It’s still going to be hard to convince me that the sum of an infinite number of days, each of which has infinite positive utility and finite negative utility, will ever be lower than zero.
Remember you’re not allowed to talk about infinity except as a limit.
Clearly the limit of this sequence is the sum you’re talking about (that is, the utility of an infinite number of immortal people who start in sphere A, where we move one each day to sphere B). At the same time, clearly the limit of this sequence is negative.
(Of course the “real” answer is that it’s not well defined; there are many sequences we can construct that come out as “infinite immortal people” in the limit, and the utility is different depending which we pick. But this is an example of why “lower than zero” is as legitimate an answer as “higher than zero”).
Except that in the original problem, there cannot be more days than people;
...each day, one more person gets permanently transferred across to... (Emphasis added.)
Then again, rephrasing the problem in equivalent ways has interesting effects:
Assume that the number of people is countable; assign each of them a natural number, but don’t tell them which one it is.
Suppose that on the Nth day you move the person with the Nth prime number to the opposite sphere; every individual prefers case 1, where they have 100% chance of infinite happiness.
Suppose that you tell everyone their number and on the Nth day you move the person with the Nth number to the opposite sphere; every individual prefers case 2, where they have a finite period of agony followed by infinite happiness.
What’s the Erdos number of an infinite number of monkeys juggling an infinite number of bananas?
Remember you’re not allowed to talk about infinity except as a limit.
Consider the sequence:
etc.
Clearly the limit of this sequence is the sum you’re talking about (that is, the utility of an infinite number of immortal people who start in sphere A, where we move one each day to sphere B). At the same time, clearly the limit of this sequence is negative.
(Of course the “real” answer is that it’s not well defined; there are many sequences we can construct that come out as “infinite immortal people” in the limit, and the utility is different depending which we pick. But this is an example of why “lower than zero” is as legitimate an answer as “higher than zero”).
Except that in the original problem, there cannot be more days than people;
Then again, rephrasing the problem in equivalent ways has interesting effects:
Assume that the number of people is countable; assign each of them a natural number, but don’t tell them which one it is.
Suppose that on the Nth day you move the person with the Nth prime number to the opposite sphere; every individual prefers case 1, where they have 100% chance of infinite happiness.
Suppose that you tell everyone their number and on the Nth day you move the person with the Nth number to the opposite sphere; every individual prefers case 2, where they have a finite period of agony followed by infinite happiness.
What’s the Erdos number of an infinite number of monkeys juggling an infinite number of bananas?