Is Mathematics Mostly Chaos or Mostly Order?
Mathematicians probe the boundaries of logic by inventing new types of infinity.
Wei-An Jin/Quanta Magazine
Introduction
Last winter, at a meeting in the Finnish wilderness high above the Arctic Circle, a group of mathematicians gathered to contemplate the fate of a mathematical universe.
It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating the nature of two new notions of infinity. The consequences, Aguilera believed, were grand. “We just don’t know what they are yet,” he said.
Infinity, counterintuitively, comes in many shapes and sizes. This has been known since the 1870s, when the German mathematician Georg Cantor proved that the set of real numbers (all the numbers on the number line) is larger than the set of whole numbers, even though both sets are infinite. (The short version: No matter how you try to match real numbers to whole numbers, you’ll always end up with more real numbers.) The two sets, Cantor argued, represented entirely different flavors of infinity and therefore had profoundly different properties.
From there, Cantor constructed larger infinities, too. He took the set of real numbers, built a new set out of all of its subsets, then proved that this new set was bigger than the original set of real numbers. And when he took all the subsets of this new set, he got an even bigger set. In this way, he built infinitely many sets, each larger than the last. He referred to the different sizes of these infinite sets as cardinal numbers (not to be confused with the ordinary cardinals 1, 2, 3…).
Set theorists have continued to define cardinals that are far more exotic and difficult to describe than Cantor’s. In doing so, they’ve discovered something surprising: These “large cardinals” fall into a surprisingly neat hierarchy. They can be clearly defined in terms of size and complexity. Together, they form a massive tower of infinities that set theorists then use to probe the boundaries of what’s mathematically possible.
But the two new cardinals that Aguilera was pondering in the Arctic cold behaved oddly. He had recently constructed them, along with Joan Bagaria of the University of Barcelona and Philipp Lücke of the University of Hamburg, only to find that they didn’t quite fit into the usual hierarchy. Instead, they “exploded,” Aguilera said, creating a new class of infinities that their colleagues hadn’t bargained on — and implying that far more chaos abounds in mathematics than expected.
It’s a provocative claim. The prospect is, to some, exciting. “I love this paper,” said Toby Meadows, a logician and philosopher at the University of California, Irvine. “It seems like real progress — a really interesting insight that we didn’t have before.”
But it’s also difficult to really know whether the claim is true. That’s the nature of studying infinity. If mathematics is a tapestry sewn together by traditional assumptions that everyone agrees on, the higher reaches of the infinite are its tattered fringes. Set theorists working in these extreme areas operate in a space where the traditional axioms used to write mathematical proofs do not always apply, and where new axioms must be written — and often break down.
Up here, most questions are fundamentally unprovable, and uncertainty reigns. And so to some, the new cardinals don’t change anything. “I don’t buy it at all,” said Hugh Woodin, a set theorist at Harvard University who is currently leading the quest to fully define the mathematical universe. Woodin was Bagaria’s doctoral adviser 35 years ago and Aguilera’s in the 2010s. But his students are cutting their own path through infinity’s thickets. “Your children grow up and defy you,” Woodin said.
The Universes of Set Theory
Most mathematicians don’t concern themselves with these kinds of questions. They work with a set of nine assumptions, or axioms, about how sets behave, known as ZFC — “Zermelo-Fraenkel set theory with the axiom of choice.” These nine rules can’t be proved. Mathematicians have simply agreed that they provide a natural foundation for the rest of mathematics. From them, mathematicians build up rigorous proofs of all their conjectures.
But in 1931, the German mathematician Kurt Gödel demonstrated that any interesting system of mathematical axioms is doomed to incompleteness. There will always be true statements that can’t be proved. To prove those true statements, mathematicians would have to add a new axiom. But then this longer list of axioms would also lead to true but unprovable statements. And so on. Anyone who wants to prove all the possible statements in the mathematical universe will be forced to keep creating new axiomatic systems forever.
