Infinities in ethics pose some difficult problems. For example, if the universe is infinite, there are already infinite numbers of good and bad things. Adding or removing finitely many of them leaves infinitely many of both. This means aggregative consequentialist theories (those that maximize the sum of the values of individual structures) will be indifferent between any acts with merely finite effects. If you save the whales, there will be infinitely many whales, but if you don’t save the whales, there will also be infinitely many whales.
Nick Bostrom wrote a paper discussing various possible solutions to this problem of “infinitarian paralysis” (as well as the “fanaticism” problem of theories that would sacrifice anything for a small chance of an infinite payoff). The solutions fall into three classes:
Modifications of the “domain rule” that determines what values are to be aggregated (e.g., discounting values far away in space and time)
Modifications of the “aggregation rule” that determines how these values are to be aggregated (e.g., representing total value as a hyperreal number)
Modifications of the “selection rule” that uses the aggregation result to recommend an action (e.g., ignoring very small probabilities)
The best-known use of infinity in ethics is probably Pascal’s wager, which has a finite variant in Pascal’s mugging.
Infinite Ethics, by Nick Bostrom