Epistemic status: I've procrastinated on a proper write-up for a number of years, so I thought it was better just to get an unpolished, rough sketch of my thoughts out there as quickly as possible. I'm playing fast and loose with terms here, apologies for that, but I thought it was better than continuing to wait for me to somehow find time to polish my old work, which doesn't seem likely to happen any time soon given how I'm focusing more on AI Safety these days.

A few years back, I spent some time at the EA Hotel trying to make progress on Infinite Ethics. I kind of got stuck, but here's a quick write-up of some of the thought paths I explored in case anyone else finds it useful.

Bostrom defines the problem of Infinite Ethics in a paper with the same name.

The obvious pragmatic response to Infinite Ethics is to just ignore the whole thing. I essentially agree with this. If our formalism disagrees with our intuitions, then it seems pretty reasonable to conclude that we're using the wrong formalism.

So I jumped on Wikipedia and in about an hour I concluded that the Surreal Numbers looked the most promising. I later learned about this paper by Eddy Chen and Daniel Rubio which provided confirmation that I had found a promising direction. At the time, there wasn't really any other work looking at applying the surreals to this problem, but I expect that if I searched now, I'd probably find more.

In any case, my initial idea was to fill out some of the details for this research direction. I was particularly focusing on the notion of bijections. I've heard people say very casually that bijections show that there are the same number of integers and even numbers (maybe mathematicians speak more precisely, though it wouldn't surprise me if they did not). This leads directly to the problems of infinite ethics, but it's also exactly the kind of view that you should have reject if you're claiming that the surreal numbers are a better measure of "size" for infinite quantities.

So how large are the evens? If you write the counting numbers as having size x, then if we assume that there are the same number of evens and odds, this gives us x/2 of both. We can imagine taking this further so that there are x/3 counting numbers that are 1 mod 3, x/3 that are 2 mod 3 and x/3 that are 0 mod 3; and so on for mod 4, 5, 6, ect. This provides us with a uniform notion of infinity in a sense and I imagine that many people would find this intuitively appealing.

On the other hand, while this is a beautiful construction, it's not obvious that this is the right thing to do. Like we could construct it so that there are x/2 odd non-negative integers and x/2 even non-negative integers including zero or excluding zero and it isn't clear why we should prefer one over the other. But it's a useful concept, so let's just define the uniform notion of infinity to exclude zero when we're defining equal sequence lengths. Oh, and we'll also add an additional condition to ensure that there are both z positive numbers and z negative numbers.

Scoping out our options at this point, we can:

• Treat all these subsequences as having equal "size" as per cardinality/standard bijections - seems philosophically dubious as it proves too much, ie. a set having equal "size" as a subset
• Claim that the sizes of these subsequences are incommensurable or there isn't really a notion of size that we can construct here - this would mean that most infinite ethics problems would have no solution
• Insist that the only valid construction of the integers is the uniform one - but why? It's the most beautiful, but that isn't exactly a reason, plus as mentioned above, different constructions can appear uniform depending on factors like including or excluding 0 in comparisons
• Accept that there are multiple possible constructions of the integers - seems the most promising direction to explore. Maybe we can make it work, maybe we can't, but only way to figure this out is to try.

So if there's multiple possible constructions of the integers, what are they? Let's look at an example. Talking very casually, we can imagine positive integers with the last number^[1] being even and positive integers with the last number being odd, however:

• This wouldn't fully define it. What about the subsequences for mod 3, 4, 5, ect?
• We don't need to talk about the concept of the "last number". We could just, for example, have one construction where there are x/2 odds and x/2 evens; and another with (x+1)/2 odds and (x-1)/2 evens

Nonetheless, thinking about it casually in this way seems like a useful intuition pump, as long as we only hold the model lightly.

Anyway, this naturally leads into a bunch of questions such as:

• How can we formalise this?
• Is pinning down the different moduli sufficient to pin down a specific infinity for the counting numbers? Does it make sense to think about the equivalent of "last number being a square" or "last number being a prime"?
• How we pin down a specific notion of infinity for the reals?

Also, this naturally leads to a desire to construct an additional, more refined notion of bijections where we ensure that the sequences have equal surreal size and the same too for each subsequence that is mapped to each other. If we had such a notion, then many of the paradoxes of infinite ethics would likely disappear.

This immediately leads to two more questions:

• How can we formalise these refined bijections?
• Are there any consequences for the foundations of mathematics if we construct it on refined bijections? The foundations of maths are built on set theory and there are a lot of bijections used there. But then again, there is a formalisation of the surreals using set theory, so maybe they don't need to change after all?

So how does this tie into the larger question? Well, firstly, if you're going to suggest using surreal numbers to measure the size of infinite sets, you're going to have to answer some questions about how to figure out these sizes. And the even numbers/odds numbers are about as easy as it gets. Secondly, once you start actually applying surreal numbers to label the sizes of sets, a lot of the paradoxes of infinity seem a lot less surprising. And so if you can make it work, that would add a lot of credibility to the rough framework.

Anyway, I didn't get far down this research direction, because I started to suspect that infinities might not exist, even in theory. I've recently started to see infinities as plausible again, but we'll get into these issues in the next post.

1. ^{^}

   When I was working on this, I was taking the notion of "last number" a lot more literally as I was imagining that even if there was an embedded perspective from which that notion of last might not make sense, there might be a non-embedded perspective from which it might, but I still don't have high confidence in how this line of reasoning plays out.