Remember the trolley problem?
There is a runaway trolley barreling down the railway tracks. Ahead, on the tracks, there are five people tied up and unable to move. The trolley is headed straight for them. You are standing some distance off in the train yard, next to a lever. If you pull this lever, the trolley will switch to a different set of tracks. However, you notice that there is one person on the side track. You have two options:
- Do nothing and allow the trolley to kill the five people on the main track.
- Pull the lever, diverting the trolley onto the side track where it will kill one person.
Which is the more ethical option? Or, more simply: What is the right thing to do?
There are now thousands of variants of this. Today I saw this one:
My first thought was, aha, finally a chance to apply Cantor’s diagonal argument to something useful.
Let’s start with an easier version. Suppose the lower rail is bounded in length, say to 99.9999… metres and there are as many people tied to the track as there as real numbers in the bounded set [0,100). That’s infinitely many reals. Each person lies at a position somewhere in [0,100).
Give each person in the heap a number 0,1, 2, … Since there are infinitely many reals in [0,100), there must be infinitely many people stacked up in some way within that almost-100 metre stretch.
Now work along that infinite philosophically imagined mound of people and construct a new real number as follows from where they are lying along the rail (in metres). (You have a very precise measuring tape.)
From person 0, take the number to the left of the decimal point on their measurement and compute 99 minus that number. From person 1, take 9 minus the 1st number to the right of the decimal point. From person 2 take 9 minus the 2nd digit, and so on. So from person i, take 9 minus the ith decimal digit (pad out the digits with zeros where necessary).
So now we have a new position on the track where by definition there is nobody tied to the track. Our original assumption is false: it was not possible to stack infinitely many people in infinitely many real-numbered positions along an almost 100 metre long stretch of track. Contradiction.
We can’t do it for [0,100). That means there’s no hope for doing it for all real numbers since [0,100) is a subset.
So it is not possible to tie heaps of people to a track so that there are as many people as there are real numbers.
It is (sort of) possible to do it for countably infinitely many people—the top track. But see the distinction between potential and actual infinity.