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Infinite ethics is notoriously difficult. Here are a few cases that we want our theory to get right.

1 Suppose we can either have a 20% chance of infinite happiness or 40% chance of infinite happiness. .2xinfinity=infinity and .4xinfinity=infinity. So it seems like mathematically they might be the same, but logic says they’re not.

2 Suppose there is already infinite happiness and suffering. This would mean that, on utilitarianism, whether or not you torture an infant, the amount of happiness and suffering is still infinite. However, infinite people should not make it okay to torture infants for fun.

3 Should we kill everyone for a 1 in a billion chance of infinite joy? Intuitively it seems like the answer is no.

4 Is a world with infinite people with 5 utility better than one with infinite people with 3 utility? If so, why?

5 Suppose that some action will in year 1 cause a person 1 unit of happiness, in year 2 cause -2 units of happiness, in year 3 cause 4 units of happiness, in year 4 cause -8 units of happiness, in year 5 cause 16 units of happiness, etc going on forever. Is that action good? If it’s neutral then would an action that caused 1 unit of happiness in year 1, 2 units of happiness in year 2, 4 in year 3, -8 in year 4, 16 in year 5, -32 in year 6, etc going to infinity also be neutral. If yes, then the actions goodness is unaffected by adding 4 extra units of utility while keeping everything else the same.

Here are my solutions. These are pretty tentative, so they might not work.  This is cross posted from my blog.  

1

The 40% chance would be better. For infinity, we can treat x like a variable and have x go to infinity. So .4x-.2x= positive infinity as x goes to infinity.

This is one way out of certain infinitarian quandries.

2

Torturing infants for fun is still bad. When comparing two infinite overlapping sets, one should ignore the overlapping elements of the set. Thus if we have two worlds with the same infinite other things and one with an infant being tortured and one with an infant not being tortured, then when comparing them, we should ignore the overlapping infinite other things, and just compare the infant being tortured to the infant not being tortured. This is both theoretically elegant and solves most issues around infinity. It also respects an intuitive principle which is that causally isolated events shouldn’t affect the morality of an action—it’s the intuition behind the Egyptology objection.

3

Yes. The reasons given are spelled out here.

4

Yes! We can do the overlapping sets method. 3 utility is overlapping in both cases and we care equally about 3 utility in both cases. Thus, the 5 utility case is better because the 3 utility is overlapping in both cases, so the difference is 2 utility for infinite people.

We can also employ the mathematical method given before. 5x-3x tends towards infinity as x goes to infinity.

5

Unsure. This is basically equivalent to -2^x as x goes to infinity. This would be undefined. However, -2^x +1 as x goes to infinity would also be undefined, though zero doesn’t equal 1.

Maybe for ethics, like mathematics this is just undefined and morality can’t evaluate it. However, adding the 1 would make it good…

I don’t have a great solution here. My tentative current view would be infinity shades for functions that are undefined like this one—so for this case, we just ignore it and then care about the 1.

It’s not a great solution but it’s something.

Anyone else have better ideas?