As stated in another comment, you have proved any ethical theory that is identical to total utilitarianism with fixed population sizes (e.g average utilitarianism).
But, you can use separability to rule out non-total versions of utilitarianism.
Separability is roughly the principle that, in comparing the value of two outcomes, one can ignore any people whose existence and welfare are unaffected.
Non-total versions of utilitarianism violate separability because they imply that the value of creating someone depends on the population or wellbeing of unaffected beings.
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.
As stated in another comment, you have proved any ethical theory that is identical to total utilitarianism with fixed population sizes (e.g average utilitarianism).
But, you can use separability to rule out non-total versions of utilitarianism.
Separability is roughly the principle that, in comparing the value of two outcomes, one can ignore any people whose existence and welfare are unaffected.
Non-total versions of utilitarianism violate separability because they imply that the value of creating someone depends on the population or wellbeing of unaffected beings.
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.