# How many numbers exist?

In May 2021, Prof Ralf Schindler, professor at the Institute of Mathematical Logic and Fundamental Research and member of our Cluster, together with Prof David Asperó (University of East Anglia) published a proof about the sizes of infinities that unites two previously rival axioms of set theory. The result has made waves among experts, as it has far-reaching implications for fundamental assumptions in this field of research.

Publication in "Annals of Mathematics": David Asperó and Ralf Schindler. Martin's Maximum++ implies Woodin's axiom (∗). *Annals of Mathematics*, 193(3):793–835, May 2021. doi:10.4007/annals.2021.193.3.3. [en]

Ralf Schindler and David Asperó have also written an article in which they discuss their result and its implications for Cantor's continuum problem. Cantor's continuum problem, namely the question of how many real numbers exist, and all its ramifications, have been one of the driving forces of set-theory research for about 150 years.

David Asperó and Ralf Schindler: "Wie viele reele Zahlen gibt es?" [de]

The new proof also aroused interest in popular science magazines. Read or listen to the contributions here:

- Quanta Magazine: "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer."
- Episode of the Quanta Science Podcast
- Contributions by David Asperó in the newspaper "El País": "¿Cuántos números reales existen?" "Axiomas naturales para las matemáticas y el problema del continuo"
- Science magazine Spektrum der Wissenschaft: "Wie viele reelle Zahlen gibt es?" [de, Paywall]
- Episode of the "Spektrum" podcast on detektor.fm: "Unendlichkeiten: Wenn Mathematik zur Glaubensfrage wird" [de]