This means that the mathematical universe, which mathematicians often call V, is fundamentally unknowable. But set theorists want to describe it as closely as possible — to create model universes that resemble the real one while being easier to study. These models provide mathematicians with the extra axioms they need to prove those elusive statements about “smaller” axiomatic systems (like ZFC) while giving them confidence that the extra axioms they’re using aren’t arbitrary. “As you strengthen these theories, you end up making low-level mathematics more concrete. It firms up,” Meadows said.
Gödel provided a starting point. He built a model, which he called L, by beginning with the empty set (which is what it sounds like) and iteratively constructing bigger sets from there. The model was a good one and easy to work with, but it was also partial. It didn’t include the large cardinals — those stranger infinities that can’t be constructed using the same methods as Cantor’s. (L is therefore referred to as an “inner” model of V, because it lives inside the bigger universe.)
Set theorists aim to expand this picture. Throughout the 20th century, they defined more large cardinals, with names like strong, compact, supercompact and huge. Each new definition requires the creation of a new axiom; set theorists then hope to show that this new axiom is consistent with ZFC, that it doesn’t violate the most fundamental rules of mathematics.
These large cardinals also seem to form a surprisingly neat hierarchy, a true tower of infinities. Each large cardinal is much, much larger than the one below it, for example, and the axiom that defines it can be used to prove far more statements than the axioms that define lower cardinals. Moreover, due to this hierarchy, if mathematicians can show that one large cardinal is consistent with ZFC, it will imply the consistency of all the cardinals below it in the tower. “You might think, ‘Oh, it’s just going to be complete chaos.’ But it doesn’t seem to be,” Meadows said.
Whenever a new cardinal is added to the tower, proving its consistency also requires the development of a larger, more sophisticated inner model. “You put in there just the minimum number of things that are necessary so that in the final model, your large cardinal will exist,” Bagaria said. With each new model, the defined mathematical universe expands.
For Woodin, the dream is to build an inner model that truly approximates V and therefore includes all the large cardinals. He calls it “Ultimate L.” It might seem like a hopeless task — after all, because of Gödel’s incompleteness results, it should require the construction of infinitely many inner models, each one containing yet another indescribably large cardinal.
Hugh Woodin has an audacious plan to map V, the entire mathematical universe.
Courtesy of Hugh Woodin
But 20 years ago, Woodin discovered a shortcut: You don’t have to build inner models for all of the large cardinals. Reach a certain point in the hierarchy — the large cardinal called “supercompact” — and the model inherits all the large cardinals above it. “You get everything,” Woodin said. “Magic happens.”
But his plan to build Ultimate L relies on the mathematical universe being nicely structured, with the large cardinals forming a neat, hierarchical tower. In mathematical parlance, they must be “hereditarily ordinal definable,” or HOD.
Woodin mapped out two possibilities. “V is either very close to HOD or very far from HOD,” Bagaria said. “There’s no middle ground.” If you could find one thing that broke the hierarchical model, many other things likely would break it as well; chaos would reign. “Maybe the universe does contain many things that are not definable,” Aguilera said.
But Woodin conjectured that the first option — that V is nicely structured, with a definable tower of cardinals — is correct. So far, the evidence suggests he’s right. No one has been able to find a large cardinal that doesn’t fit into the tower while staying consistent with ZFC.
Now Aguilera and his collaborators are complicating the picture.
Order Versus Chaos
There’s some circumstantial evidence in favor of mathematical chaos. For one thing, despite many decades of work, mathematicians’ progress on the Ultimate L program has been slow. Woodin himself has experienced periods of doubt, though he currently believes that achieving Ultimate L is possible.
Set theorists have also identified very large cardinals that seem to break away from HOD, though they are extreme outliers. Defining these cardinals involves getting rid of one of the nine axioms of ZFC — the so-called axiom of choice. Throwing out a fundamental axiom is not, to many mathematicians, an appealing approach.
That’s where Aguilera, Bagaria and Lücke entered the fray. In their recent work, they’ve produced two novel types of infinity, which they named exacting and ultraexacting cardinals. These cardinals, crucially, don’t violate the axiom of choice.
