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How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.
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.
The number of numbers was long thought unknowable. But mathematicians now feel they may be closing in on an answer.
“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.
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. (Continued)