Paradoxes, Infinity, and Probability

I’ve been loving the videos that SpikedMathGames has been posting on youtube. Check out their channel here. In particular, I’ve enjoyed Paradox Tuesday. Here’s one from a few weeks ago which really interested me (if you go to the youtube page, you’ll see I’ve been active in the comments!):

They also cover some famous paradoxes like the Unexpected Hanging Paradox or the  Barber Paradox. If you’ve never heard of these, go watch these now.

I’m especially interested in paradoxes that deal with infinity, countability, and probability. Here’s another great paradox that deals with just those issues that my friend Matthew Wright shared with me a few months ago (thanks Matthew!). It’s called the Grim Reaper paradox (can’t link to the Wikipedia article–it doesn’t yet exist), proposed in 1964 by José Amado Benardete in his book Infinity: an essay in metaphysics, and I first read about it on Alexander Pruss’s blog here, and I quote:

Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you’re alive, it instantaneously kills you, and if you’re not alive, it doesn’t do anything. Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you’re not going to be resurrected that day.

Then, you’re going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you’re guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:


(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

Here’s a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t1, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t1, say at t0. But if so, then you’re going to be dead right after t0, and hence the Grim Reaper who woke up at t1 is not going to do anything, since you’re dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I’ve shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

Go visit his blog post for a discussion of why this seems unresolvable, and how it may actually put forward a case for time being discrete rather than continuous. Crazy thought.

There’s something deeply unsettling about this paradox and also the Unexpected Hanging paradox. Anytime we deal with probabilities and certainty, paradox seems to be lurking nearby.

I sometimes ask my students this somewhat related question–perhaps you’ve heard it too:

How many positive integers have a 3 in them? (That is, in their decimal representation. 6850104302 has a 3 but 942009947 does not.)

If you haven’t ever considered this question, take the time to do it now.

Though I actually once worked out the result using limits (like Alexander Bogomolny does marvelously here), it’s easy enough to work out the result in our heads:

First ask yourself how many digits a randomly selected integer has. The number of digits is almost certainly greater than 2, right? There are only 90 two-digit positive integers, a finite number, and there are an infinite number of integers with more than two digits. It follows that if you were to pick one at random from among all positive integers*, it would be almost certain to contain more than two digits.

The same argument could be applied to a larger number of digits. By the same logic as above, we can convince ourselves that ‘most randomly selected integers have more than a trillion digits’. It’s a bit of an incredible statement, really. We rarely ever work with the ‘most-common’ kind of numbers (the big ones!).**

What is the probability that a number with a trillion digits has a 3 in it? Well, it’s almost certain. The probability approaches 100%. If we consider ALL numbers, the probability IS 100% (or is it?). This is a real dilemma. How can we say that 100% of numbers have a 3 in them when this is clearly not true?

We’ve been pretty sloppy here, but regardless, this kind of fast-and-loose infinite probability question is unsettling.

Do you want to try taking a crack at these? Feel free to comment below.

Oh, and Happy Birthday Euler!


* Picking a number from the set of all positive integers requires the axiom of choice.
** My comment that the ‘most-common’ kind of numbers are the big ones reminds me of Ronald Graham’s quote: “The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.” Love that quote, especially considering it comes from Ronald Graham, an expert in Ramsey Theory, and the creator of one of the largest named numbers :-). The fact that we have only ever studied the most common kinds of numbers is also confirmed by the fact that most numbers are irrational. Worse, most numbers are indescribable!

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