Utilitarianism, if it is to be a proper paradigm of ethical reasoning, must apply in an abstract sense. It cannot be that there exist certain cases, certain states of the world, where the paradigm falls apart, because if so, the theory cannot be said to be an ultimate foundation for all moral reasoning.
In this argument, I will argue that there do exist certain cases where Utilitarianism becomes useless. That is, within these scenarios, Utilitarianism cannot be use to perform any sort of moral reasoning, or, if it is attempted to be used in such a way, it produces useless, trivial gibberish.
The argument is simple, but its genius mustn't be denied. We need two propositions, from which the argument then follows readily.
The first proposition, which is just the definition of Utilitarianism rephrased.
First proposition. Let A be some action and B be some other action. We are to decide which one is morally preferable. Let U(A) and U(B) be the total utility functions, both of which consists of sums of the individual utilities that each person gets under actions A and B. The utilitarian proceeds by comparing U(A) and U(B) and picks the action which the largest possible value. If the values are equal, both actions are equally defensible from a moral standpoint.
The second proposition.
Second proposition. It is possible that infinitely many people exist with strictly positive utilities while there does not exist infinitely many people with strictly negative utilities.
This is the certain case mentioned earlier. Indeed, it is certainly possible that infinitely many people or "organisms" exist and, under both actions A and B, these people have strictly positive utilities, while the opposite (i.e. infinitely many people with strictly negative utilities) does not occur. There's no particular reason why this is impossible, at least not within the framework of Utilitarianism. I am not saying it is likely, or that I believe this to be true, but its possibility is all that is needed.
Now, combining these two propositions, a knockdown argument of Utilitarianism emerges. It is the following.
Argument. If infinitely many people with strict utilities can exist, then both U(A) and U(B) are in this case equal to infinity, since they are the (countably) infinite sum of individual utilities, and infinitely many of these utilities are strictly positives, and hence the sum is necessarily infinity. Hence, Utilitarianism collapses. It cannot perform moral reasoning, because to ask whether A is better than B corresponds to asking whether infinity is bigger than infinity: this is gibberish. In fact, it's worse than gibberish, because even if we add to action A the murder of a person, then no matter how much of a loss of utility this is, the Utilitarian still cannot say that B is now better than A, because infinity minus a billion is still infinity. So U(A) is still equal to U(B).