Infinity is Counter-intuitive

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Even people who love studying and thinking about it get it wrong. I stumbled into this sneaky example recently.

Facebook meme: You're in control of a switch that can flip a runaway trolley between two tracks. One of the tracks has aleph-null people along it, but the trolley will travel twice as fast; the other has aleph-one people. The trolley will never stop. Which track will you put it on?
A fair number of people would argue for the aleph-one track, because of the speed difference. While that’s a good fit for intuition, your intuition has only dealt with finite amounts and is led astray by them. Maybe you’ll see that if we do a simple tweak.

tweaked_trolley

All I’ve done is asked the people on the aleph-null track to shuffle over a bit. The second person wiggles until they’re where the third person was, that third person rolls to where the fifth one was, and so on down the line. Have we changed the number of people? No, we’ve just made them half as dense as they once were. This cancels out the speed increase, so now people are run over at the same rate on both tracks. The two choices wind up being equal.

Or so I thought, at first.

I should explain this aleph-what stuff. “aleph-null” refers to the countable infinity, the infinity you get when just counting without end. If you can assign each member of a set exactly one natural number (0, 1, and so on down the line), with no omission or repetition, it’s countable. If you can keep doing that without limit, it’s countably infinite.

Aleph-one is an uncountable infinity, the least such infinity in fact. Try as hard as you can, but you’ll never be able to assign a natural number to every possible real number. It’s a very different beast.

People are countable, though. We occupy a fixed space for a finite length of time. You can easily put a number on us. So if you could lay down an aleph-one number of people, you could assign each of them an integer and prove that aleph-one is countable. This is a contradiction! So it must be the case that you cannot lay down aleph-one people in a line, even in theory.

The original question is incoherent, without an answer at all. It makes as much sense as asking for the set of all sets which don’t contain themselves.

[HJH 2016-04-29: Atrocious grammar is atrocious.]

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