Imagine that you are standing in front of two buttons, deciding which one to press. If you press the first button, a new universe – call it Universe A – will come into existence, in addition to our current universe. If you press the second button, Universe B will come into existence instead. Universe A contains a set of people, each with their own desires, personalities, circumstances, and levels of wellbeing. Universe B contains an entirely different set of people. Assuming that the new universe will be totally independent of our own, which universe should you bring into existence?

Here is one simple response: you should bring into existence whichever universe has a higher total wellbeing across all of its inhabitants. This is an appealing solution, but it runs into problems if we allow Universe A and Universe B to be infinitely large. Suppose, for example, that Universe A contains infinitely many people, each with 1 unit of wellbeing. Then the “total wellbeing” in Universe A is undefined – it is infinitely large. If Universe B also contains an infinite amount of wellbeing, how should we compare the two?

One option is to throw up our hands and say “both universes contain infinite amounts of wellbeing, so they are both equally good”. But this leads to absurd judgements in some cases. For example, if Universe A contains infinitely many people each with 1 unit of wellbeing, and Universe B contains infinitely many people each with 100 units of wellbeing, it is clear that Universe B is better than Universe A, despite the fact that both contain infinite amounts of wellbeing.

We can represent the total wellbeing in an infinite universe by using infinite sums. In the example we are considering, the total wellbeing in Universe A would be 1 + 1 + 1 + 1…, and the total wellbeing in Universe B would be 100 + 100 + 100 + 100…. We can justify our intuitive judgement that Universe B is better than Universe A by saying that this is because its infinite sum dominates Universe A’s infinite sum – every term of Universe B’s sum is greater than every term of Universe A’s. But what if Universe A’s infinite sum was 1 + 1 + 101 + 1 + 1 + 101… instead?

Now, not all infinite universes will generate these problems, because some infinite sums converge to finite values. For example, a universe whose wellbeing is represented by the infinite sum 1/2 + 1/4 + 1/8 + 1/16… will have a total wellbeing of 1.

But these universes have their own special problems.

It seems obvious that the moral value of a universe should not depend on the order in which we sum up the wellbeings of its inhabitants. Unfortunately, order matters for infinite sums. Consider the alternating harmonic series, 1 – 1/2 + 1/3 – 1/4…. This series converges to a value of about 0.7. But by rearranging its terms, we can actually get it to converge to any finite value we want!

At this point things are just getting ridiculously weird.

What should we make of a universe that contains people whose levels of wellbeing correspond to the terms of the alternating harmonic series? It seems that it is both better than any finite universe (since we can rearrange it to make its limit bigger than any finite number) and worse than any finite universe (since we can rearrange it to make its limit smaller than any finite number). This is absurd.

As these examples have hopefully illustrated, ethics becomes very difficult when applied to infinite universes. We are not yet sure how to approach aggregating infinite amounts of wellbeing. But you can be sure that philosophers are working on the problem.