The Grim Reaper Paradox (Benardete’s Paradox)

Page last updated 01 Mar 2026

The Grim Reaper paradox is a distillation of various paradoxes that occur in the book “Infinity: An Essay in Metaphysics” by José A Benardete. José Benardete, Infinity: An Essay in Metaphysics, Oxford: Clarendon Press, 1964. You can read an online copy of the book at Internet Archive: Infinity: An Essay in Metaphysics.

The paradox describes the following situation: there is an infinite sequence of Grim Reapers where each Reaper is associated with a specific time on a specific day; at that time they must kill a specific person if that person is alive at that time. One Reaper is assigned to kill the victim at 1pm. Another Reaper is assigned to kill the victim at 12:30pm, another at 12:15pm, and so on. On the specific day, the victim must die; the victim cannot survive at any point in time after 12 pm, therefore he must die at precisely 12pm. The paradox is that there cannot be any particular Reaper that kills the victim, since for any specific Reaper, there is always another Reaper that is associated with a point in time that is closer to 12 pm.

For the situation as described above to apply, the victim must be alive before 12 pm. But if the situation as described above does apply, and the victim is dead after 12 pm, then either:

1. the victim is both alive and dead at 12 pm, or
2. the Reaper kills the victim an “infinitesimal” amount of time immediately before 12 pm, or
3. the Reaper kills the victim an “infinitesimal” amount of time immediately after 12 pm.

None of these possibilities are plausible; the latter two rely on the notion that there is a point of time closer to 12 pm than any other point of time. The notion that the victim becomes a quantum superposition of alive and dead for at 12 pm leads to the problem that the single point has no duration and so the victim has to be in a superposition of being both alive and dead for zero time.

Analysis

In the real world, there are no infinite sets of reapers, and in the real world, there can be no such scenario as described above; each reaper would have to be composed of at least a certain number of atoms, and hence take up a certain amount of space, so infinitely many reapers could not fit in the same locality as the person to be killed, i.e: there cannot be infinitely many space occupying things within a finite volume of space.

But we can set aside the question of physical impossibility, and observe that the points of time, as described, are simply points that have a mathematical relationship; the points are 1, ½, ¼, ⅛, etc, and we can note that the points approach every closer to zero without ever reaching zero - but at the same time there cannot be any definable point that is between zero and the sequence of points. Much of Benardete’s book meanders while repeatedly stating that there cannot be any specific point between zero and the points of the sequence, and yet the sequence cannot ever reach zero. A large part of the book consists of Benardete conjuring up various impossible physical situations in an attempt to convince us that this has some sort of profound significance.

The mathematical solution to this question is straightforward and was established many years ago. It has been used very successfully in calculus in applications in the real world. And that solution is to use the limiting value of such sequences for calculations involving such sequences. Similar considerations apply to infinite series, where the individual values of the sequence are added together. While the limit point and the points of the sequence can never coincide, the mathematical value of the limit point is a value that has been shown, time after time, to generate useful non-contradictory results applicable to real world scenarios.

Problems only arise when people assume that the limit value is the “final” value of such infinite sequences (or series) - that such non-terminating sequences somehow have an “end” or can be “completed”. This carries with it the notion that there is a definitive yet infinitesimal “distance” between the points of the sequence and the limit value. You will see that sort of muddled thinking in cases like the Balls in the Urn paradox, the Platonist Rod paradox and Alexander’s Horned Sphere.

Regarding the Grim Reaper scenario, one can attach various quasi-physical interpretations to the underlying mathematics of the sequence. One could argue that each point has an associated reaper, and that the reapers are enumerable, and that the limit point also has an associated reaper, and hence that there is a reaper that kills the person, and that it is simply the case that the limit reaper is not any of the enumerated reapers. But that leads back to the problem of the three possibilities listed above, where none of them is plausible. The alternative is to observe that there is no mathematical analogy to the notion of a reaper associated with a number, and hence one can simply dismiss the scenario as an absurd imaginary and impossible situation that is bereft of any deep meaning or significance.

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