Baking is not my strong point. So when I have houseguests, I dash to a bakery to grab dessert and am often spoiled for choice. Given the huge selection of appetizing cakes and tarts, I find it difficult to decide. My strategy is to say, “Oh, why don’t you pack me one piece of each?”
This approach is actually tied to a famous debate in mathematics. It’s not my lack of decisiveness that would provoke a mathematician’s ire (unless they were standing behind me in line). No, what really causes trouble is the idea that I can, in fact, choose exactly one piece or slice of any number of different cakes and tarts and take them home with me. That idea links to an unproven fundamental truth, the so-called axiom of choice.
At first, one would not expect this approach to violate any mathematical principles. But the conclusions that follow from the axiom of choice once sparked the biggest controversy in mathematics. That’s because this axiom leads to apparently contradictory results: for example, it can “magically” double a sphere or imply that there are finite objects that cannot be measured.
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For that reason, some experts specifically indicate when they use the axiom of choice in a proof—and there are mathematicians who have wanted to overhaul the subject without this axiom. Yet a world without the axiom of choice is even stranger.
The Foundation of Mathematics
To understand the debate, we first need to consider what distinguishes mathematics from the other natural sciences. At the end of the 19th century, mathematicians realized that they had to agree on a common foundation—that is, a few basic truths—including a set of rules. The idea is that all mathematical results—whether 1 + 1 = 2 or a complicated integral—can derive from a common foundation. Every statement and every proof can be unambiguously checked using the same set of rules.
At the time when mathematicians were creating these common laws, set theory seemed to be a good starting point. The experts then had to agree on the fundamental truths, or axioms, that were accepted as true even thought they could not be proven. For example, “there is an empty set” is one of these truths. It fulfills all the requirements of an axiom: it is short, precise, defines an unambiguous object and is unquestionably true.
Mathematicians sought other axioms in the hope of finding the simplest, shortest possible set of rules from which the entire subject could ultimately be built. And they were successful. Their efforts resulted in the so-called Zermelo-Fraenkel axiom system, consisting of eight fundamental truths. All of these axioms state that there are certain sets: for example, the empty set or the power set (the set of all subsets) of a set. And these are always clearly defined by the axioms.
Mathematician Ernst Zermelo quickly realized, however, that these eight fundamental truths were not enough. In 1904 he therefore introduced the axiom of choice. And thus began the conflict.
You Always Have a Choice
The axiom of choice allows you to select one element at a time from a series of nonempty sets—much as I can have a slice of several cakes at the bakery. At first, it seems only natural that this is possible. The axiom of choice is not limited to finite cases, however: even if there are an infinite number of cakes, the axiom of choice allows you to pick one piece at a time.
The axiom essentially states that a rule exists that allows you to make that request. For example, one such rule would be: “Please give me an edge piece of each cake.” This gives the person behind the bakery counter a specific instruction that they can follow. But finding “an edge piece” is admittedly a much easier directive for a rectangular cake than a circular one. I can only say that I would like a piece of each cake—but I cannot specify exactly which one I want.
Lack of precision is exactly what bothered many experts. The other axioms predict a clearly defined set. But in the axiom of choice, the “selection function” (or the instruction that I give to the baker) will lead to my receiving something that I cannot totally describe in advance.
Then, later in 1904, Zermelo, who had introduced the axiom of choice, found an extremely counterintuitive result that he could only prove with the help of the axiom of choice. He showed that any quantity can be well ordered. From this so-called well-ordering theorem, it follows, among other things, that every set has a smallest element according to that ordering.
But this contradicts the usual principles of mathematics. If you consider the set (0, 1), for example, it contains all real numbers that are greater than 0 and smaller than 1. You are constantly calculating with such sets in mathematics. It is important to note that 0 and 1 are not part of (0, 1). The well-ordering theorem implies that this set has a smallest element according to a given ordering. But this is not possible with typical approaches to ordering numbers: there is no smallest element in this set according to standard mathematics. In fact, there is no agreed-upon answer to the question of which ordering gives the smallest number in (0,1).
Zermelo’s result sparked a worldwide debate that was almost philosophical in nature: When does a mathematical object (such as the selection function or the smallest element of a set) exist? Do we always have to be able to specify how an object can be constructed—or is it enough to prove its existence indirectly? “From 1905 to 1908 eminent mathematicians in England, France, Germany, Holland, Hungary, Italy and the United States debated the validity of [Zermelo’s proof]. Never in modern times have mathematicians argued so publicly and so vehemently about a proof,” writes mathematics historian Gregory Moore in his 1982 book Zermelo’s Axiom of Choice.
