In my previous installment of this column, I looked at what happens when some outlandish mathematical results collide with the restrictions of the physical world. The Banach–Tarski paradox—the startling theorem that a three-dimensional sphere can be cut up into finitely many pieces and reassembled into two spheres, each the size of the original—is of limited applicability to real spheres because the “cuts” require space to be separated into points of zero size. And the infinite monkey theorem, the idea that enough monkeys typing randomly will eventually produce any given text, is undone by the finiteness of time, space, and number of monkeys.
In a way, the finite age of the universe is also relevant to the Banach–Tarski paradox. That’s because it’s not possible, even in principle, to catalog which points in the original sphere belong to which piece in the decomposition. Defining the pieces necessarily requires infinitely many arbitrary choices—which presumably would take an infinitely long time to make.
To see what I mean by arbitrary, consider a thought experiment laid out by Bertrand Russell in his 1919 book, Introduction to Mathematical Philosophy. Suppose you had infinitely many pairs of shoes, and you needed, for some reason, to pick one shoe from each pair. You could do that in a way that’s not arbitrary by formulating a rule that prescribes which shoe to pick from each pair. For example, choose all the left shoes. Or all the right ones. Or, if the pairs of shoes have a natural order to them, choose the left shoe of the first pair, then the right shoe of the second, and so on.
But if you had infinitely many pairs of socks, you couldn’t devise such a rule. The socks in a pair don’t come in left and right, so there’s no way to reliably specify which one to pick. The only way to choose one sock from each pair is to do it arbitrarily.
Miller’s Diary
Physics Today editor Johanna Miller reflects on the latest Search & Discovery section of the magazine, the editorial process, and life in general.
For a more rigorous example that’s more relevant to the Banach–Tarski paradox, imagine partitioning the real numbers into subsets so that within each subset, the difference between any two numbers is rational. (So one subset would contain all the rational numbers themselves. Another would contain all the numbers of the form a rational number plus π. A third, all the numbers of the form a rational number plus .) There are infinitely many such subsets; can you pick exactly one element from each of them? If you can, you must have a lot of time on your hands, because there’s no way to specify the choice by any rule or pattern.
In the mathematics of set theory, the principle that making infinitely many arbitrary choices is a valid way to construct a set is known as the axiom of choice. And invoking it is an essential step in the proof of the Banach–Tarski paradox. Without the axiom of choice, the proof stalls.
The axiom of choice is inherently unphysical. If a monkey can never perform a typing task that takes a mere 10164000 times the age of the universe to complete, then you can never define a set whose construction necessarily takes infinitely long. And though mathematicians grapple with unphysical concepts all the time—they’ve never had any qualms, as far as I know, about doing geometry with lines of zero width and points of zero size—they did agonize for a few decades in the early 20th century over whether the axiom of choice was appropriate to use in proofs. Even today it’s regarded with suspicion in a way that most mathematical axioms aren’t.
The problem isn’t the unphysicality of the axiom per se but the subtlety of the foundations of set theory and the need to be scrupulously careful about what, exactly, counts as a set.
Russell, in 1901, had discovered one way that too incautious a definition can lead to trouble. If a set can be defined from any conceivable collection of objects or ideas, then there exists a set of all sets. The set of all sets, being itself a set, therefore contains itself as an element. One can then partition the set of all sets into two subsets: those that contain themselves as elements, and those that don’t. Therein, Russell found, lies the problem: Does the set of all sets that don’t contain themselves, contain itself? If it does, then it’s one of those sets that don’t contain themselves, so it doesn’t contain itself. Likewise, if it doesn’t, then it does.
This is what’s known as Russell’s paradox. (Russell wrote about many paradoxes over the years, but this is the one that got his name attached to it.) Unlike the Banach–Tarski paradox, which is a logically sound theorem that happens to be counterintuitive, Russell’s paradox is a real paradox: If the set of all sets that don’t contain themselves exists, it both does and doesn’t contain itself.
Paradoxes in math have devastating consequences. If it’s ever possible to deduce, within a mathematical system, that two contradictory statements are both true, then it follows by the rules of logic that every statement in that system is both true and false—and the system, therefore, is useless. The only way out is to challenge the rules. If set theory was to be taken seriously as a useful branch of mathematics, as early-20th-century set theorists very much wanted it to be, they needed to choose its axioms in such a way that “the set of all sets that don’t contain themselves” never emerges as a coherent concept. That is, they needed to establish that not every collection of objects that can be imagined or described in words is a set.
The axiom of choice postulates the existence of sets that can’t even be explicitly imagined or described. So if “the set of all sets that don’t contain themselves” raises such problems, does “the set of one sock from each of infinitely many pairs” raise problems too?
In 1938 Kurt Gödel showed that it doesn’t: He proved that the axiom of choice can’t be disproved from the other axioms of set theory. And in 1963, Paul Cohen proved that it can’t be proved from the other axioms either. So although the axiom of choice doesn’t have the millennia-long history of Euclid’s parallel postulate—only because set theory is a much younger branch of mathematics than geometry is—the ending of the story is the same. Just as there are Euclidean and non-Euclidean geometric worlds, there are versions of set theory in which the axiom of choice holds, and versions in which it doesn’t. And they’re equally consistent.
