it would be trivial for finite generating groups… That would mean only finitely many utility levels and then the result is obvious
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 ”, 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.
a partially ordered group is a group (G,+) equipped with a partial order “≤” that is translation-invariant; in other words, “≤” has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b
So if a > 0, a+a > a etc. which results means the group has to be torsion free.
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 ”, 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.