## How many real numbers are there?

For more than 50 years, mathematicians have assumed that the number of points on the number line cannot be determined. But new evidence is making experts rethink their position.

During a vacation in October 2018, David Asperó was looking out of the car window at the Italian countryside, lost in thought, when he suddenly had a brilliant idea for the missing piece of a puzzle that had been bothering him for a long time: he had just completed the last step of one groundbreaking new proof revealing the magnitudes of infinities.

The mathematician at the University of East Anglia in Great Britain immediately contacted his colleague Ralf Schindler from the University of Münster, with whom he had been pursuing the topic for several years, and described his findings."It all sounded completely incomprehensible to me," Schindler recalls. But eventually both researchers managed to unravel Asperó's thoughts and turn them into understandable logical conclusions.

Their proof, which they published in May 2021 in the renowned journal "Annals of Mathematics", combines two competing basic statements, so-called axioms, each of which should expand the foundation of mathematics. Until now, experts had assumed that one had to choose between the two assumptions because they would be mutually exclusive. But as Asperó and Schindler were able to show, one of the statements leads to the other. This suggests that both are true - and therefore everything they say about the world of infinities.

The result makes waves because it reinforces the arguments against the famous continuum hypothesis…

Spektrum der Wissenschaft's translated and edited version of the article "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer" from "Quanta Magazine", a content-independent magazine of the Simons Foundation, which supports the distribution of research results from mathematics and the natural sciences.