Musings

Axiom of Choice

To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.
— Bertrand Russell

Antonio Luis’s latest post points to Eric Schechter’s absolutely wonderful Axiom of Choice Homepage. The latter discusses the AC, and a whole range of related topics.

Here, for instance, is his discussion of the Banach-Tarski Paradox:

Banach and Tarski used the Axiom of Choice to prove that it is possible to take the 3-dimensional closed unit ball, $$B = \{(x,y,z)\in \mathbb{R}^3: x^2 + y^2 + z^2 \leq 1\}$$ and partition it into finitely many pieces, and move those pieces in rigid motions (i.e., rotations and translations, with pieces permitted to move through one another) and reassemble them to form two copies of $B$.

At first glance, the Banach-Tarski Decomposition seems to contradict some of our intuition about physics – e.g., the Law of Conservation of Mass, from classical Newtonian physics. Consequently, the Decomposition is often called the Banach-Tarski Paradox. But actually, it only yields a complication, not a contradiction. If we assume a uniform density, only a set with a defined volume can have a defined mass. The notion of “volume” can be defined for many subsets of $\mathbb{R}^3$, and beginners might expect the notion to apply to all subsets of $\mathbb{R}^3$, but it does not. More precisely, Lebesgue measure is defined on some subsets of $\mathbb{R}^3$, but it cannot be extended to all subsets of $\mathbb{R}^3$ in a fashion that preserves two of its most important properties: the measure of the union of two disjoint sets is the sum of their measures, and measure is unchanged under translation and rotation. Thus, the Banach-Tarski Paradox does not violate the Law of Conservation of Mass; it merely tells us that the notion of “volume” is more complicated than we might have expected.

By the way, the sets in the Banach-Tarski Decomposition cannot be described explicitly; we are merely able to prove their existence, like that of a choice function. One or more of the sets in the decomposition must be Lebesgue unmeasurable; thus a corollary of the Banach-Tarski Theorem is the fact that there exist sets that are not Lebesgue measurable. The existence of unmeasurable sets has a much shorter and easier proof, which can be found in every introductory textbook on measure theory. That proof also uses the Axiom of Choice, but doesn’t mention the Banach-Tarski Decomposition.

Great stuff!