The axioms of set theory, part 3

The subject of this post is the axiom of choice, probably the most interesting axiom in set theory. While the axiom of extensionality is essentially just a definition of the concept of “set”, and the other axioms of set theory are just about which sets exist, the axiom of choice is of a different kind: it essentially permits a technique that can be used to make proofs that would be impossible otherwise.

When the axiom of choice is stated in its usual form, it seems fairly obvious. Basically it says that there is a way of picking exactly one member out of each set in a collection of sets. Or in formal logical terms, for every set A there is a function f such that for every member B of A, f(B) is a member of B. In the case where A is a finite set, this can actually be proved; it’s only in the infinite case that it must be stated as an axiom. The reason the axiom is so controversial is that it can be proved to be equal to certain statements which seem much less plausible.

One of these statements is the well-ordering theorem. This says simply that every set can be well-ordered, although that probably isn’t very helpful to you if you don’t know what well-ordering means. Essentially a set is well-ordered if you can arrange its members into a sequence by first picking out a “smallest” member, then picking out the “smallest” member among the ones remaining, and continuing along in the same way, always being able to pick out the “smallest” member among the remaining ones. All finite sets can obviously be well-ordered, but it’s not obvious that all infinite sets can, and in fact for sets like \mathbb R, the set of all real numbers, it’s hard to imagine how a well-ordering could be done. But the axiom of choice can be used to prove that there is a way of well-ordering the set.

An even less plausible consequence is the Banach-Tarski paradox. This says that starting with a sphere (in precise terms, the set of all points in 3D space whose distance from a given point is less than a particular value), one can use rotations and translations to split the sphere into a finite number of shapes, and then use rotations and translations to re-assemble the shapes into two spheres, each of exactly the same size as the original sphere. So even though all that has happened to the sphere is that particular portions of it have been rotated and translated, somehow the total volume has doubled. It’s known as a “paradox” because this seems ridiculous.

However, most mathematicians are perfectly happy with the existence of the Banach-Tarski paradox. The thing is, the paradox depends on the notion of volume, which can easily be applied to objects like spheres, but cannot be applied to all 3D shapes (if we define a shape as a set of points in 3D space). Volume is a hard notion to define. It isn’t simply a correlate of the number of points in the set; all shapes whose volume we can easily define would contain an infinite number of points, and even though mathematicians can distinguish between different kinds of infinities, there is no difference in kind between the infinite number of points in a sphere of radius 1 and a sphere of radius 2. In fact, only a small portion of the subsets of 3D space can be given a definite volume. The problem with the Banach-Tarski paradox is that the shapes into which the first sphere is split are very exotic, and their volumes cannot be defined. Rotation and translation will always preserve volume, but only if it makes sense to speak of the volume of the new shape produced by these transformations.

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