It occurred to me that Kurzweil’s singularity might be better characterized as an event horizon. The difference is that a singularity is where something actually infinite happens, while an event horizon is a surface beyond which one can never know anything. An event horizon looks like a singularity from the perspective of a fixed external observer, but from the perspective an observer falling it in always appears to be further down.
Of course, in the case of a black hole anyone who gets near the event horizon is ripped apart by the massive gravitational gradient. Hopefully this won’t happen in the Kurzweil version.