Infinity is big. You just won't believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to infinity.

And there are a lot of paradoxes connected with infinity. Here we'll be looking at a small selection of them, connected with infinite ethics.

Suppose that you had some ethical principles that you would want to spread to infinitely many different agents - maybe through acausal decision making, maybe through some sort of Kantian categorical imperative. So even if the universe is infinite, filled with infinitely many agents, you have potentially infinite influence (which is more than most of us have most days). What would you do with this influence - what kind of decisions would you like to impose across the universe(s)'s population? What would count as an improvement?

There are many different ethical theories you could use - but one thing you'd want is that your improvements are actual improvements. You wouldn't want to implement improvements that turn out to be illusionary. And you certainly wouldn't want to implement improvements that could be undone by relabeling people.

How so? Well, imagine that you have a countable infinity of agents, with utilities (..., -3, -2, -1, 0, 1, 2, 3, ...). Then suppose everyone gets +1 utility. You'd think that giving an infinity of agents one extra utility each would be fabulous - but the utilities are exactly the same as before. The current -1 utility belongs to the person who had -2 before, but there's still currently someone with -1, just as there was someone with -1 before the change. And this holds for every utility value: an infinity of improvements has accomplished... nothing. As soon as you relabel who is who, you're in exactly the same position as before.

But things can get worse. Subtracting one utility from everyone also leaves the outcome the same, after relabeling everyone. So this universal improvement is completely indistinguishable from a universal regression.