¹ The sum is meaningless because it fails to converge to any finite or infinite value.

² Journal of Philosophy, xciv (1997)Google Scholar. We will follow Vallentyne and Kagan's use of location as the generic term for the things with which local goodness is associated, such as people, points in space or times, etc.

³ In a footnote, they give two such principles (p. 20, n. 18). Also, they particularly emphasize that GM states only sufficient conditions for one world to be judged better than another and state that they ‘would be very surprised if it cannot be strengthened’ (p. 21). And again: ‘we certainly do not think that GM is the strongest plausible metaprinciple for dealing with unbounded locations’ (p. 25). They also point out that GM is silent in certain cases.

⁴ In the case of finite worlds the usual theory of utilitarianism is complete, because the better of any two worlds is simply the one with the greater total utility; and if two worlds have the same total utility, then they are equally good. When comparing finite worlds, therefore, utilitarianism is never silent.

⁵ In a forthcoming paper, ‘With Infinite Utility, More Isn't Always Strictly More’, we take the stronger position that the Basic Idea itself contradicts natural principles that utilitarians would want to accept.

⁶ They use the term isometric counterpart function.

⁷ One can define the notion of boundedness in any metric space. All the example worlds in this paper, however, will have a particularly simple structure, making them isomorphic to the natural numbers 0,1,2,…, and so the one-dimensional idea of intervals will suffice.

⁸ While Vallentyne and Kagan refer to their principle as a metaprinciple, we prefer to call it a principle, reserving the term metaprinciple for principles about principles. We have taken the liberty of reformulating GM to an equivalent statement.

⁹ Here are two even stronger principles. The Subworld Principle: If every location in U is in V, and every location in V which is not in U has non-negative goodness, then V is at least as good as U. And the Strong Subworld Principle: If A and B have no locations in common and B is at least as good as its zero world, then AUB is at least as good as A. Since they both imply AG, our argument that GM contradicts AG will also show that GM contradicts them.

¹⁰ Vallentyne and Kagan, 20, n. 18.

¹¹ This is the hotel with ω many rooms which can always accommodate a new guest, even when all rooms are taken, by asking all the guests to move to the next room, thereby vacating the first. Indeed, when a bus arrives with to many new guests, all can be accommodated by asking each current guest to move to the room whose number is twice the number of his or her current room, and putting the new guests in the odd-numbered rooms. It is a fun exercise to figure out how the hotel can handle ω many buses, each with ω many new guests.

¹² In our forthcoming paper, ‘With Infinite Utility, More Isn't Always Strictly More’, we argue that each of the worlds U and V is at least as good as the other; that is, they are equally good. Thus, while the extra money does not make you any poorer, neither in this case does it make you any richer.

¹³ We assume both here and in AB and AG, of course, as do Vallentyne and Kagan (see, e.g., 7), the comparability and equivalence of zero goodness at different locations.

¹⁴ Their example 3, at 10.

¹⁵ Vallentyne and Kagan, 20, n. 18.

¹⁶ This means, in the case that negative goodness is added, that if what is added to U is less bad than what is added to V, then U' remains at least as good as V'.