Conway Can Divide by Three, But I Can’t Patrick Lutz June 29, 2021 Abstract to describe a construction which requires innitely many choices then we must describe explicitly how these choices Conway and Doyle have claimed to be able to divide by are to be made, rather than just assuming they can be made three. We attempt to replicate their achievement and fail. any-which-way when the time comes. In the process, we get tangled up in some shoes and socks There is a well-known example, due to Bertrand Russell, and forget how to multiply. which illustrates this. Suppose there is a millionaire who loves to buy shoes and socks. Every day, he buys a pair of 1 Introduction shoes and a pair of socks and after innitely many days have passed, he has amassed innitely many pairs of each. In the paper “Division by Three,” Conway and Doyle show He then asks his butler to pick out one shoe from each pair that it is possible to divide by 3 in cardinal arithmetic, even for him to display in his foyer. The butler wants to make without the axiom of choice [DC94]. Actually, they show sure he is following the millionaire’s instructions precisely that it is possible to divide by = for all natural numbers so he asks how to decide which shoe to pick from each pair. = ¡ 0; they called their paper “Division by Three” rather The millionaire replies that he can pick the left shoe each than “Division by =” because the case = = 3 seems to capture time. The next day, the millionaire decides he would also all the diculty of the full result. More precisely, they give like to display one sock from each pair and so he asks the a proof of the following theorem. butler to do so. When the butler again asks how he should decide which sock to pick from each pair, the millionaire is Theorem 1. It is provable in ZF that for any natural number stymied—there is no obvious way to distinguish one sock = = = ¡ 0 and any sets and , if j × j = j × j then j j = j j. in a pair from the other1. Here we are using the notation × = to denote the set The point of this example is that if we have a sequence × f1, 2, . , =g and the notation jj = jj to mean that f 8 g8 2N of sets of size 2 then there is no way to prove with- there is a bijection between and . out choice that Π8 2N 8 is nonempty. Doing so would require The purpose of this article is to question whether the explicitly constructing an element of Π8 2N 8 , which is anal- statement of Theorem 1 is really the correct denition of ogous to giving a way to choose one sock from each pair in “dividing by = without choice.” We will propose an alternate the millionaire’s collection. On the other hand, if we have a statement, show it is not provable without the axiom of xed ordering on each set 8 in the sequence then we can choice and explain what all this has to do with Bertrand show without choice that Π8 2N 8 is nonempty, just as it Russell’s socks. was possible to choose one shoe from each pair. Of course, none of this should be taken too seriously. I’m Russell’s story about the shoes and socks may seem like not really here to argue about what “division by = without just a charming and straightforward illustration of the ax- choice” means. Instead, the goal is to have fun with some in- iom of choice, but we will return to it a few times throughout teresting mathematics and the question of what “division by this article and see that there is more to it than is initially = without choice” should really mean is merely an inviting apparent. jumping-o point. 1When Russell introduced this example, he was careful to point out Mathematics Without Choice that in real life there actually are ways to distinguish between socks—for instance, one of them probably weighs slightly more than the other—but What does it mean to do math without the axiom of choice? he asked for “a little goodwill” on the part of the reader in interpreting In brief, it means that if we are proving something and want the example. 1 2 Failing to Divide by Three pairs of socks, we will refer to the standard version as “shoe division” and the alternate version as “sock division.” You Can Divide by Three Denition 2. Suppose = ¡ 0 is a natural number. Shoe As we mentioned earlier, in the paper “Division by Three,” division by n is the principle that for any sets and , if Conway and Doyle prove without the axiom of choice that j × =j = j × =j then jj = jj. for any natural number = ¡ 0 and any sets and , if j ×=j = j ×=j then jj = jj. What this requires is giving Denition 3. Suppose = ¡ 0 is a natural number. Sock an explicit procedure to go from a bijection between × = division by n is the principle that for any sets and and - . = and × = to a bijection between and . any collections f 0 g02 and f 1 g1 2 of disjoint sets of size , Ð - Ð . This result has a long history. It was (probably) rst if j 02 0 j = j 1 2 1 j then j j = j j. proved by Lindenbaum and Tarski in 1926 [LT26], but the Since we know that shoe division by = is provable without proof was not published and seems to have been forgot- the axiom of choice, it is natural to wonder if the same is ten. The rst published proof was by Tarski in 1949 and is true of sock division. regarded as somewhat complicated [Tar49]. Conway and By the way, this is not the rst time that someone has Doyle gave a simpler (and more entertainingly exposited) asked about the necessity of having collections of ordered proof, which they claimed may be the original proof by rather than unordered sets when dividing by = in cardinal Lindenbaum and Tarski. Later, the proof was simplied arithmetic. In the paper “Equivariant Division,” Bajpai and even more by Doyle and Qiu in the paper “Division by Four” Doyle consider when it is possible to go from a bijection [DQ15]. There is also a charming exposition of Doyle and × = ! × = to a bijection ! when the bijections Qiu’s proof in the article “Pangalactic Division” by Schwartz are required to respect certain group actions on , , and [Sch15]. = [BD17]. Since the axiom of choice can be considered a way to “break symmetries,” the question of whether sock Can You Divide by Three? division is provable without choice is conceptually very similar to the questions addressed by Bajpai and Doyle. Does the statement of Theorem 1 really capture what it means to divide by = without choice? To explain what we mean, we rst need to say a little about how division by = is You Can’t Divide By Three = proved. Recall that we are given a bijection between × In this section we will show that sock division by 3 is not = and × and we need to construct a bijection between provable without the axiom of choice. In fact, neither is = = and . We can think of both × and × as unions of sock division by 2 or, for that matter, sock division by = for = collections of disjoint sets of size . Namely, any = ¡ 1. Ø × = = f¹0, 1º, ¹0, 2º,..., ¹0, =ºg Theorem 4. For any natural number = ¡ 1, the principle of 02 sock division by = is not provable in ZF. Ø × = = f¹1, 1º, ¹1, 2º,..., ¹1, =ºg. Proof. We will show that if sock division by 2 is possible 1 2 then it is also possible to choose socks for Bertrand Russell’s A key point, which every known proof uses, is that we millionaire. The full theorem follows by noting that the can simultaneously order every set in the two collections proof works not just for human socks but also for socks for by using the ordering induced by the usual ordering on octopi with = ¡ 1 tentacles. f1, 2, . , =g. More precisely, suppose sock division by 2 does But if we are already working without the axiom of hold and let f 8 g8 2N be a sequence of disjoint sets choice, this seems like an unfair restriction. Why not also of size 2. We will show that Π8 2N 8 is nonempty allow collections of unordered sets of size =? This gives us by constructing a choice function for f 8 g8 2N. We an alternate version of “division by = without choice” where can picture f 8 g8 2N as a sequence of pairs of socks. we replace the collections × = and × = with collections of unordered sets of size = (we will give a precise statement of this version below). Since collections of ordered sets of size = behave like the pairs of shoes from Russell’s example while collections of unordered sets of size = behave like the 2 Now consider taking a single pair of socks—say 0 = • First, each set in the collection of rows has the form G ,~ G, , G, G Ð f 0 0g—and forming the Cartesian product of this pair f¹ 0º ¹ 1ºg for some 2 8 2N 8 , so we can think , Ð with the set f0 1g. This gives us a 4 element set, depicted of the collection of rows as being indexed by 8 2N 8 by the grid below.