# 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
bijectiong 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.

^{^}I liked listening to

__this__episode of Amanda on The 80,000 Hours Podcast.^{^}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.

^{^}The most common axiomatic set theory according to

__Wikipedia__.^{^}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.^{^}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.

As far as I can tell you don't even need to go as far as rejecting all "possibility" of infinite worlds to respond Amanda paradox / infinite worlds paradoxes. All you need to do is put a non-zero possibility on the world being finite.

As long as you have a non-zero chance of finitenes (even if that chance limits to zero, even if it's only a chance in a trillion years time) then you can apply a preference ordering for sooner (or closer events). So the paradoxes I've seen do not apply (or are simply issues to do with using very large non-infinite numbers). See my comments here https://forum.effectivealtruism.org/posts/CeeeLu9PoDzcaSfsC/on-infinite-ethics?commentId=Aw7W2LNxbtqLiMPe7 and other comments on that article. (Maybe I've missed something but I've yet to see a way in which "infinite worlds wreaks havoc in ethical decisions").

Sorry I've not read Amanda's thesis so don't know her language. My best guess is in Amanda's language it means that for any action you can consider that shifts the world from being w1 to w2 where w1 and w2 appear to be (non-identical) qualitative duplicates to an impartial observer, because you are applying some chance of finitenes you are saying there is some chance that they are not truly qualitative duplicates and can then favour one world above the other based on how close to you the positive utility is, so they are not incomparable to you and there are no problems for any actual decision makers.

That said I also think it is reasonable to think the world is finite based on current physics so that's also a good take if it works for you.

I tend towards believing that there almost certainly is infinity existing, due to both the universe being flat and inflation having fairly good observational evidence.

So as far as the infinities I think likely exist, it is the type 1 multiverse due to flatness, and the type 2 multiverse, which is eternal inflation. And in eternal inflation, various universes have quantities that with the right tools can create infinities of what you want. The type 3 multiverse is essentially the many worlds interpretation of quantum mechanics, and I'd assign a 50-55% probability existing. The big type of multiverse that I don't place much probability on existing is the mathematical multiverse/type 4 multiverse, aka all logically/mathematical universes exist. I'd only place a 1-5% probability on it existing.

May be there is some nuance needed. My perhaps out of date understating of physics is that:

I don’t know the types 1-4 of which you talk.

Either way even if you still don’t believe 1 - 3 above the main point I was making about even the possibility of the universe being finite remains sufficient.

I definitely agree that for the most part, we should probably stick to finite realms, because even if it is possible to get infinity, we have no plan of attack on how to do so, and the physics of our universe without changes only allows us finite utility.

Thanks for clarifying.

Even if that is the leading theory, we may be wrong. So I think we cannot rule out non-null utility being possible forever, and therefore there is a chance cumulative utility tending to infinity. However, I think there is an important distinction between something tending to infinity, and something being infinite. If utility tends to infinity, it is still finite at any given time, which means expected value calculations will not lead to unresolvable indeterminate forms.

Yes, in the sense that a physical theory can only suggest infinities as limits. There will never be data pointing towards infinities because these are not (physically) measurable.

This is very much in agreement with

Ellis 2018.Hi Sharmake,

I used to think along these lines, but I no longer think it makes sense to extrapolate this way. When I say a surface (e.g. of a table) is flat, I do not mean it is perfectly flat in a geometrical sense, just that its curvature is sufficiently small for me to call it flat in everyday language.

Similarly, when physicists claim the universe is flat, they just mean there is a very low likelihood that (the absolute value of) its curvature is higher than a very small value. However, that upper bound for the curvature is still infinitely larger than 0. In this sense we have zero evidence about the universe being exactly flat. The density parameter (see here) being Omega = 1 + VS = 1 + 10^-10^10^10^10^10^10^10^10^10 satisfies our data exactly as well as Omega = 1.

"As long as you have a non-zero chance of finitenes (even if that chance limits to zero, even if it's only a chance in a trillion years time) then you can apply a preference ordering for sooner (or closer events)."

Do you mean moral time discounting, so rejecting impartiality (and so Amanda's permutation principle)?

I haven't read Amanda's work so I cannot say for certain but yes this sounds correct. My view would basically equate to moral time discounting. (If you think in a trillion trillion years just maybe the universe might not exist you should discount any good done in a trillion trillion years.)

Since you are not advocating for a pure time/space preference, I do not think you are rejecting impartiality (nor Amanda's Permutation Principle).

Thanks for commenting!

I like the comment you linked, and heartily agree with:

Regarding:

This also seems to work, but I do not think it is fair to apply such preference ordering. I have very high credence on expectational total hedonistic utilitarianism. Impartiality is one of its elements, and I do not think distance in time or space is intrinsically morally relevant.

