A critical appraisal of the singularity theorems

The 2020 Nobel prize in Physics has revived the interest in the singularity theorems and, in particular, in the Penrose theorem published in 1965. In this short paper, I briefly review the main ideas behind the theorems and then proceed to an evaluation of their hypotheses and implications. I will try to dispel some common misconceptions about the theorems and their conclusions, as well as to convey some of their rarely mentioned consequences. In particular, a discussion of space–time extensions in relation to the theorems is provided. The nature of the singularity inside black holes is also analysed. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.