Natalie Wolchover on July 15, 2021
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.
In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said.
Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d long pursued the proof, Ralf Schindler of the University of Münster in Germany, and described his insight. “It was completely incomprehensible to me,” Schindler said. But eventually, the duo turned the phantasm into solid logic.
Their proof, which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Asperó and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true.
“It’s a fantastic result,” said Menachem Magidor, a leading mathematical logician at the Hebrew University of Jerusalem. “To be honest, I was trying to get it myself.”
Most importantly, the result strengthens the case against the continuum hypothesis, a hugely influential 1878 conjecture about the strata of infinities. Both of the axioms that have converged in the new proof indicate that the continuum hypothesis is false, and that an extra size of infinity sits between the two that, 143 years ago, were hypothesized to be the first and second infinitely large numbers.
“We now have a coherent alternative to the continuum hypothesis,” said Ilijas Farah, a mathematician at York University in Toronto.
It’s one of the most intellectually exciting, absolutely dramatic things that has ever happened in the history of mathematics.
Juliette Kennedy
The result is a victory for the camp of mathematicians who feel in their bones that the continuum hypothesis is wrong. “This result is tremendously clarifying the picture,” said Juliette Kennedy, a mathematical logician and philosopher at the University of Helsinki.
But another camp favors a different vision of infinite mathematics in which the continuum hypothesis holds, and the battle between these sides is far from won.
“It’s an amazing time,” Kennedy said. “It’s one of the most intellectually exciting, absolutely dramatic things that has ever happened in the history of mathematics, where we are right now.”
An Infinity of Infinities
Yes, infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.
Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number ℵ0 (“aleph-zero”).
But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.
Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.
Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.
His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ℵ1 real numbers. In other words, the cardinality of the continuum immediately follow ℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.
But to Cantor’s immense distress, he couldn’t prove it.
In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.
To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.
The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove.
As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.
These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.
In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.
In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.
Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question.
Gödel, for his part, believed that the continuum hypothesis is false — that there are more reals than Cantor guessed. He suspected there are ℵ2 of them. He predicted, as he wrote in 1947, “that the role of the continuum problem in set theory will be this, that it will finally lead to the discovery of new axioms which will make it possible to disprove Cantor’s conjecture.”
Source of Light
Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible. “There was always this tension,” Schindler said.
To understand them, we have to go back to Paul Cohen’s 1963 work, where he developed a technique called forcing. Starting with a model of the mathematical universe that included ℵ1 reals, Cohen used forcing to enlarge the continuum to include new reals beyond those of the model. Cohen and his contemporaries soon found that, depending on the specifics of the procedure, forcing lets you to add however many reals you like — ℵ2 or ℵ35, say. Aside from new reals, mathematicians generalized Cohen’s method to conjure up all manner of other possible objects, some logically incompatible with one another. This created a multiverse of possible mathematical universes.
“His method creates an ambiguity in our universe of sets,” said Hugh Woodin, a set theorist at Harvard University. “It creates this cloud of virtual universes, and how do I know which one I’m in?”
What was virtual and what was real? Which of two conflicting objects, dreamed up by different forcing procedures, should be permitted? It wasn’t clear when or even whether an object, just because it could be conceived of with Cohen’s method, really exists.
To address this problem, mathematicians posed various “forcing axioms” — rules that established the actual existence of specific objects rendered possible by Cohen’s method. “If you can imagine an object to exist, then it does; this is the guiding intuitive principle that leads to forcing axioms,” Magidor explained. In 1988, Magidor, Matthew Foreman and Saharon Shelah took this ethos to its logical conclusion by posing Martin’s maximum, which says that anything you can conceive of using any forcing procedure will be a true mathematical entity, so long as the procedure satisfies a certain consistency condition.
For all the expansiveness of Martin’s maximum, in order to simultaneously permit all those products of forcing (while satisfying that constancy condition), the size of the continuum jumps only to a conservative ℵ2— one cardinal number more than the minimum possible value.
