Pareto: If two worlds (w1 and w2) contain the same people, and w1 is better for an infinite number of them, and at least as good for all of them, then w1 is better than w2.
As far as I can see, the Pareto principle is not just incompatible with the agent-neutrality principle, it’s incompatible with set theory itself. (Unless we add an arbitrary ordering relation on the utilities or some other kind of structure.)
Let’s take a look at, for instance, N∪0N vs 2N∪0N, where nN is the multiset containing n,2n,3n,… and ∪ is the disjoint union. Now consider the following scenarios:
(a) Start out with N∪0N and multiply every utility by 2 to get 2N∪0N. Since infinitely many people are better off and no one is worse off, N∪0N≺2N∪0N.
(b) Start out with 2N∪0N and take every other of the 0-utilities from 0N and change them to 1,3,5,…=2N−1. Since a copy of 0N is still left over, this operation leaves us with N∪0N. Again, since infinitely many are better off and no one worse off, 2N∪0N≺N∪0N.
In conclusion, both 2N∪0N≺N∪0N and N∪0N≺2N∪0N, a contradiction.
In (b) the remaining copy of 0N is specifically missing those “upgraded individuals”. They might contain the same number of people but it is not clear to me that they contain the same people. Thus (b) is not an instance of applying pareto.
I don’t understand what you mean. The upgraded individuals are better off than the non-upgraded individuals, with everything else staying the same, so it is an application of Pareto.
Now, I can understand the intuition that (a) and (b) aren’t directly comparable due to identity of individuals. That’s what I mean with the caveat “(Unless we add an arbitrary ordering relation on the utilities or some other kind of structure.)”
Okay the pareto thing applies but the formal contradiction has a problem in the (b) prong. Consider 1N which is 1,2,3,4,5,6,7… if you took each other out from that yu would get 1,3,5,7… There is no 2 in there so there is no copy of 1N remaining in there. Sure if you have 0N which is 0,0,0,0,0… you can have it as a multiset but multisets track amounts. It is not sufficient that the members are of the same object the amount needs to match too. And in that dropping the amounts (atleast ought to) change. So 0N2 is not the same as 0N. So you get 2N∪0N≺N∪0N2 which is not an exact mirror of the (a) prong.
The number of elements in 0N won’t change when removing every other element from it. The cardinality of 0N is countable. And when you remove every other element, it is still countable, and indistinguishable from 0N. If you’re unconvinced, ask yourself how many elements 0N with every other element removed contains. The set is certainly not larger than N, so it’s at most countable. But it’s certainly not finite either. Thus you’re dealing with a set of countably many 0s. As there is only one such multiset, 0N equals 0N with every other element removed.
That there is only one such multiset follows from the definition of a multiset, a set of pairs (a,c), where a is an element and c is its cardinality. It would also be true if we define multisets using sets containing all the pairs (a,1),(a,2),… -- provided we ignore the identity of each pair. I believe this is where our disagreement lies. I ignore identities, working only with sets. I think you want to keep the identities intact. If we keep the identities, the set {(0,1),(0,2),(0,3),…} is not equal to {(0,1),(0,3),(0,5),(0,7),…}, and my argument (as it stands) fails.
To my mind the reduced set has ω2 elements which is less than ω. But yeah its part of a bigger pattern where I don’t think cardinality is a very exhaustive concept when it comes to infinite set sizes. But I don’t have that much knowledge to have a good alternative working conception around “ordinalities”.
Pareto explicitly says that you have to keep identities intact, because the definition stipulates that w1 and w2 “contain the same people.” If you don’t preserve identities, you can’t verify that that condition is met, in which case Pareto isn’t applicable.
Yeah, so Pareto seems to require that we don’t just think about the people in the universe in terms of set theory as you do, but instead maybe have something like a canonical order in which to compare people between universes… that seems to work for comparing worlds where (roughly) the people are the same but their utilities change; I’m not sure how to compare universes with people arranged differently using something like this set theory. Ideally we could think of infinite utilities as hyperreal numbers rather than in terms of sets; then there’s no contradiction of this form.
I think full agent-neutrality/multisets will make it basically impossible for anything to matter in practice assuming the universe is infinite (maybe you can only care about finite universes, though). You’d need to change the number of individuals at some utility level to make any difference, but if the universe is infinite, the number of individuals at any given utility level is probably infinite, and you probably won’t be able to change its cardinality predictably through normal acts that don’t predictably affect weird possibilities of jumping between different infinite cardinals.
If a hyperreal approach gets past this, then it probably assumes additional structure, effectively an order.
Multisets don’t track identity and effectively bake agent-neutrality into them, so they don’t have enough structure to express the Pareto principle properly. For Pareto, it’s better to represent your worlds/universes/outcomes as vectors with a component for each individual, or functions mappings identities (or labels) to real numbers. Your set of identities can just be the natural numbers, or integers or whatever.