And things got worse. From the axiom of choice follows the so-called Vitali theorem, according to which you can form a set of real numbers between 0 and 1 that is not measurable. The axiom of choice makes it possible to group the numbers into individual subsets and to select an element from each, whereby the resulting set is so jagged that it is no longer measurable.
Another counterintuitive result is the magical doubling of a sphere, better known as the Banach-Tarski paradox. With the help of the axiom of choice, a sphere with volume V can be broken down into complicated individual parts and reassembled in such a way that two spheres with respective volumes V are created. These and other results have increased distrust in the axiom of choice.
An Alternative Mathematics
Some experts were therefore determined to reject the axiom of choice and instead work only with the eight fundamental truths of Zermelo-Fraenkel set theory. But they didn’t get very far. In fact, Zermelo examined the work of some of the most vehement critics of the axiom of choice and was able to prove that his colleagues—without realizing it—had actually made use of the axiom again and again.
Without the axiom of choice, for example, it is not possible to ensure that every vector space has a basis. This property is implied by another statement called Zorn’s lemma and sounds abstract, but physicists and mathematicians refer to it time and again. You can visualize this with the help of a sheet of paper (which from a mathematical point of view is nothing other than a vector space). If you draw two arrows on this sheet, one pointing horizontally and one pointing vertically, you can reach any point on the sheet from these arrows. For example, you can move 0.5 times the length of the first arrow horizontally and add 1.65 times the length of the second vertically to arrive at a specific point x.
Zorn’s lemma therefore makes it possible to draw a coordinate system in every vector space that can be used to describe every point in the space unambiguously. If you dispense with the axiom of selection, there are also vector spaces without such a coordinate system—which can lead to serious problems, especially in physics.
As it turns out, the well-ordering theorem, Zorn’s lemma and the axiom of choice are not only related but they are also equivalent. From a mathematical point of view, they are on the same level. This seems astonishing, as mathematician Jerry Bona aptly put it: “The axiom of choice is obviously true; the well-ordering principle is obviously false; and who can tell about Zorn’s lemma?”
Is the Axiom of Choice True or False?
The problem is that you can’t prove axioms. Zermelo introduced the axiom of choice because the eight axioms of the Zermelo-Fraenkel set theory were not powerful enough and could not be used to construct a selection function. In other words: if you exist in a universe that only uses the eight accepted fundamental truths, you cannot try every cake at the bakery.
But some experts thought perhaps they could demonstrate that adding the axiom of choice to the eight fundamental axioms would invariably lead to contradictions. That is, in combination with one of the eight Zermelo-Fraenkel axioms, a contradictory statement such as 1 = 2 could arise. If so, the argument went, the very foundations of mathematics would be flawed, and the entire subject would collapse like a house of cards. As mathematicians determined in the 1960s, however, that’s just not the case. If you add the axiom of choice to the eight fundamental truths, no such problems arise.
But the reverse is also true. You can add the negation of the axiom of choice to the eight Zermelo-Fraenkel axioms without encountering any contradictions. This means that the axiom of choice can be regarded as either true or false, and mathematicians are free to choose one of the two possibilities.
A World without Choice
“One commonly hears the axiom of choice described as useful for certain mathematical arguments, but problematic in light of the Banach-Tarski paradox and other counterintuitive consequences. To my way of thinking, however, a more balanced pro/con discussion arises when we also highlight the counterintuitive situations that can occur when the axiom of choice fails,” writes mathematician Joel David Hamkins of the University of Notre Dame in his book Lectures on the Philosophy of Mathematics.
If the axiom of choice is regarded as false, paradoxical results arise with regard to real numbers. Suppose you want to divide the real numbers into different buckets; number x goes into bucket A, number y into bucket B, and so on. Each number can end up in exactly one bucket, and the buckets each contain at least one number.
If you reject the axiom of choice, you can prove that the number of buckets exceeds the number of real numbers. There are therefore an infinite number of real numbers (and buckets), but the infinity of buckets is greater than the infinity of real numbers—at least if the axiom of choice is false.
Worse still, if the negation of the axiom of choice were true, there wouldn’t just be a few (very contrived) examples of sets that can’t be measured—the whole theory of measurement would collapse! “This is much worse than just having Vitali’s non-measurable sets,” writes Hamkins in his book.
For this reason, the axiom of choice has become accepted in mainstream mathematics. Although there is a small mathematical community that is trying to revamp the subject completely without the axiom of choice, most mathematicians have now accepted it as true. This is fortunate because it means it is still possible to order a sample of any number of cakes from the baker—and I don’t have to start learning to bake.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
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