Besides settling the continuum problem, Martin’s maximum has proved to be a powerful tool for exploring the properties of infinite sets. Proponents say it fosters many sweeping statements and general theorems. By contrast, assuming that the continuum has cardinality ℵ1 tends to yield more exceptional cases and roadblocks to proofs — “a paradise of counterexamples,” in Magidor’s words.
Martin’s maximum became massively popular as an extension of ZFC. But then in the 1990s, Woodin proposed another compelling axiom that also kills the continuum hypothesis and pins the continuum at ℵ2 but by a totally different route. Woodin named the axiom (*), pronounced “star,” because it was “like a bright source — a source of structure, a source of light,” he told me.
(*) concerns a model universe of sets that satisfies the nine ZF axioms plus the axiom of determinacy, rather than the axiom of choice. Determinacy and choice logically contradict each other, which is why (*) and Martin’s maximum seemed irreconcilable. But Woodin devised a forcing procedure by which to extend his model mathematical universe into a larger one that is consistent with ZFC, and it’s in this universe that the (*) axiom holds true.
What makes (*) so illuminating is that it lets mathematicians make statements of the form “For all X, there exists Y, such that Z” when referring to properties of sets within the domain. Such statements are powerful modes of mathematical reasoning. One such statement is: “For all sets of ℵ1 reals, there exist reals not in those sets.” This is the negation of the continuum hypothesis. Thus, according to (*), Cantor’s conjecture is false. The fact that (*) lets mathematicians conclude this and assert many other properties of sets of reals makes it an “attractive hypothesis,” Schindler said.
With two highly productive axioms floating around, proponents of forcing faced a disturbing surplus. “Both the forcing axiom [Martin’s maximum] and the (*) axiom are beautiful and feel right and natural,” Schindler said, so “which one do you choose?”
If the axioms contradicted each other, then adopting one would mean sacrificing the other’s nice consequences, and the judgment call might feel arbitrary. “You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said.
Instead, his new work with Asperó shows that Martin’s maximum++ (a technical variation of Martin’s maximum) implies (*). “If you unify these theories, as we did,” Schindler said, “I would say that you can take it as a case in favor of: Maybe people got something right.”
Missing Link
Asperó and Schindler were young researchers together at an institute in Vienna 20 years ago. Their proof germinated several years later, when Schindler read a manuscript, handwritten as usual, by the set theorist Ronald Jensen. In it, Jensen invented a technique called L-forcing. Schindler was impressed by it and asked a student of his to try to develop it further. Five years later, in 2011, he described L-forcing to Asperó, who was visiting him in Münster. Asperó immediately suggested that they might be able to use the technique to derive (*) from Martin’s maximum++.
They announced that they had a proof the next year, in 2012. Woodin immediately identified a mistake, and they withdrew their paper in shame. They revisited the proof often in the years that followed, but they invariably found that they lacked one key idea — a “missing link,” Asperó said, in the logical chain leading from Martin’s maximum++ to (*).
Their plan of attack for deriving the latter axiom from the former was to develop a forcing procedure similar to L-forcing with which to generate a type of object called a witness. This witness verifies all statements of the form of (*). So long as the forcing procedure obeys the requisite consistency condition, Martin’s maximum++ will establish that the witness, since it can be forced to exist, exists. And thus (*) follows.
“We knew how to build such forcings,” Asperó said, but they couldn’t see how to guarantee that their forcing procedure would meet the key requirement of Martin’s maximum. Asperó’s “flash experience” in the car in 2018 finally showed the way: They could break up the forcing into a recursive sequence of forcings, each satisfying necessary conditions. “I remember feeling very confident that this ingredient would in fact make the proof work,” he said, though it took further flashes of insight from both Asperó and Schindler to work it all out.
Other Stars
The convergence of Martin’s maximum++ and (*) creates a solid foundation for a tower of infinities in which the cardinality of the continuum is ℵ2. “The question is, is it true?” asks Peter Koellner, a set theorist at Harvard.
According to Koellner, knowing that the strongest forcing axiom implies (*) can count as evidence either for or against it. “Really that depends on what your take on (*) is,” he said.
