How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer - I - Quanta

Subhotosh Khan

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Jun 18, 2007
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer - I. Olena Shmahalo
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.”

It’s one of the most intellectually exciting, absolutely dramatic things that has ever happened in the history of mathematics.

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.

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 all the 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.

The German mathematician Georg Cantor would, in the decade following this 1870 portrait, develop set theory and discover the infinite hierarchy of infinite sets.

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.

Cantor first posed the continuum hypothesis in this manuscript, dated July 11, 1877. The paper was published the following year.

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.

(Continued to How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer - II.)