Can you explain the "for any reasonable treatment" part here? How is the least upper bound property (or something equivalent to it) getting used in quantitative ethical theories?
Because even if we assume that we can assign numbers to a single person's utility for many different things, we cannot assume that the ratios between these utilities will be integers or rational numbers. We have to consider the possibility of utility ratios that are irrational numbers. In treatments like that of Von Neumann, where numerical utilities are assigned meaning by asking hypothetical agents to accept or decline a potentially infinite series of bets, the possibility of utility ratios that are irrational numbers arises as the least upper bound property requirement--because the final utility ratio from the infinite series of bets will be the least upper bound of the series of ratio estimates that arise from the bets.
That also raises a different question about people's introspection and ability to accurately answer questions about their utility and preferences, but that's a bit far afield from the mathematical question.