The convergence result will focus scrutiny on (*)’s plausibility, since (*) allows mathematicians to make those powerful “for all X, there exists Y” statements that have consequences for the properties of the real numbers.
Despite (*)’s extreme usefulness in permitting those statements, seemingly without contradiction, Koellner is among those who doubt the axiom. One of its implications — a mirroring of the structure of a certain large class of sets with a much smaller set — strikes him as strange.
Notably, the person who might have been most enthusiastic about (*)’s correctness has also turned against it. “I’m considered a traitor,” Woodin said in one of our Zoom conversations this summer.
Twenty-five years ago, when he posed (*), Woodin thought the continuum hypothesis was false, and thus that (*) was a source of light. But about a decade ago, he changed his mind. He now thinks that the continuum has cardinality ℵ1 and that (*) and forcing are “doomed.”
Woodin called Asperó and Schindler’s proof “a fantastic result” that “deserves to be in the Annals” — the Annals of Mathematics is widely considered to be the top math journal — and he acknowledged that this kind of convergence result “is usually taken as evidence of some kind of truth.” But he doesn’t buy it. There’s the issue mentioned by Koellner, and another even bigger problem that he identified in a flash experience of his own in 2019, shortly after reading the preprint of Asperó and Schindler’s paper. “It’s an unexpected twist in the story,” Woodin said.
When he posed (*), Woodin also posed stronger variants called (*)+ and (*)++, which apply to the full power set (the set of all subsets) of the reals. It’s known that, in various models of the mathematical universe if not in general, (*)+ contradicts Martin’s maximum. In a new proof, which he began to share with mathematicians in May, Woodin showed that (*)+ and (*)++ are equivalent, which means (*)++ contradicts Martin’s maximum in various models also.
(*)+ and (*)++ far outshine (*), for one reason: They permit mathematicians to make statements of the form “There exists a set of reals …” and thus to describe and analyze properties of any and all sets of reals. (*) does not provide such an “existential theory” of sets of reals. And because Martin’s maximum seems to contradict (*)+ and (*)++, it seems that existential statements about sets of reals might not be possible in the Martin’s maximum framework. For Woodin, this is a deal breaker: “What this is saying is, it’s doomed.”
The other main players are all still digesting Woodin’s proof. But a few stressed that his arguments are conjectural. Even Woodin acknowledges that a surprising discovery could change the picture (and his opinion), as has happened before.
Many in the community await the results of Woodin’s attempt to prove the “ultimate L” conjecture: that is, the existence of an all-encompassing generalization of Gödel’s model universe of sets. If ultimate L exists — Woodin has good reason to think it does, and he is 400 pages into a proof attempt now — he’ll consider it obvious that the “dream axiom” to add to ZFC must be the ultimate L axiom, or the statement that ultimate L is the universe of sets. And in ultimate L, Cantor is right: The continuum has cardinality ℵ1. If the proof works out, the ultimate L axiom will be, if not an obvious choice of extension for ZFC, at least a formidable rival for Martin’s maximum.
Ever since Gödel and Cohen established the independence of the continuum hypothesis from ZFC, infinite math has been a choose-your-own-adventure story in which set theorists can force the number of reals up to any level — ℵ35, or ℵ1000, say — and explore the consequences. But with Asperó and Schindler’s result pointing compellingly to ℵ2, and Woodin building the case for ℵ1, a clear dichotomy has established itself, and an outright winner seems newly possible. Most set theorists would like nothing more than to exit the mathematical multiverse and coalesce behind a single picture of Cantor’s paradise, one that’s beautiful enough to call true.
Kennedy, for one, thinks we may soon return to that “prelapsarian world.” “Hilbert, when he gave his speech, said human dignity depends upon us being able to decide things in mathematics in a yes-or-no fashion,” she said. “This was a matter of redeeming humanity, of whether mathematics is what we always thought it was: to establish the truth. Not just this truth, that truth. Not just possibilities. No. The continuum is this size, period.”
Clarification July 23, 2021:
This article was changed to eliminate a possible misimpression about the number of order types of natural numbers.
Collected at: https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/