The Grim Reaper Paradox
There is a long tradition, in philosophy, of employing paradoxical thought experiments in order to show that our understanding of some subject is either incomplete or incorrect. Quite famously, the paradoxes of Zeno of Elea puzzled philosophers and mathematicians for millennia. These enigmas can be, at once, immensely entertaining and thoroughly maddening to contemplate.
About a year ago, I was introduced to one such thought experiment which I had not previously encountered. It is known as the Grim Reaper Paradox, and the version with which I will interact today is presented by philosopher Alexander Pruss. The thought experiment proceeds as follows:
Fred is sitting in a room at 8:00 am. There exists an infinite number of Grim Reapers along with Fred, each of which is currently dormant. When any individual Grim Reaper becomes activated, if Fred is still alive, then that Reaper will instantaneously kill Fred; however, if Fred is not alive, the Reaper will return to a dormant state and continue to do nothing. Each of the Grim Reapers is timed to activate at a specific time after 8:00 am. The last Reaper will activate at 9:00 am. The second to last activates at 8:30 am. The third from last at 8:15 am. In general, the nth from last Grim Reaper will activate after hours have passed.
Now, we are guaranteed that Fred will not survive past 9:00 am. After all, if he is alive at 9:00 am, then the last Grim Reaper will activate and kill him. However, he can’t have lasted that long, either, since the previous Grim Reaper would have killed him if he had survived until it activated. In fact, we can generalize this: the nth from last Grim Reaper cannot have killed Fred, because if he had survived until hours after 8:00 am, then the (n+1)st from last Grim Reaper would have killed him.
Therefore, we see that Fred cannot survive until 9:00 am, and yet we have also shown (by mathematical induction) that none of the Grim Reapers can have been the one which killed Fred. Thus, we have come to a paradox.
One of the fun things about a paradox is that it often illustrates that one or more of our assumptions in defining the problem must have been incorrect. Often, the resolution of a paradox comes by abandoning one of the assumptions on which the problem was founded, even if that assumption feels entirely intuitive. In the case of the Grim Reaper paradox, Pruss finds that there are two assumptions whose abandonment could lead to a resolution of the paradox (emphasis added):
So either there are only finitely moments of time between 8 and 9 am, or no combination of Grim Reaper alarm clock settings is possible. In the latter case, it basically follows that it’s just impossible to have infinitely many Grim Reapers, whether their wakeup times are arranged so as to result in a paradox or not. So why can’t there be infinitely many Grim Reapers? It seems that the only reason to suppose there can’t be infinitely many Grim Reapers, even in cases where no paradox is generated, is if one thinks there can’t be an actual infinity of objects in existence.
The idea that time contains some finite number of total moments (where a “moment” is the smallest possible subdivision of time) is known in philosophy and in physics as Discrete Time. If time is discrete, then the setup for our paradox cannot actually occur, since there would not be an infinite number of subdivisions of the hour between 8:00 am and 9:00 am. Re-formulating the problem to fit with the notion of Discrete Time leads to an obvious answer: the Grim Reaper timed to activate in the first moment after 8:00 am is the one which kills Fred. If time is Continuous (the opposite of Discrete; that is, infinitely divisible) then there is no smallest possible subdivision of time, and therefore there is no first moment after 8:00 am, leading to our paradox.
The second assumption which Pruss illustrates is pretty much just a generalization of the first. Rather than limiting our scope to time, the second assumption is that anything can be actually infinite. Obviously, if there cannot be an infinite number of any object, then the setup for our Grim Reaper problem is in error, as there can be neither an infinite number of Grim Reapers nor an infinite number of moments for those Grim Reapers to occupy. While this is a natural extension of the first assumption, I don’t think that there is any warrant to reject the possibility of any and all actual infinites, from this paradox alone. Since the more narrow case of Discrete Time resolves the paradox comfortably, one would need some other method to show that infinites are impossible more generally.
Though Alexander Pruss discusses this Grim Reaper Paradox in a few of his other blog posts, I have not seen him discuss any other assumptions which might underly the problem. He seems to have focused upon these as being the prime constituents. However, it occurs to me that the problem includes another assumption, which is a bit more subtle. The Grim Reaper Paradox, as formulated, seems to presume the Tensed Theory of Time. I have discussed, elsewhere, the reasons that I believe the Tensed Theory of Time does not hold, so I’ll simply focus here on how Tenseless Time resolves the Grim Reaper Paradox.
On Tensed Time, the future is not yet actual, and actions in the present are what give shape and form to the reality of the future. As such, the actions of each individual future Grim Reaper, in our paradox, can be contingent upon the actions of the Reapers which precede them. However, this is not the case on Tenseless Time. If we look at the problem from the notion of Tenseless Time, then it is not possible that a future Reaper’s action is only potential and contingent upon the Fred’s state at the moment of activation. Whatever action is performed by any individual Reaper is already actual and cannot be altered by the previous moments of time. At 8:00 am, before any Reapers activate, Fred’s state at any given time between 8:00 am and 9:00 am is set. It is not dependent upon some potential, but not yet actual, future action as no such thing can exist.
Given that I thoroughly reject the Tensed Theory of Time, and given that I see no other reason why we should reject actual infinities, I am inclined to believe that the Grim Reaper Paradox’s assumption of Tensed Time is incorrect.