Juan Aguilera recently defined two new notions of infinity that, he claims, complicate our picture of how the mathematical universe might be structured.
Ulrich Zinell
Defining them was relatively straightforward: They’re essentially bigger analogues of other types of large cardinals. But these new infinities exhibit strange properties. At first, they appear to fit into the usual hierarchy: It’s clear where in the tower they should live, in terms of size and complexity. But then the trio tried combining them with other, smaller cardinals.
Usually, when you add up large cardinals that are compatible with HOD — yes, it’s possible to add infinities to each other — you get a sum that’s roughly the same size as the largest cardinal in the mix. It’s not possible to jump to a higher part of the hierarchy. (Consider how, if you add 100 to a quintillion, the sum will be on the same order as a quintillion; you won’t end up with a much, much bigger number.)
But add a smaller cardinal to one of the new infinities, and “they kind of blow up,” Bagaria said. “This is a phenomenon that had never appeared before.” It implied the existence of a much, much larger cardinal — an infinity larger than anything mathematicians had imagined to be consistent with ZFC.
“This challenges the intuition we had of how large cardinals relate to each other and how they’re ordered,” Aguilera said.
“We have discovered this new region where the universe becomes so wild, so complicated, that it can no longer be equal to HOD,” Bagaria said. And there are probably lots of other bizarre infinities dwelling out there, waiting to be explored, he added.
According to Aguilera, the mathematical universe, like our physical one, may be made up mostly of dark matter. “It seems now that most of the universe somehow consists of things that we can’t see,” he said.
Mathematical Experiments
Still, this does not necessarily disprove the HOD conjecture, or render Woodin’s Ultimate L program moot. In fact, Woodin had previously considered a possible large cardinal that was, in retrospect, equivalent to the ultraexacting cardinals. But he discounted it because he did not think it was consistent with ZFC, which would make it irrelevant to our mathematical universe.
The team’s next step is to gather evidence to the contrary, to show that both of the cardinals they have defined are consistent.
But the math of set theory, which so often dwells in a world beyond proof, tends to look more like physics or biology, as Aguilera put it. You come up with a principle and test it experimentally; you see if it leads to contradictions, and analyze how it compares with things you already know. You might not be able to prove it true just yet, but with the right experiments, you can give good evidence for your claim.
Aguilera, Bagaria and Lücke have provided evidence that their new cardinals are consistent with ZFC — that’s what the bulk of their new paper is about. For instance, they’ve shown that exacting and ultraexacting cardinals have similar structure and behavior to other large cardinals. Their definition also implies that if another, more established type of infinity is consistent with ZFC, then ultraexacting cardinals should also be consistent — and there’s been a lot of recent evidence for the consistency of that other type of infinity.
This, the trio argues, makes the new cardinals compelling candidates for showing that HOD is far from V, that the mathematical universe isn’t nicely ordered.
Not everyone is convinced. Gabe Goldberg, a set theorist at the University of California, Berkeley, points out that while the trio might feel confident in their evidence for consistency with ZFC, surprises frequently happen in the study of the infinite. Certain assumptions might no longer hold, causing the mathematicians’ arguments to crumble. It will take much more to truly refute the HOD conjecture, he said.
Woodin agreed. “I think we have to be very, very careful,” he said. He still thinks that HOD is close to V — and plans to continue his decades-long journey toward building a description of the mathematical universe that’s as complete as possible. He’s currently writing a “very long manuscript” on his latest approach to Ultimate L.
Other set theorists are excited by the potential of the new infinities. To them, chaos is the more exciting option. “I mean, in some ways … if Ultimate L worked out, it would kind of end a really interesting chapter in set theory,” Meadows said. “It would be like Alexander when he gets to India, finding there’s no more world to conquer.”
Perhaps there’s much more left for mathematicians to conquer. It’s time, Aguilera said, to explore what else is out there. “I always tell people that mathematics is infinite, but time is not,” he said.
