The curvature becomes infinite. This is not the same as following the geodesic indefinitely. Having it hit a place where the curvature is infinite is one way to get a geodesic to terminate. I would also be careful with books such as these that are written by physicists that are not relativists; they tend to use weird, imprecise, inaccurate, or downright wrong phrases such as “the fabric of space-time is torn” or “the surface has collapsed to a singularity”, or making statements like “the cone extends to infinity” without proper care on the coordinates used. The proper time for a particle to fall into the singularity of a Schwarzschild black hole, for example, is finite, and not infinite as you might be mislead from looking at their infinite cone.
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I had suspected as much. As a biologist, I sometimes cringe when reading articles and books by non-biologists discussing biology.
Thanks for the explanations!