Three Conflicting Principles in Infinite Worlds

When we try to compare infinitely-large populations of people with non-zero utility, principles that coincide in the finite case are sometimes not jointly satisfiable. For example, consider the following three highly plausible, restricted principles (I can expand on why each of them is plausible when I consider which we should reject)::

Agential Pareto: If the populations of $w_1$ and $w_2$ contain exactly the same people and (i) for each person $P_i$ in $w_1$ and $w_2$, $U_{w_1}(P_i) \succcurlyeq U_{w_2}(P_i)$, and (ii) there exists a person $P_j$ in $w_1$ and $w_2$ such that $U_{w_1}(P_j) \succ U_{w_2}(P_j)$, then $w_1$ is strictly better than $w_2$ ($w_1 \succ w_2$).

Agential Isomorphism: If the populations of worlds $w_1$ and $w_2$ are completely disjoint and there is a one-to-one mapping from $w_1$ to $w_2$ such that each person at $w_1$ has a corresponding person in $w_2$ that has at least as much value in $w_2$ as they do in $w_1$, then $w_1$ is not strictly better than $w_2$ ($w_1 \nsucc w_2$).

Transitivity: If $w_1 \succcurlyeq w_2$ and $w_2 \succcurlyeq w_3$ then $w_3 \nsucc w_1$

But if worlds can contain infinitely many people then we can construct a problem case for these three principles. Suppose that $x$'s and $y$'s are different groups of agents, and we are to that there are infinitely many of these agents (i.e. the worlds contain infinitely many $x_1$ agents, infinitely many $x_2$ agents and so on). Now consider the following three worlds:

	$x_0$	$x_1$	$x_2$	$y_0$	$y_1$
$w_1$	1	1	2	0	0
$w_2$	0	0	0	1	2
$w_3$	1	2	2	0	0

We can see that, by Agential Pareto, $w_3 \succ w_1$. But Agential Pareto is silent on the relations between $w_1$ and $w_2$ since they contain distinct populations.

But since the populations of $w_1$ and $w_2$ are distinct, and so are the populations of $w_2$ and $w_3$, Agential Isomorphism applies. And it is easy to see that $w_1$ and $w_2$ are isomorphic copies of one another, and that $w_2$ and $w_3$ are also isomorphic copies of one another. So $w_1$ is at least as good as $w_2$ by Agential Isomorphism. And $w_2$ is at least as good as $w_3$ by Agential Isomorphism. But now we have it that $w_1 \succcurlyeq w_2$, $w_2 \succcurlyeq w_3$ and $w_3 \succ w_1$, which violates Transitivity.

In the finite population cases, these three principles are jointly satisfiable. But in infinite population cases, any theory that attempts to aggregate over people - which come in no natural order - will violate one of these three principles in the kind of case given above. I'll try to expand on this problem in the next post.