I think I understand what the axiom is doing. I'm not sure it's strong enough, though. There is no guarantee that there is any N that is >= M_i for all i (or for all large enough i, a weaker version which I think is what is needed), nor an N that is <= them.
The M_i's can themselves be lotteries. The idea is to group events into finite lotteries so that the M_i's are >= N.
Personally, I am not convinced that bounded utility is the way to go to avoid Pascal's Mugging, because I see no principled way to choose the bound.
There is no principled way to chose utility functions either, yet people seem to be fine with them.
My point is that if one takes the VNM theory seriously as justification for having a utility function, the same logic means it must be bounded.
There is no principled way to chose utility functions either, yet people seem to be fine with them.
The VNM axioms are the principled way. That's not to say that it's a way I agree with, but it is a principled way. The axioms are the principles, codifying an idea of what it means for a set of preferences to be rational. Preferences are assumed given, not chosen.
My point is that if one takes the VNM theory seriously as justification for having a utility function, the same logic means it must be bounded.
Boundedness does not follow from the VNM axioms. ...
Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.