@johncarlosbaez Thanks! I think the picture is getting clearer. So what Penrose calls a singularity in his theorem is geodesic incompleteness from the point of view of differential geometry. So the geodesic can’t go on forever which means it’s encountering what. It’s terminating into what? The end of spacetime? One could take Kerr’s view and say physically singularities aren’t real and the existence of singularities indicates there’s a problem with the theory. This is quite interesting. Also sounds like there’s a conjecture here: “every black hole solution contains a curvature singularity.” I will read the physics stackexchange for the details. This is strange and interesting.