@jadagul (splitting off the infinite trolley thread which is getting long)
So there’s a lot to unpack there.
My post was assuming the usual real line, not the long line. In the regular real line, you just can’t get aleph-1 people into it without having aleph-1 people in some arbitrarily small interval. (Sketch of proof: choose an epsilon > 0. The line is a union of countably many intervals of length less than epsilon, and a countable union of countable sets is countable, so at least one interval must have uncountably many people in it).
On the long line things are different. We do have to back up and define things more carefully, because typically a “measure” only outputs a real (or complex) number or infinity, and so you can’t have a set with measure aleph-1. But it’s easy enough to route around that (as long as you can define the idea of adding infinite cardinals sensibly, which I think you can).
Then sure, you can take the long line and put one person in each [0,1) interval. There will be aleph-1 people on the track at “the same density” as the aleph-0 track.
But this is where we get back to my original complaint, which is that some (almost all!) people will never get hit by the train. Not only will they not get hit in finite time, they won’t get hit in countably infinite time. The train doesn’t have a way to get off the “usual” line onto the “long” part of the long line.
Basically, we say something is countable if for every element, you know how many elements you have to go through to get there. The set may be infinite but each individual element is only finitely far into the list. Which means you can ask questions like “when does the train hit this person?” Or “how many people will get hit before this person gets hit?” and have well-defined answers.
In uncountable sets, by definition, you can’t ask those questions. You can order an uncountable set (which means that for any two elements, you can ask which comes first). And if you believe in choice you can well-order an uncountable set (which means that for each element, there’s a “next” element–but not necessarily a “previous” element! There is, however, a “first” element). But none of that is enough to ask questions like “who is the fifth person to get hit?” or “How long will it take before John gets run over?”
So yeah, if your set is “the set of points on the real line” then there’s a natural “total order”. (Although there isn’t really a natural well-order). So you could say “after time t=1 these (uncountably many people) have been run over.” But you couldn’t ask who gets hit first or anything.
My understanding is that “countable” is a description of a cardinality of a set, not of its order type, which is (confusingly) the aspect that is relevant for questions of counting one by one.
So, for instance, consider ω+1. ω is just the order type of the natural numbers:
0 < 1 < 2 < 3 < …
Then ω+1 corresponds to taking the natural numbers and adding on an extra element that’s greater than all of them:
0 < 1 < 2 < 3 < … < 0′
This thing has the same cardinality as the usual natural numbers, so it’s countable. But with order type ω, for any number, we can define how many steps it takes to get there. With order type ω+1, we can’t do this with 0′. (And since the original picture just stated the cardinality, we can’t tell which of these we are dealing with.)
Of course, since these are both countable, we can put them in one-to-one correspondence, and thus rearrange the one with 0′ to the one where you can count to anything, including the numbers what 0′ gets mapped to. So at worst this seems like an “ordering issue” rather than some issue with 0′ being “so far out” that it can’t be counted to.
But now consider the uncountable case. Specifically, consider the first uncountable ordinary ω_1 (used to make the long line). It orders all the countable ordinals, of which there are uncountably many. So from one perspective, this contains all sorts of things you can “never count to.” On the other hand, for any x∈ ω_1, the set of elements of ω_1 less than x is countable. So wherever you are in ω_1, any inability to “count to that point” is again just an “ordering issue,” no worse than in the above case.
I guess this is just another way of stating your point that the train can’t ever get to “the ‘long’ part of the line.” However, I don’t think this makes the original picture or problem somehow ill-defined. There’s nothing wrong with depicting the uncountable people on the second track as a first one followed by a second one etc. – this implies that they are well-ordered, but ω_1 is well-ordered (even without the axiom of choice, I think?).
On either track, you can have various order types, implying various people who can’t be reached by counting – you can even have countably many of these on the first track. But on the second track, no matter what the order is, you must have uncountably many people who can never be hit. I’m not sure the “uncountably many” is the important part here – it also just seems important that you can have zero people who can never be hit on the first track, but can’t on the second. But “people who can’t be hit” isn’t a concept that applies only to the second track – it’s just that on the first track you have somewhere between 0 and aleph_0 of them, while on the second you have aleph_1 of them.
Of course none of this is at all in the spirit of the original problem, since it gives you no way to hit “more people” on the second track. Which is I guess what you’ve been saying. But I don’t think this is because you can’t possibly answer “how long until John gets hit?” for the second track. As on the first track, the answer will either be “[some finite time t]” or “never.”
(I should say, again, that I am a a complete newb at all of this and am basically just thinking through the process of learning basic stuff about infinite cardinals and ordinals. So I apologize if this is tedious or irritating for that reason.)
(On another, uh, track entirely – it’s funny how utilitarianism breaks down with infinities. You could add another person to one of these tracks without changing the cardinality, like in Hilbert’s hotel, but surely that’s bad and not neutral, right?))
But on the second track, no matter what the order is, you must have uncountably many people who can never be hit.
Assuming the “second track” here is the one with aleph 1 cardinality, why is this the case? Say the track is the real line, then for any person on it we can find a time (in terms of X) after which that person is hit, no? Or say even that just the first meter of the track is the real interval (0,1], then everyone on it is hit in finite time!
Here we were talking over whether there can be aleph_1 people on a track (and what that means) in the way it’s depicted in the picture, where they’re lined up one by one, and it takes some finite amount of time to hit each one.
(In other words, the question is what happens when we have a well-order on the people. The Axiom of Choice implies that there is a well-order on the real numbers, but it’s not the usual order and we have no idea what it looks like. My statement isn’t true for the reals with the usual order, but that’s because it isn’t a well-order.)
I suggest we avoid counting people with Reals