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Salutator comments on A Pure Math Argument for Total Utilitarianism - Less Wrong
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Comments (36)
Two points:
I don't know the Holder theorem, but if it actually depends on the lattice being a group, that includes an extra assumption of the existence of a neutral element and inverse elements. The neutral element would have to be a life of exactly zero value, so that killing that person off wouldn't matter at all, either positively or negatively. The inverse elements would mean that for every happy live you can imagine an exactly opposite unhappy live, so that killing off both leaves the world exactly as good as before.
Proving this might be hard for infinite cases, but it would be trivial for finite generating groups. Most Less Wrong utilitarians would believe there are only finitely many brain states (otherwise simulations are impossible!) and utility is a function of brain states. That would mean only finitely many utility levels and then the result is obvious. The mathematically interesting part is that it still works if we go infinite on some things but not on others, but that's not relevant to the general Less Wrong belief system.
(Also, here I'm discussing the details of utilitarian systems arguendo, but I'm sticking with the general claim that all of them are mathematically inconsistent or horrible under Arrow's theorem.)
Z^2 lexically ordered is finitely generated, and can't be embedded in (R,+). [EDIT: I'm now not sure if you meant "finitely generated" or "finite" here. If it's the latter, note that any ordered group must be torsion-free, which obviously excludes finite groups.]
But your implicit point is valid (+1) - I should've spent more time explaining why this result is surprising. Just about every comment on this article is "this is obvious because <some proof which is invalid>", which I guess is an indication LWers are so immersed in utilitarianism that counter-examples don't even come to mind.
I'm a bit out of my depth here. I understood an "ordered group" as a group with an order on its elements. That clearly can be finite. If it's more than that the question would be why we should assume whatever further axioms characterize it.
from wikipedia:
So if a > 0, a+a > a etc. which results means the group has to be torsion free.