In this post I will use revealed preference and assume people are rational. Sorry. I’m an economist and I can’t entirely resist.
I will argue that our joy and suffering is bounded, that we can’t be infinitely happy or infinitely miserable.
The first argument is standard, the second is something Peter Mollgaard thought of the instant I explained the first.
Both discussions are typically limited (as economists tend to be) to selfish swinish agents who care only about consumption. This has nothing to do with the argument.
The proof that we are not rational utility maximizers with utility unbounded above follows. It is called the St Petersburg paradox.
If we were capable of unbounded joy, for any positive epsilon no matter how small there is a lottery which we would rationally play which gave us horrible pain with probability 1-epsilon and good outcomes with probabilities adding up to epsilon. The trick is an infinite series of good outcomes. goodoutcome(i) for i a natural number. goodoutcome(i) occurs with probability epsilon/2 and gives us happiness 2/epsilon. Goodoutcome occurs with probability 2^(-i)epsilon and gives us happiness 2^(i)/epsilon
So expected happiness if we take the bet is (1-epsilon)(a very large negative number) + the sum from one to infinity of 1. That is infinite expected happiness.
We don’t make such bets. What if epsilon = 1/google that is 10^(-100) ? 10^100 is the original meaning of the word “google”. The company chose the word for that number exactly as the Apple corporation chose the name of a fruit.
what if epsilon = 1/(google plex)= 10^(-google) = 10^(-10^100). or one over a google plex plex and so on.
Similarly, we are not rational utility maximizers capable of infinite misery. For any X>0 no matter how huge and any epsilon >0 no matter how tiny, we would accept happiness of -X with certainty rather than + X with probability 1-epsilon and a series of increasingly horrible outcomes
getting -2/epsilon with probability epsilon/2 and -2^i/epsilon with probability 2^(-i) so if we didn’t take -X with certainty our expected welfare is minus infinity.
Again epsilon can be so low that if something happened with probability epsilon every millisecond, the chance that it has happened since the big bang is 1/(google plex). I guess I should call the second paradox the Mollgaard paradox, but I want to call it the Leningrad paradox, because Leningrad is St Petersburg and because the limits of human misery were well explored by the Stalinist purge of Leningrad followed soon after by the Nazi (and Finnish) seige of Leningrad. We are talking about two of the most massive losses of human life ever (and much slow death by borderline fatal malnutrition become fatal after long suffering). But it would be worth a 1/google plex risk of that happening again even if there were a series of risks of it happening once, twice, four times etc.
This has been another explanation of why the concept of infinity is pernicious following in the tradition of Zeno’s paradox but never reaching the end of the list of errors due to the concept of infinity (which list is infinite). Borges tried to catalogue them (he does that sort of thing) in “Avatars of the Tortoise.”
I think it is also not needed. But I am confident with probability epsilon (not a type not 1-epsilon) that every counterintuitive result in mathematics has something to do with infinity and could not be explained if instead of natural numbers we started with the cyclical group with N elements (no matter how big N is) and then went on to define rational and real numbers as we do.