FWIW, if you extend the rationality axioms to prospects (probability distributions) with infinitely many possible outcomes in their natural ways, Harsanyi’s theorem + separability leads to contradiction. In general, unbounded utility functions violate extensions of standard rationality axioms to prospects with infinitely many possible outcomes, and these extensions can be motivated pretty much the same ways as the versions here, in the vNM utility theorem and Savage’s theorem. See my post here.
FWIW, if you extend the rationality axioms to prospects (probability distributions) with infinitely many possible outcomes in their natural ways, Harsanyi’s theorem + separability leads to contradiction. In general, unbounded utility functions violate extensions of standard rationality axioms to prospects with infinitely many possible outcomes, and these extensions can be motivated pretty much the same ways as the versions here, in the vNM utility theorem and Savage’s theorem. See my post here.