Interesting point, and what's interesting is that it doesn't rely on the cardinality arguments above. Both the set of points visited by the walker and the set o

Interesting point, and what's interesting is that it doesn't rely on the cardinality arguments above. Both the set of points visited by the walker and the set of points in the universe have the same cardinality, but that doesn't mean that the walker visits all points. (In fact I suspect he can never visit more than a vanishing subset of them, but I could be wrong, and I'm sure somebody has figured this out already...)