Summary

• The possibility of infinite worlds wreaks havoc in ethical decisions.
• However, I prefer rejecting the existence of infinite worlds because I believe:
  • They imply intuitively nonsensical claims to be true.
  • There is no evidence for them.
  • They are not intuitively appealing.

Acknowledgements

Thanks to Michael St. Jules, who wrote this comment which ended up motivating me to think about infinite ethics.

The threat of infinitarian paralysis

Infinite ethics studies the ethical implications of living in an infinite universe. One seemingly common theme is claims being both strongly believed to be true based on intuitions, and logically impossible together. For example, Amanda Askell’s PhD thesis shows that we cannot jointly accept the following 5 axioms^[1] (see p. 180):

• Transitivity of ≽. If w1 ≽ w2 and w2 ≽ w3 then w1 ≽ w3 [w1 is better than or as good as w3].
• Permutation Principle. For any world pair ⟨w1, w2⟩ and any bijection g from the population of ⟨w1, w2⟩ onto any population, there exists a world pair ⟨w3, w4⟩ that is a qualitative duplicate of ⟨w1, w2⟩ under bijection g.
• Qualitativeness of ≽. If the pair ⟨w3, w4⟩ is a qualitative duplicate^[2] of the pair ⟨w1, w2⟩, then w3 ≽ w4 if and only if w1 ≽ w2.
• Pareto Principle. If w1 and w2 have identical populations and for all agents x in w1 and w2, uw1(x) ≥ uw2(x) [utility of w1 greater or equal than that of w2], then w1 ≽ w2. If there is also some agent x in w1 and w2 such that uw1(x) > uw2(x), then w1 ≻ w2.
• Minimal Completeness of ≽. Comparability between infinite worlds (w1 ≽ w2 or w2 ≽ w1) is not incredibly rare.

Moreover, if any action has a non-null chance of leading to both positive and negative utility, expected value calculations will always result in an indeterminate form of the type inf - inf. Not ideal in case we want to compare anything at all!

The implicit acceptance of the possibility of infinite worlds

Axioms cannot be proved to be true or false. Rather, they provide a framework for assessing what is true or false. That being said, one can add/drop/update axioms so that the truthfulness of some claims matches our strongest intuitions.

Somewhat obviously but crucially, all problems in infinite ethics arise from the implicit acceptance of the possibility (i.e. non-null chance) of infinite worlds. Rejecting such worlds, one can, for instance, hold as true the aforementioned 5 axioms studied in Amanda’s thesis.

Rejecting the possibility of infinite worlds

You may be thinking that rejecting the possibility of infinite worlds is not reasonable because we have non-null evidence of their existence. You would be in good company. According to Bostrom 2011:

We do not know for sure that we live in a canonically infinite world. Contemporary cosmology is in considerable flux, so its conclusions should be regarded as tentative. But it is definitely not reasonable, in light of the evidence we currently possess, to assume that we do not live in a canonically infinite world.

However, I actually believe we have zero evidence about the existence of infinite worlds:

• In maths, infinity is one of the axioms of ZMC set theory^[3]. So it is assumed true by definition, and accepting/rejecting it is not supported by any evidence.
• In physics, all measurements have a finite sensitivity (smallest detectable variation) and range (minimum and maximum detectable value). So neither zero nor infinity are measurable. In other words, their existence is not falsifiable.

One can produce infinities in physical laws by setting some variables to zero (1/0 = inf, and ln(0) = -inf), but that does not mean we have evidence for them. As explained in Ellis 2018, zeros and infinities are placeholders for very small and large quantities. We could explain reality just as well by replacing all zeros we have in physical laws by the very small number VS = 10^-10^10^10^10^10^10^10^10^10. Of course, I do not think there is a need for that. 0 is more concise and aesthetically appealing.

I can try to illustrate the point above. Suppose I hypothesise that all current human adults have a mass between 0.01 kg and 100 t. I am pretty confident all data we have corroborates this hypothesis^[4], and therefore the probability of it being true is essentially 1. Widening the mass interval of the hypothesis would make it more likely to be true, but this would not continue forever. We have a finite amount of evidence. For example, expanding the interval beyond VS kg to 1/VS kg would arguably not cover any additional evidence^[5].

Furthermore, infinities are compatible with the size of the whole being equal to that of each of its parts (e.g. inf = inf/2), and I find that quite unintuitive. Some types of infinity allow for the Banach–Tarski paradox.

This is often stated informally as “a pea can be chopped up and reassembled into the Sun”

All in all, I prefer rejecting the existence of infinite worlds because I believe:

• They imply intuitively nonsensical claims to be true. For instance, I cannot see how one of Amanda’s 5 axioms would be false.
• There is no evidence for them.
• They are not intuitively appealing. My senses, as physical instruments, are only able to perceive finite quantities.

1. ^{^}

    I liked listening to this episode of Amanda on The 80,000 Hours Podcast.

2. ^{^}

    One world pair is a ‘qualitative duplicate’ of the other if and only if it has all of the same qualitative properties and relations as the original world pair.

3. ^{^}

    The most common axiomatic set theory according to Wikipedia.

4. ^{^}

    According to Guinness World Records: the lightest person with an age over 16, Lucia Xarate, weighed 2.13 kg at 17; the heaviest human ever, Jon Minnoch, weighed 635 kg.

5. ^{^}

    The mass of the observable universe is 1.5*10^53 kg, which is only 10^10^1.73 kg << 1/VS kg. The upper limit presented in Workman et al. 2022 for the mass of the photon is 1.78*10^-44 kg (= 1*10^-18*1.602176634*10^-19/299792458^2), which is still 10^-10^1.64 kg >> VS kg.