As far as I can see, the Pareto principle is not just incompatible with the agent-neutrality principle, it’s incompatible with set theory itself. (Unless we add an arbitrary ordering relation on the utilities or some other kind of structure.)
Let’s take a look at, for instance, N∪0N vs 2N∪0N, where nN is the multiset containing n,2n,3n,… and ∪ is the disjoint union. Now consider the following scenarios:
(a) Start out with N∪0N and multiply every utility by 2 to get 2N∪0N. Since infinitely many people are better off and no one is worse off, N∪0N≺2N∪0N.
(b) Start out with 2N∪0N and take every other of the 0-utilities from 0N and change them to 1,3,5,…=2N−1. Since a copy of 0N is still left over, this operation leaves us with N∪0N. Again, since infinitely many are better off and no one worse off, 2N∪0N≺N∪0N.
In conclusion, both 2N∪0N≺N∪0N and N∪0N≺2N∪0N, a contradiction.
In (b) the remaining copy of 0N is specifically missing those “upgraded individuals”. They might contain the same number of people but it is not clear to me that they contain the same people. Thus (b) is not an instance of applying pareto.
I don’t understand what you mean. The upgraded individuals are better off than the non-upgraded individuals, with everything else staying the same, so it is an application of Pareto.
Now, I can understand the intuition that (a) and (b) aren’t directly comparable due to identity of individuals. That’s what I mean with the caveat “(Unless we add an arbitrary ordering relation on the utilities or some other kind of structure.)”
Okay the pareto thing applies but the formal contradiction has a problem in the (b) prong. Consider 1N which is 1,2,3,4,5,6,7… if you took each other out from that yu would get 1,3,5,7… There is no 2 in there so there is no copy of 1N remaining in there. Sure if you have 0N which is 0,0,0,0,0… you can have it as a multiset but multisets track amounts. It is not sufficient that the members are of the same object the amount needs to match too. And in that dropping the amounts (atleast ought to) change. So 0N2 is not the same as 0N. So you get 2N∪0N≺N∪0N2 which is not an exact mirror of the (a) prong.
The number of elements in 0N won’t change when removing every other element from it. The cardinality of 0N is countable. And when you remove every other element, it is still countable, and indistinguishable from 0N. If you’re unconvinced, ask yourself how many elements 0N with every other element removed contains. The set is certainly not larger than N, so it’s at most countable. But it’s certainly not finite either. Thus you’re dealing with a set of countably many 0s. As there is only one such multiset, 0N equals 0N with every other element removed.
That there is only one such multiset follows from the definition of a multiset, a set of pairs (a,c), where a is an element and c is its cardinality. It would also be true if we define multisets using sets containing all the pairs (a,1),(a,2),… -- provided we ignore the identity of each pair. I believe this is where our disagreement lies. I ignore identities, working only with sets. I think you want to keep the identities intact. If we keep the identities, the set {(0,1),(0,2),(0,3),…} is not equal to {(0,1),(0,3),(0,5),(0,7),…}, and my argument (as it stands) fails.
To my mind the reduced set has ω2 elements which is less than ω. But yeah its part of a bigger pattern where I don’t think cardinality is a very exhaustive concept when it comes to infinite set sizes. But I don’t have that much knowledge to have a good alternative working conception around “ordinalities”.
Pareto explicitly says that you have to keep identities intact, because the definition stipulates that w1 and w2 “contain the same people.” If you don’t preserve identities, you can’t verify that that condition is met, in which case Pareto isn’t applicable.
Yeah, so Pareto seems to require that we don’t just think about the people in the universe in terms of set theory as you do, but instead maybe have something like a canonical order in which to compare people between universes… that seems to work for comparing worlds where (roughly) the people are the same but their utilities change; I’m not sure how to compare universes with people arranged differently using something like this set theory. Ideally we could think of infinite utilities as hyperreal numbers rather than in terms of sets; then there’s no contradiction of this form.
I think full agent-neutrality/multisets will make it basically impossible for anything to matter in practice assuming the universe is infinite (maybe you can only care about finite universes, though). You’d need to change the number of individuals at some utility level to make any difference, but if the universe is infinite, the number of individuals at any given utility level is probably infinite, and you probably won’t be able to change its cardinality predictably through normal acts that don’t predictably affect weird possibilities of jumping between different infinite cardinals.
If a hyperreal approach gets past this, then it probably assumes additional structure, effectively an order.
Multisets don’t track identity and effectively bake agent-neutrality into them, so they don’t have enough structure to express the Pareto principle properly. For Pareto, it’s better to represent your worlds/universes/outcomes as vectors with a component for each individual, or functions mappings identities (or labels) to real numbers. Your set of identities can just be the natural numbers, or integers or whatever.