Bayesian statistics (i.e. "Carter Catastrophe") only works when applied to finite sets. When applied to potentially infinite sets, such as total number of humans that ever will live, it produces meaningless results. In particular, it produces same result (we are nearing the end of the set) no matter when it is applied -- today, 1000 years ago, or 10,000 years ago.

__________________
Fiction has to be plausible. Reality is under no such constraint.