Welcome to Quantum Magazine's podcast.

Each episode, we bring you stories about developments in science and mathematics.

I'm Susan Vallett.

For a half century, mathematicians have believed that the total number of real numbers is unknowable.

A new proof suggests otherwise. That's next.

Quantum Magazine is an editorially independent online publication supported by the Simons Foundation to enhance public understanding of science.

In October of 2018, David Aspero was on holiday in Italy, gazing out a car window when it came to him.

The missing step of what's now a landmark new proof about the sizes of infinity.

It was like this flash experience.

Aspero is a mathematician at the University of East Anglia in the United Kingdom.

He contacted the collaborator with whom he'd long pursued the proof, Ralph Schindler, of the University of Moonster in Germany,

and he described his insight. Schindler says it was completely incomprehensible to him,

but eventually the duo turned the incomprehensible into solid logic. Their proof appeared in May

in the annals of mathematics. It unites two rival

axioms that have been posited as competing foundations for infinite mathematics. Aspero and

Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms and all they hint at

about infinity are true.

Manakim Magador is a leading mathematical logician at the Hebrew University of Jerusalem.

It's a fantastic result.

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.

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