I think you missed my point a bit. Nothing I said was to challenge impartiality. I am not at all saying that people further away in time and space are any less

intrinsicallymorally relevant, only that if you ascribe some non-zero probability of the universe being finite then you can ascribe a preference ordering. Like all else being identical helping someone now is better than helping someone after the universe might no longer exist, because, you know, the universe might not exist then (they are no less morally relevant). And so ta-da all paradoxes to do with infinite ethics go away as you can not longer shift utility infinitely into the future.You can ignore infinite cases if you assume only the aggregate matters and infinite/undefined cases can't be predictably affected. But, if you're a risk-neutral expected value maximizing total utilitarian, you should be trying to increase the probability of a positive infinity aggregate or reduce the probability of a negative infinity aggregate (or both), and at any finite cost and fanatically.

I don’t see why that is different from saying "But, if you're a risk-neutral expected value maximizing total utilitarian, you should be trying to increase the probability of a [very large number]* aggregate or reduce the probability of a negative [very large number]* aggregate (or both), and at [essentially] any finite cost and fanatically."

I don’t think you need infinities to say that very small probabilities of very big positive (or negative) outcomes messes up utilitarian thinking. (See Pascal's Mugging or Repugnant Conclusion.)

My claim is that any paradox with infinites is very easily resolvable (e.g. by noting that there is some chance the universe is not infinite, etc) or can be reduced to an well known existing ethical challenge (e.g. utilitarianism can get fanatical about large numbers) .

I hope that explains where I am coming form and why I might say that actually you "can ignore infinite cases".

* E.g. TREE(3)

I agree. One does not even need large numbers nor small probabilities. Complex cluelessness is enough to make the result of any expected value calculation quite unclear. However, not totally arbitrary, so I still endorse expectational total hedonistic utilitarianism.

(I've edited this comment somewhat.)

It is pretty much the same, but I don't see why that justifies ignoring infinities, if you maximize total utility risk neutrally. I personally assign <50% weight to fanatical decision theories, so I mostly don't maximize total utility risk neutrally. Maybe you mean something similar (or less tha 100% weight to fanatical views)?

Some people have proposed specific responses to Pascal's mugging and the RC that are more specific to the structures of those problems, but they can't be used to ignore infinities in general.

I am not sure that we disagree here / expect we are talking about slightly different things. I am not expressing any view on fanaticism issues or how to resolve them.

All I was saying is that infinites are no more of a problem for utilitarianism/ethics than large numbers. (If you want to say "infinite" or "TREE(3)" in a thought experiment, well either works.) I am not 100% sure, but based on what you said, I don’t think you disagree on that.

Doesn't infinity make aggregating utilities

undefined, in a way that's not true for just very large numbers? Maybe I'm missing something here though.So what? What thought experiment does this lead to that causes a challenge for ethics? If infinite undefined-ness causes a problem for ethics please specify it, but so far the infinite ethics thought experiments I have seen either:

Are trivially the same as non-infinite thought experiments.For example undefined-ness is a problem for utilitarianism even without infinity. For example think of the Pascal's mugger who offers to create "an undefined and unspecified but very large amount of utility, so large as to make TREE(3) appear small"Make no sense.They require assuming two things that physics says are not true – let us assume that we know with 100% certainty that the universe is infinite and let us assume that we can treat those infinites as anything other than limits in a finite series. This make no more sense than though experiments about what if time travel was true make sense and are little better than what if "consciousness is actually cheesy-bread".Maybe I am missing something and there are for example some really good solutions to Pascal's mugging that don’t work in the infinite case but work in the very large but undefined cases or some other kind of thought experiment I have not seen yet in which case I am happy to retract my skepticism.

Hi Linch,

I would say both "very large unknown positive number x" - "very large unknown positive number y" and inf - inf are undefined. However, whereas the value of 1st difference can in theory be determined by looking into what is generating x and y, the 2nd difference cannot be resolved even in principle.

inf - inf can sometimes be resolved under certain assumptions with richer representations of infinite outcomes, e.g. if both infinities are the result of infinite series over a common ordered index set (e.g. spacetime locations by distance from a specific location, moral patients with some order), you can rearrange the difference of series as a series of differences. This doesn’t always work, because the series of differences may not always have a limit at all.

See:

Right, but I would classify these cases as resolving "very large unknown positive number x" - "very large unknown positive number y". It looks to me that infinite series are endless in the sense that we cannot point to where they end, but they do not contain infinity.

For example, the natural numbers 1, 2, ... go on indefinetely, but any single one of them is still finite, so I would say they can be represented by 1, 2, ..., N, where N is a very large unknown number. From the point of view of physics, I am pretty confident we could assume N = TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3)^TREE(3) while explaining exactly the same evidence.

