Which infinity is bigger? Under alternative mathematical rules, a 2D map emerges

It is one of the great questions of mathematics: What does "infinity" actually mean? Which infinities are larger than others? Can infinities be arranged in a sequence, like the rungs of an endless ladder, so that for any two of them we can always say which one lies above the other?

In the 20th century, mathematicians discovered that some of these questions are, in principle, impossible to answer. William Chan of TU Wien has now taken a somewhat more modest approach: He abandons the requirement that there must be a single, clear hierarchy of infinities. And precisely by doing so, he arrives at something remarkable. If infinities are not arranged along a single line but placed in a two-dimensional system, apparent difficulties disappear and a surprisingly simple order emerges—a kind of two-dimensional map of infinities.

The research is published in the journal Proceedings of the National Academy of Sciences.

Infinitely many cats

How can two numbers be compared in the first place? Quite simply: We match every element of one set with an element of another. If there are five cats and five food bowls, we can place one cat in front of each bowl. We then know that the number of cats and the number of bowls are the same. If some bowls are left over, there are more bowls than cats.

"The same idea can be applied to infinite sets," explains Chan from the Institute of Discrete Mathematics and Geometry at TU Wien. "For example, the set of natural numbers is the same size as the set of even numbers."

That may sound strange, because only half of all natural numbers are even. But the even numbers can all be numbered: 2 gets number 1, 4 gets number 2, 6 gets number 3, and so on. No even number is ever left out.

The mathematician Georg Cantor proved that the same trick does not work for the real numbers. The set of numbers with infinitely many decimal places cannot be put into such a list. It is larger than the set of natural numbers—even though both sets are infinite.

A great question—and its failure

This led to a major question that generations of mathematicians struggled with: Are there infinities of arbitrary sizes, including arbitrary intermediate sizes? Is there a set with more elements than the natural numbers but fewer than the real numbers?

To investigate such questions, mathematicians usually rely on a basic framework known as Zermelo–Fraenkel set theory, together with the so-called axiom of choice. It says that if we have any collection of nonempty sets, we can select one element from each of them. That may sound obvious: If there is a bowl of strawberries, of course I can pick a strawberry from it. But with infinitely many sets, and with no explicit rule telling us which element to choose, it is no longer obvious that such a "choice function" must exist.

The axiom of choice makes it possible to arrange infinite sets by size—one infinity after another, like the rungs of an infinitely long ladder. We can then ask: If the first rung corresponds to the infinity of the natural numbers—the smallest infinity we know—on which rung does the infinity of the real numbers appear? On the very next one? Or much higher up?

After decades of research, this question was answered—but not in the way mathematicians had expected. It was shown that within the usual mathematical axioms—Zermelo–Fraenkel set theory together with the axiom of choice—the question cannot be answered. The generally accepted basic rules of mathematics do not determine an answer.

Mathematics without the axiom of choice

"But what happens if we leave out the axiom of choice?" Chan asked. In doing so, he drew on ideas developed in the 1960s. It had already been shown that instead of the axiom of choice, one can work with the axiom of determinacy—a different kind of regularity principle that is incompatible with the axiom of choice.

The axiom of determinacy says that if two players follow certain rules in an infinitely long process, together producing an infinite sequence of numbers, then one of them must always have a strategy that allows them to force the outcome of the game. This may be somewhat more technical and less intuitive than the axiom of choice, but it too provides mathematical structure.

In a mathematical system based on the axiom of determinacy rather than the axiom of choice, one can investigate how new infinite sets arise from other infinite sets.

A typical construction is to consider all possible mappings from one set to another. The real numbers, for example, can be thought of in terms of numbers with infinitely many decimal places. At every decimal position, there are 10 possible digits, from 0 to 9, and there are infinitely many such positions. So we repeatedly choose one of 10 digits, once for every natural number. In this sense, the real numbers arise from infinitely many choices among 10 possibilities. Symbolically speaking, 10 to the power of aleph-zero gives the cardinality of the continuum.

Instead of taking the finite set of 10 digits and raising it to an infinite power, one can also take one infinite set to the power of another infinite set. The infinitely many possible mappings between them then define new infinities.

Ordering infinities: Not on a line, but on a map

Chan now asked: What if we do not try to place all such infinities in a single sequence, but instead arrange them in a two-dimensional scheme, like points on a map?

Let A, B, C and D be different infinities. Can we compare A to the power B with C to the power D? It depends, says Chan. If A is larger than C and B is larger than D, the comparison is straightforward: A to the power B is larger.

But what happens if we compare something smaller raised to something larger with something larger raised to something smaller? Which one is bigger?

Chan's answer is: These two sizes need not be comparable at all. There is no single linear ordering of these infinities—they have to be described using two coordinates. A and B define one point on the map of infinities; C and D define another.

And remarkably, once the idea of a single "ladder of infinities" is abandoned, the algebra of infinities suddenly becomes strikingly simple and reveals a rich structure. Results that are already known about infinities can be incorporated into this map without producing contradictions. All of this becomes possible once the axiom of choice—and with it the idea of one universal ladder of infinities—is set aside.

Does this solve the age-old puzzle of the sizes of infinite sets? Probably not for a very long time. "But it is a remarkably simple and fruitful new way of thinking about how infinite sets can be organized," says Chan.