricable if we can separate them physically by breaking them into finitely many pieces. If is extricable, then is extricable when for each . We want to find a class of sets that are always extricable. So we let consist of bounded sets such that any two disjoint subsets of are extricable. Equivalently, such that any finite pairwise disjoint family of subsets of is extricable. We intend to show that . To do so, we’ll need to know that is closed under taking subsets and under unions of extricable families, and that there is a partition of into sets and such that are codense. We will assume both of these, and the second requires the use of the Axiom of Choice. We can now prove that . First, we let . Using the sets , , we define . Also, define . Let the diameter of , and now we know that is extricable, because we can translate by in the second dimension and then by in the first. It is enough to show that . Actually, for any and there is a singe path in such that it can separate any disjoint pair defined as follows: Let and be sequences in and respectively, and define a sequence in by , , Then, we let linearly interpolate between and during the time interval and with . For we have , and so . And so, for all , and moreover lies strictly to the right of , and so extricates . And so we have established that . This theorem has many consequences. For one thing, if , any finite partition of a bounded subset of is extricable by translations. However, here’s the coup de grace: If and is a path-connected group of isometries of containing all translations in two dimensions, then any two -equidecomposable bounded subsets and of are continuously -equidecomposable. If we suppose that and are discretely equidecomposable using the partitions and and isometries . Choosing a path from to in gives , and so we have . And now, we apply the traditional Banach-Tarski Paradox to obtain the following: If , then any two bounded subsets of with nonempty interior are continuously equidecomposable using proper isometries. This is the strong form of the Banach-Tarski paradox, in which the pieces are pulled apart continuously. I’ve been told that logicians who thought about this were trying to show that you couldn’t do this (or that it was consistent that you couldn’t). Well, ZFC is far richer than expected, it seems.