Recently, I was thinking about some rather interesting generalizations of the trolley problem to tracks containing an infinite number of people, and I was wondering how to formulate a moral argument around these problems.

The general setup is this: Suppose you have a trolley that is traveling down a track at a finite speed, and suppose there is an upcoming fork between one of two branches. Suppose each of the branches is a one-dimensional topological space with a person at each of several points.

I am interested in three scenarios in particular:

Trolley Problems

Scenario 1:

Branch 1: A long line with exactly one person in every subinterval [0, 1).

Branch 2: The real line with a person at every real number.

Scenario 2

Branch 1: The interval [0, 1] with a person at every real number between 0 and 1.

Branch 2: The real line with a person at every real number.

Scenario 3

Branch 1: A person at every integer.

Branch 2: A person at every rational number.

All of these scenarios have the same cardinality of people on each branch, so at first glance, it seems like it doesn't matter to which track you divert the trolley.

However, I soon came up with some arguments for each side. For instance, in Scenario 2, there are arguments for diverting to both Branch 1 and Branch 2:

  • The cardinality of [0, 1] is the same as the cardinality of the reals. If you divert to branch 1, you kill all of the people in [0, 1] in infinite time, while it at least takes you infinitely long to kill all of these people in branch 2. More precisely, on branch 2, there will at least be people who are still alive at any finite time, while everyone will be dead after sufficiently long on branch 1. Thus, you should divert to branch 2.
  • On the other hand, [0, 1] is contained in the reals. Thus, you kill all of the people in branch 1 and then some on branch 2, so branch 2 is worse. Thus, you should divert to branch 1.

In addition, it feels intuitively clear that in Scenario 3, killing a person at every rational is worse than just killing a person at every integer, since you kill an infinite number of people in any finite time on the rationals. However, the cardinalities of the rationals and the integers are the same, which seems to contradict this moral intuition. For the first scenario, the issue is similar: Is a killing people in a continuous uncountable set morally worse than killing people in a discrete uncountable set?

The question is this: is there a moral framework for dealing with these infinite trolley problems? I am also interested in the case where the trolley moves at an infinitesimal speed.

Edit: To be completely rigorous, we need to assume the continuum hypothesis for Scenario 1 to ensure that the cardinalities of the number of people on each branch are the same.

  • Edited some typos and emphasized the questions.
    – J D
    Nov 3, 2021 at 20:21
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    What does it even mean to talk about uncountably many people? None of these scenarios, except the first branch of Scenario 3, make sense given my understanding of personhood. In particular, every person I've met has had non-zero volume, so I'm not sure how you would pack an infinite number of people into a finite interval, without changing our notion of what a person is.
    – Sandejo
    Nov 3, 2021 at 22:53
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    In this scenario, I'm not necessarily considering each point as representing a real physical person per se. I am merely attaching the same moral weight to the trolley passing a point $x$ as I am to a person getting killed. Nov 3, 2021 at 23:48
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    Trolley problems are already criticized for being contrived and non-indicative of actual moral choices, and this takes it one more step in a wrong direction. Moral philosophers have their hands full with more realistic moral dilemmas to venture into "problems" where the number of people is measured by alephs. So no, there is no framework for this for lack of a point to it. One can set up some axiomatic "moral rules" and play with this as a mathematical toy, but that belongs on Math SE.
    – Conifold
    Nov 4, 2021 at 0:21
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    Well, I find the question interesting. It really stress-tests the limit of valuation. I think people are getting too caught up in the details, like how to get an uncountable number of people or whether or not it is realistic. None of that is relevant to thought experiments. Nov 4, 2021 at 2:33

1 Answer 1


I love this question. Is it philosophy less than mathematics, or equal to it here? I don't know for sure. The Stanford Encyclopedia of Philosophy article on infinity has a subsection on something called "God's lottery", for example. At worst, your reasoning illustrates the inadequacy of the utility aggregation scheme generally, maybe. Not a bad logical trade-off by any means.

That being said, re: the constructed scenarios... You know, I actually just read a blog post or something by a Christian writer trying to argue that at any given time, a sinner in Hell will only have been in Hell for a finite amount of time, so God ought not be accused of ever inflicting an actually infinite punishment on a finite creature. I'm not going to add to or subtract from that argument; I just mean to show that it is not an immediately clear thing for us, to say what effect variations on the infinitary descriptive parameters has for our understanding of the moral of this abstract story. In other words, if the cardinal disvalue of two actions is equal even unto some level of infinity, yet they "ordinally" differ in an important way nevertheless, would this allow us to decide whether to choose one of the options instead of the other? If so, then at least we would conclude: so aggregating cardinal value is not the only operation in moral algebra. Another legitimate argument, it turns out.

EDIT: indeed, this exact issue of one cardinal corresponding to multiple ordinals, does not arise among the finite numbers. Only by showcasing the abstract structure of the situation on the level of infinity can one best represent a case where we could use an ordinal judgment to solve an equation in deontic algebra as such. For example, you could just pick the taller ordinal at stake.

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