Saved to watch later. Thanks for sharing!

Ah, sorry for misinterpreting!

I think you are saying that, although utility may exist arbitrarily far away (in time/space), the likelihood of it existing tends to zero as it is gets further and further away from us. So the expected utility of the utility which is further and further away will approach zero (e.g. in the same way that x e^-x^2 tends to zero as x tends to infinity, which explais the finite mean of a normal distribution).

I think allowing for infities is still a problem. My understanding is that, when people talk about infinite worlds, they do not mean finite worlds tending to infinite worlds. They mean literally infinite world. In this case, the expected utility will be infinite, not just tend to infinity as interpreted in physics.

Hi Vasco, No I am not saying that at all. Sorry I don’t know how best to express this. I never said utility should approach zero at all. I said your discount could be

infintesimallysmall if you wish. So utility declines over time but that does not mean it needs to approach zero In fact in the limit it can stay at 1 but still allow a preference ordering.For example consider the series where you start with 1 and then minus a quarter and then minus an 1/8 from that then minus a 1/16 from that and so on*, which goes like: 1, 3/4, 5/8, 9/16, 17/32, 33/64, ... . This does not get closer to zero over time – it gets closer to 0.5. But also each point in the series is smaller than the previous so you can put them in order.

Utility could tend to zero if there was a constant discount rate applied to account for a small but steady chance that the universe might stop existing. But it would make more sense to apply a declining discount rate, so there is no need to say it tends to zero or any other number.

In short if there is a tiny tiny probability the universe will not last forever then that should be sufficient to apply a preference ordering to infinite sequences and resolve any paradoxes involving infinity and ethics.

You can only get those weird infinite ethics paradoxes if you say

lets pretend for a minute that with 100% certainty we live in an infinite world, and it is"literally infinite ... not just tend to infinity as interpreted in physics". Which is just not the case!I mean you could do that thought experiment but I don’t see how that makes any more sense than saying:

lets pretend for a minute that time travel is possible and then point out that utilitarianism doesn’t have a good answer to if I should go back in time and kill Hitler if doing so would stop me from going back in time and killing Hitler.In my view such though experiments are nonsense, all you are doing is pointing out that in impossible cases where there are paradoxes your decision theory breaks down – well of course it does.

Hope that helps.

"You can only get those weird infinite ethics paradoxes if you say lets pretend for a minute that with 100% certainty we live in an infinite world, and it is"literally infinite ... not just tend to infinity as interpreted in physics". Which is just not the case!"

EU maximization will typically mean all expected values are undefined under naive treatment of infinities like real numbers, because every option should have a nonzero probability of +infinity and a nonzero probability of -infinity (e.g. there's a chance there's a god and you will worship the right one, or there's a chance the universe is infinite and the aggregate utility is undefined). So, even if you assign tiny probabilities to infinities, they break naive EU maximization.

There are extensions that break less, though, but each framework for handling both finite and infinite cases seems to have serious problems or require pretty arbitrary assumptions.

"In my view such though experiments are nonsense, all you are doing is pointing out that in impossible cases where there are paradoxes your decision theory breaks down – well of course it does."

Infinities haven't been proven to be impossible, though.

What do you think is the meaning of possibility? In my view, it only makes sense to talk about conditional probabilities, i.e. we can only say something is more or less likely conditional on a set of assumptions.

For example, when I say the probability of getting heads in a coin flip is about 50 %, I am assuming I flip it in a non-strategic way (e.g. not aiming for only 0.5 rotations such that I get my desired outcome), and that the coin has heads on one face, and tails on the other, among others. The probability of getting heads will tend to 0 or 1 as I specify more and more of the conditions of the coin flip.

Similarly, when we talk about the probability of an ongoing chess match between Magnus and Alireza being won by Magnus, we will conditionalise on the current state of the board, and that the game will continue following chess rules as we know it, among others.

I see axioms as the propositions which are always playing the role of assumptions. They are like the rules of a table game which allow us to determine which player is most likely to win. In this sense, asking what is the probability of infinite worlds sounds similar to asking what is the probability of the rules of chess being correct. It is meaningless to say the rules of chess are correct or incorrect, all I can do is talking about the likelihood of certain board states conditional on the rules of chess being followed. In reality, we can only say what is the probability of a given world state conditional on some axioms.

I am still a little confused about how to decide on what should be defined as axioms, but I think 2 important criteria are:

I am happy to reject the possibility of infinite worlds because:

Got it, thanks!

I agree.

