That's not correct. Elliptic geometry fails to satisfy some of the other postulates, depending on how they are phrased. I'm not too familiar with the standard ways of making Euclid's postulates rigorous, but if you're looking at Hilbert's axioms instead, then elliptic geometry fails to satisfy O3 (the third order axiom): if three points A, B, C are on a line, then any of the points is between the other two. Possibly some other axioms are violated as well.

Notably, elliptic geometry does not contain any parallel lines, while it is a theorem of neutral geometry that parallel lines do in fact exist.

Hyperbolic geometry was actually necessary to prove the independence of Euclid's fifth postulate, and few would call it a "fairly simple counterexample".

I agree that introducing elliptic geometry (and other simple examples like the Fano plane) earlier on in history would have made the discussion of Euclid's fifth postulate much more coherent much sooner.

It's a bit worse than that. Even if we defined the "k-successions" operator (which is basically addition), it doesn't actually let us do what we want. "For all x, there exists a number k such that 0 after k successions is equal to x" is always satisfied by setting k=x, even if x is some weird alternate-universe number like 2*. Granted, I have no clue what "taking 2* successions of 0" means, but...

Another option is to replace "50 utils" or "50 degrees utility" with "50 utils". Yes, always. The wiki link would have to be updated to address the affine caveat (as well as some others) but it might be worth it.

Edit: yet another option is to explicitly include a constant: "50+C utils". This has a fine tradition stemming from calculus. If necessary, it could be combined with my previous suggestion.

Of all fictional treatments of this question, the one that stood out to me the most is the one in Three Worlds Collide because of its restraint from turning a psychological question into a moral question.

"Once upon a time," said the Kiritsugu, "there were people who dropped a U-235 fission bomb, on a place called Hiroshima. They killed perhaps seventy thousand people, and ended a war. And if the good and decent officer who pressed that button had needed to walk up to a man, a woman, a child, and slit their throats one at a time, he would have broken long before he killed seventy thousand people."

"But pressing a button is different," the Kiritsugu said. "You don't see the results, then. Stabbing someone with a knife has an impact on you. The first time, anyway. Shooting someone with a gun is easier. Being a few meters further away makes a surprising difference. Only needing to pull a trigger changes it a lot. As for pressing a button on a spaceship - that's the easiest of all. Then the part about 'fifteen billion' just gets flushed away. And more importantly - you think it was the right thing to do. The noble, the moral, the honorable thing to do. For the safety of your tribe. You're proud of it -"

"Are you saying," the Lord Pilot said, "that it was not the right thing to do?"

"No," the Kiritsugu said. "I'm saying that, right or wrong, the belief is all it takes."

Oh. This seems unnecessarily treading over previously covered ground. My short answer is "no".

My long answer would probably be some sort of formalization of "no, but I understand why they'd do it". I'd be happy with the cognitive algorithm that would make the other person flip the switch. But my feeling is that when you do the calculations, and the calculations say I should die, then demanding I should die is one thing... demanding I be happy about it is asking a bit much.

The constant depends on the two languages, but not on the number. As army1987 points out, if you pick the number first, and then make up languages, then the difference can be arbitrarily large. (You could go in the other direction as well: if your language specifies that no number less than 3^^^3 can be entered as a constant, then it would probably take approximately log(3^^^3) bits to specify even small numbers like 1 or 2.)

But if you pick the languages first, then you can compute a constant based on the languages, such that for all numbers, the optimal description lengths in the two languages differ by at most a constant.

Suppose in case 3 someone else, not you, is tied to the track but can reach the switch. What now?

I'm confused. If I'm not the one flipping the switch, what's the question you're asking?

You could say this -- doing so would be like describing your own language in which things involving 12,345,346,437,682,315,436 can be expressed concisely.

So Kolmogorov complexity is somewhat language-dependent. However, given two languages in which you can describe numbers, you can compute a constant such that the complexity of any number is off by at most that constant between the two languages. (The constant is more or less the complexity of describing one language in the other). So things aren't actually too bad.

But if we're just talking about Turing machines, we presumably express numbers in binary, in which case writing "3" can be done very easily, and all you need to do to specify 3^^^3 is to make a Turing machine computing ^^^.

Well, those used to be the three questions we asked, but now you've gone and ruined the Turing test for everyone. Way to go.

If you offer me a $1:$1 bet that a six-sided die doesn't land on a six, I take it. But now you tell me that the die landed on a six, and want to make the same bet about its outcome. Of course I don't give the same answer!

My life is worth more to me than other lives; I couldn't say by how much exactly, so I'm not prepared to answer any of the dice-rolling questions. However, I am aware that to person C, person A and person B have equal-worth lives, unless one of them is C's spouse or child, and this provides an opportunity to make deals that benefit both me and other people who value their own lives more than mine.

So, for example, I would endorse the policy that bridges be manned in pairs, each of the two people being ready to push the other off. This is, effectively, a commitment to following the unselfish strategy, but one that applies to everyone. TDT offers a solution that doesn't require commitments; but there, we need the vague assumption that I'm implementing the same algorithm as everyone else in the problem, and I'm not too sure that this applies.

Oh and also I think I would jump for a hypothetical wife and daughter (or even a hypothetical son, imagine that), but at that point the question becomes less interesting.