Decision theory, economics and ethics all involve comparing different options. Sometimes the relevant sums or integrals that we'd hope would provide the value of an option instead diverge to +∞, making comparison between such options impossible. 2/

This problem arises because the standard approach to evaluating infinite sums and integrals uses a very coarse-grained system of infinite numbers, where there is only one positive infinite number (+∞). To assign values to such options, we need a more fine-grained system. 3/

The most well-known alternatives (Cantor's cardinal & ordinal numbers) are also very coarse-grained, as they are generalisations of natural numbers. But if we use the hyperreal numbers (a system including a wealth of infinite and infinitesimal numbers) we can get it to work. 4/

We just need to generalise the standard finite sum and integral from 𝑎 up to 𝑏 such that 𝑎 and 𝑏 can be hyperreal numbers, and then use this alternative kind of summation/integration to evaluate infinite sums and integrals (taking them up to the hyperreal number ω). 5/

This gives us a new way of evaluating infinite sums and integrals that refines the standard method, such that two sums which were standardly equal to +∞ can now have different positive infinite values. And the values they get are very intuitive: 6/

I then show how we can apply this new technical solution to evaluate troublesome options such as the St Petersburg gamble, infinite streams of utility, and infinitely large universes. 7/

Basically, we just find the relevant sum or integral that previously diverged and evaluate it using the new method. But there are lots of interesting things we encounter along the way. 8/

My method applies to cases where we are evaluating an infinite whole in terms of its parts. This could be a prospect with infinitely many possible outcomes, a future with infinitely many years, or a universe with infinitely many inhabited planets. 9/

All theories assessing infinite wholes via their parts face a dilemma — they can only choose one: Pareto: improving every part of the whole must make the whole better. Unrestricted Permutation: a whole with the same parts but in a different order would be equally valuable. 10/

So theories tackling this subject can be divided into two classes based on which of these principles they keep. Mine sides with Pareto. 11/

A major theme in my results is that most of the counterintuitive aspects of the infinite are revealed to be artefacts of infinite number systems that are too coarse-grained and so are forced to lump together quite different things. 12/

On my theory, there is none of this "∞+1 = ∞" business where finite changes don't make outcomes better or worse, where a 50% chance of an infinite outcome is worth the same as a guarantee of it, or where Hilbert-hotel-like rearrangements produce counterintuitive effects. 13/

And making a situation better for someone (without changing anything else) always matters just as much, regardless of whether you are in an infinite situation or a finite one. 14/

Indeed, on my theory the infinite behaves very much like the finite. Infinite values are much like any other value, except that they happen to be larger than all finite ones. (Like Leibniz's conception of the infinite, not Cantor's.) 15/

My technique also has consequences for finite cases. Many decision theorists claim that an agent’s utilities must be bounded as this helps avoid the problem of infinities. And economists argue for discounting future utility on the grounds that this avoids their infinities. 16/

Changing our own values about finite cases in order to sidestep a technical issue about infinite cases should have been an absolute last-resort. Especially as the technical problem now appears to have a technical solution. 17/

Using my approach to evaluating infinite options, we can restore our natural values, where suffering lasting twice as long can always be twice as bad (without limit) and someone's suffering doesn't matter less just because it happens later. 18/

That said, my method doesn't attempt to solve every problem that can arise from infinite value, and it does have some remaining issues which I clearly outline in the paper. I think of it more as a proof-of-concept that there is a technical solution, than as a completed one. 19/

Please do check out the paper if you're interested! 20/20 https://arxiv.org/abs/2509.193...

@tobyordoxford I may have missed it, but do you offer a take on the theory of choosing between utility 0 and expected utility (1/2 * 1 + 1/4 * -4 + 1/8 * 16 + 1/16 * 64 + ...)?

@jamespayor (I assume that was meant to be a -64, such that the positive and negative contributions to the EV are undergoing exponentially large oscillations) Sums like this that don't converge to a finite real or to -∞ or +∞ do indeed depend on the ultrafilter.

@tobyordoxford Interesting!

@DavidDeutschOxf A couple of aspects that could be of particular interest to you: 1) I sketch how one might be able to develop this into an alternative to renormalisation in quantum field theory. It isn't there yet, but there are promising signs.

@tobyordoxford Nice to see a use for the hyperreals. It’s a fascinating and simple mathematical structure. I always thought it must be good for something. Of course, in this case, what it’s good for is certain abstruse puzzles in moral philosophy, so some would say it’s still useless.

@tobyordoxford Interesting. According to this take is the ‘sum of natural numbers’ then greater than the ‘sum of primes’?

@tobyordoxford Would a present value fit in your framework? Basically a variety of streams are growing to infinity so we choose to evaluate them at a particular point in time to make comparisons.

@tobyordoxford As it concerns ethics, it’s not clear that the search for convergent sums is a genuine problem of value as much as it is an artifact of over-mathematization

@tobyordoxford Wow, this looks infinitely better than previous solutions such as artificial discounting or assumptions of boundedness!