Banach-Tarski is a consequence not just of infinities but also uses non-measurable sets, which depend on the axiom of choice. You can just limit or reject the AoC to prevent it. There are weaker versions of the AoC that don't result in Banach-Tarski.

I actually think Banach-Tarski paradox could actually be done if Conservation of Energy is allowed to be removed. The real issue is that conservation of energy prevents you from making new energy, which is likely key to why Banach-Tarski's works.

Interesting. I also find lack of conservation of energy quite unintuitive, so it looks my intuitions are internally consistent if that is the case.

Thanks for clarifying. I find the axiom of infinity much more unintuitive than the axiom of choice. So, for me, rejecting Banach-Tarski implies rejecting the axiom of infinity.

The axiom of choice is already true in ZF for finite families of sets without the full axiom of choice: https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf

Confusingly (to me, at least; I haven't spent the time to understand this), AoC and Banach-Tarski are also true generally in the constructible universe, which is nice as a model of ZF, but reasonably defined sets are still measurable, so you don't get the Banach-Tarski paradox if you only use them: https://math.stackexchange.com/questions/142499/are-sets-constructed-using-only-zf-measurable-using-zfc

Also note that Banach-Tarski uses a different kind of (and larger) infinity, specifically the continuum (real numbers) to model objects. If the universe is discrete but infinite in extent, then you can still avoid Banach-Tarski.

Thanks! I have replaced "They allow for the

Banach–Tarski paradox" by "Some types of infinity allow for theBanach–Tarski paradox".Huh, this is interesting to me - can you go into a bit more detail here?

Hi Bruce,

Thanks for asking. Not sure I have much more to say, but:

"Furthermore, I find the idea that the whole is the sum of its parts quite intuitive, and infinities are not compatible with that (inf/2 = inf)."

Note that infinite sets in standard math satisfy

They don't in general satisfy the size of the difference is the difference of the sizes.

*Except for uncountable sums, where size=measure

Thanks for clarifying! I have changed the sentence you quoted to:

It's not exactly the whole being equal to its parts, since they will be different sets, but they can have the same size.

Thanks, again! I have updated the sentence to:

"They imply intuitively nonsensical claims to be true. For instance, I cannot see how one of Amanda’s 5 axioms would be false."

Amanda's axioms are moral claims, so reasoning backwards from them to the claim that infinities are impossible seems like motivated reasoning.

Hi Michael,

What else can I use besides my strongest intuitions (which I guess can be called motivated reasoning) to know which axioms to accept/reject?

Generally, I don't think you should reject any possibility with 100% certainty unless you can demonstrate it's contradictory, or inconsistent with current evidence. You can assign it low probability based on intuition and things like Occam's razor, but not 0. Furthermore, we do have consistent models of the universe with infinities that are consistent with current evidence, so it really should be treated like a live possibility on this basis. Lots of domain experts (probably most physicists) think infinities are possible, too.

I'd go even further, since all evidence is in principle questionable, so maybe nothing gets ruled out with certainty other than logical impossibilities. However, any or every specific logical inference you've made could be mistaken, too, so you can't really know with certainty that you've demonstrated that something is logically impossible. People make invalid inferences all the time, and in the worst case, you could be under a delusion constantly convincing you of invalid inferences, although this seems unlikely.

This is probably very controversial, but maybe even the rules of classical logic are questionable. However, I'd take the rules of logic as "working assumptions", because I'm totally clueless/confused about what to do without them. Similarly if you doubt your specific logical inferences too much even for very basic ones when extremely careful and explicit. Similarly if you doubt your own experiences too much.

So, I would take the rules of logic for granted and treat them as if they’re true with certainty, and that's basically it. Then I'd assign very high weight to my own experiences, simple logical inferences and basic standard probability theory (maybe 100% to probability theory, but I think there are competitors for dealing with uncertainty). I pretty strongly endorse Cromwell's rule: https://en.m.wikipedia.org/wiki/Cromwell's_rule

I think this is a great principle for almost everyting, but does not apply to axioms. These cannot be falsified, and therefore are neither supported nor rejected by evidence. I do not even know what to think about the concept of probability without axiomatically accepting/rejecting some stuff. As far as I can tell, when we talk about probability of X, we are really talking about probability of X if a certain set of axioms (e.g. rules of logic) are true.

I agree. However, this does not seem to apply to axioms, whose acceptance does not depend on inference (otherwise they would be theorems).

Ok, I guess my rejection of infinities can also be interpreted as a working assumption. I understand there are tools to deal with infinities, but these also rely on their own definitions/axioms, so they would also require working assumptions (while adding complexity).

Me too. It is just that I consider infinities to be at the same level as logical impossibilities (e.g. A>B and A<B being true simultaneously). There will never be evidence for or against them, so I can just set my prior for their existence to 0.