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The standard modern foundation of mathematics is constructed using set theory. With these foundations, the mathematical universe of objects one studies contains not only the “primitive” mathematical objects such as numbers and points, but also sets of these objects, sets of sets of objects, and so forth. (In a pure set theory, the primitive objects would themselves be sets as well; this is useful for studying the foundations of mathematics, but for most mathematical purposes it is more convenient, and less conceptually confusing, to refrain from modeling primitive objects as sets.) One has to carefully impose a suitable collection of axioms on these sets, in order to avoid paradoxes such as Russell’s paradox; but with a standard axiom system such as Zermelo-Fraenkel-Choice (ZFC), all actual paradoxes that we know of are eliminated. Still, one might be somewhat unnerved by the presence in set theory of statements which, while not genuinely paradoxical in a strict sense, are still highly unintuitive; Cantor’s theorem on the uncountability of the reals, and the Banach-Tarski paradox, are perhaps the two most familiar examples of this.
One may suspect that the reason for this unintuitive behaviour is the presence of infinite sets in one’s mathematical universe. After all, if one deals solely with finite sets, then there is no need to distinguish between countable and uncountable infinities, and Banach-Tarski type paradoxes cannot occur.
On the other hand, many statements in infinitary mathematics can be reformulated into equivalent statements in finitary mathematics (involving only finitely many points or numbers, etc.); I have explored this theme in a number of previous blog posts. So, one may ask: what is the finitary analogue of statements such as Cantor’s theorem or the Banach-Tarski paradox?
The finitary analogue of Cantor’s theorem is well-known: it is the assertion that for every natural number , or equivalently that the power set of a finite set of elements cannot be enumerated by itself. Though this is not quite the end of the story; after all, one also has for every natural number , or equivalently that the union of a finite set and an additional element cannot be enumerated by itself, but the former statement extends to the infinite case, while the latter one does not. What causes these two outcomes to be distinct?
On the other hand, it is less obvious what the finitary version of the Banach-Tarski paradox is. Note that this paradox is available only in three and higher dimensions, but not in one or two dimensions; so presumably a finitary analogue of this paradox should also make the same distinction between low and high dimensions.
I therefore set myself the exercise of trying to phrase Cantor’s theorem and the Banach-Tarski paradox in a more “finitary” language. It seems that the easiest way to accomplish this is to avoid the use of set theory, and replace sets by some other concept. Taking inspiration from theoretical computer science, I decided to replace concepts such as functions and sets by the concepts of algorithms and oracles instead, with various constructions in set theory being replaced instead by computer language pseudocode. The point of doing this is that one can now add a new parameter to the universe, namely the amount of computational resources one is willing to allow one’s algorithms to use. At one extreme, one can enforce a “strict finitist” viewpoint where the total computational resources available (time and memory) are bounded by some numerical constant, such as ; roughly speaking, this causes any mathematical construction to break down once its complexity exceeds this number. Or one can take the slightly more permissive “finitist” or “constructivist” viewpoint, where any finite amount of computational resource is permitted; or one can then move up to allowing any construction indexed by a countable ordinal, or the storage of any array of countable size. Finally one can allow constructions indexed by arbitrary ordinals (i.e. transfinite induction) and arrays of arbitrary infinite size, at which point the theory becomes more or less indistinguishable from standard set theory.
I describe this viewpoint, and how statements such as Cantor’s theorem and Banach-Tarski are interpreted with this viewpoint, below the fold. I should caution that this is a conceptual exercise rather than a rigorous one; I have not attempted to formalise these notions to the same extent that set theory is formalised. Thus, for instance, I have no explicit system of axioms that algorithms and oracles are supposed to obey. Of course, these formal issues have been explored in great depth by logicians over the past century or so, but I do not wish to focus on these topics in this post.
A second caveat is that the actual semantic content of this post is going to be extremely low. I am not going to provide any genuinely new proof of Cantor’s theorem, or give a new construction of Banach-Tarski type; instead, I will be reformulating the standard proofs and constructions in a different language. Nevertheless I believe this viewpoint is somewhat clarifying as to the nature of these paradoxes, and as to how they are not as fundamentally tied to the nature of sets or the nature of infinity as one might first expect.
Notational convention: In this post only, I will colour a statement red if it assumes the axiom of choice. (For the rest of the course, the axiom of choice will be implicitly assumed throughout.)
The famous Banach-Tarski paradox asserts that one can take the unit ball in three dimensions, divide it up into finitely many pieces, and then translate and rotate each piece so that their union is now two disjoint unit balls. As a consequence of this paradox, it is not possible to create a finitely additive measure on that is both translation and rotation invariant, which can measure every subset of , and which gives the unit ball a non-zero measure. This paradox helps explain why Lebesgue measure (which is countably additive and both translation and rotation invariant, and gives the unit ball a non-zero measure) cannot measure every set, instead being restricted to measuring sets that are Lebesgue measurable.
On the other hand, it is not possible to replicate the Banach-Tarski paradox in one or two dimensions; the unit interval in or unit disk in cannot be rearranged into two unit intervals or two unit disks using only finitely many pieces, translations, and rotations, and indeed there do exist non-trivial finitely additive measures on these spaces. However, it is possible to obtain a Banach-Tarski type paradox in one or two dimensions using countably many such pieces; this rules out the possibility of extending Lebesgue measure to a countably additive translation invariant measure on all subsets of (or any higher-dimensional space).
In these notes I would like to establish all of the above results, and tie them in with some important concepts and tools in modern group theory, most notably amenability and the ping-pong lemma. This material is not required for the rest of the course, but nevertheless has some independent interest.