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

Natalie Wolchover in Quanta:

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.

More here.