# Boxing Pythagoras

## 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 $\frac {1}{2^{n+1}}$ 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 $\frac {1}{2^{n+1}}$ 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.

## 29 thoughts on “The Grim Reaper Paradox”

1. I don’t have much faith in the idea of tenseless time either.

• How do you mean? Do you prefer Tensed Time? Or are you saying that you prefer something more along the lines of Julian Barbour’s entirely timeless model?

2. Dylan on said:

I am not sure I understand your tenseless solution here. Sure in tenseless time, the events are fixed, but how a Grim Reaper would react in each time would still be dependent upon what happens earlier. And if that is the case, then there is still the question of when, exactly, Fred gets killed and by which Grim Reaper.

• Causality, on Tenseless Time, is descriptive and not prescriptive. Whether or not any individual Reaper kills the subject is set and unchangeable. As such, the ontology of any individual Reaper’s action is not dynamic.

In this case, the Paradox simply tells us that it does not seem to be possible to arrange an infinite set with the causal relationship which the thought experiment demands.

• Dylan on said:

So are you saying that such a scenario is impossible to begin with?

• Yes. It would be like trying to set up a thought experiment with a triangle that has two internal right angles, or trying to discuss a negative number greater than 2.

3. Dylan on said:

Thanks for the response. In that case I am still not sure where tenselessness fits into this. A tensed theorist can just as well say something along the same lines: you can’t define an infinite sequence of grim reapers because it would lead to a contradiction. I am not saying that it is wrong to say that (the response sounds similar to Yablo’s in fact), but just that it doesn’t seem to have anything to do with tense.

• A tensed theorist can just as well say something along the same lines: you can’t define an infinite sequence of grim reapers because it would lead to a contradiction.

Alexander Pruss actually mentions this possibility, in the original article to which I was replying. He says that one solution to the problem might be that this particular configuration of infinite Grim Reapers is impossible. However, he saw no reason why this particular configuration should be impossible while other infinite configurations should be possible.

Tenseless Time makes the setup of the paradox impossible, as the setup requires that the ontology of any particular Grim Reaper’s action is not set until the time of that Reaper’s activation. The thought experiment is dependent upon the concept of temporal becoming. If this concept is invalid, the whole thought experiment crashes.

• Dylan on said:

For Yablo, the reason why something like the Grim Reaper paradox is impossible is because it is setup with an inconsistent set of rules. We simply can’t set up a system of Grim Reapers in that way, and expect them all to work as intended. It is like trying to program a machine to produce a triangle with four sides and expecting it to actually do so. The logic just doesn’t allow for it.

As I see it, the setup of the paradox doesn’t require temporal becoming at all. It only requires that the Grim Reaper’s actions at a particular time be dependent upon what happens at earlier times. This still makes sense under tenseless time as in tensed time, so I’m afraid I don’t see how temporal becoming is important here.

4. For anything to kill Fred, it must perform physical work on his physical body, so the scenario seems to fall foul of both quantum mechanics (the uncertainty principle) and relativity (that much energy concentrated around Fred collapses into a black hole, and, as on the Orient Express, they all did it.) There’s no need to invoke speculation about the nature of time.

• I think you are taking the thought experiment a little too literally. The idea of “killing Fred” is simply a stand-in concept for some general change in state over time. This need not necessarily be matter or energy changing, in which case neither quantum mechanics nor conservation of energy need come into play.

However, given this revelation, it becomes clear that the Grim Reaper Paradox is really no different than Zeno’s Dichotomy Paradox, except for the fact that we are discussing a temporal dimension of measure rather than a spatial one.

5. Nicola Prigg on said:

Maybe I’m misunderstanding this “paradox” but I really don’t see a paradox, either he’s killed by the first reaper or he’s killed by an effective infinite number of reapers.
You’ve dealt with the idea of the first reaper.
But if you break 8.00am to 8.01am in to an infinite with an infinite numbers of reapers to go with it. or even the first 30 seconds into an infinite. Also depending on how they kill Fred, are they stabbing him or shooting him or what? It will take him a specific amount of time for him to die which you could also divide into an infinite. So assuming it takes a minute for Fred to die, in that first minute after 8am an infinite amount of reapers activate, see Fred is alive and so take an action to kill him.

• Thanks for taking the time to read and reply!

It’s actually not the case that “either he’s killed by the first reaper or he’s killed by an effective infinite number of reapers.” For one thing, there is no “first reaper” in the sequence. Every individual reaper is preceded by an infinite number of reapers.

The paradox is that no individual reaper can be the one which kills Fred and yet Fred must be killed by an individual reaper.

For the purposes of the thought experiment, we assume that a reaper kills Fred instantaneously as soon as it activates, such that Fred’s death does not actually occur over time.

• araybold on said:

Surely there is a first actvated reaper, though? There are infinite ordinal numbers, yet there is a first.

• There is no first reaper in this sequence. The way the thought experiment is set up, the reapers map in reverse order to the ordinals so that the final reaper to activate maps to the least ordinal. And just as there is no largest ordinal, there is no first reaper.

6. araybold on said:

To expand on my post a few minutes ago, there are many countably infinite series of rational numbers that converge on a finite value, and the reciprocals of the powers of 2 are as good as any to use as an example. Suppose the reapers activate on the schedule of 7 + 1/2^n for n in {0,1,2…}, then Fred is killed by reaper 0 at exactly 8:00, and, so long as the reapers are each assigned a distinct activation time, then regardless of the exact schedule, we can put them in 1:1 correspondence with the reapers in this particular schedule. Reaper 0 always kills Fred.

The point is that, just because one numbering scheme does not yield an answer, it does not show that that there isn’t one, and if we pick a suitable one, it yields the commonsense answer. Compare this to the putting of the set of integers into a 1:1 correspondence with the natural numbers, to show that they are countably infinite: there are many ways that do not yield this result, but the point is that there is at least one that does.

Zeno’s Achilles-tortoise race paradox has the same problem: just because his model of the race never reaches the time when Achilles catches up to the tortoise, it does not mean that there is not one in which he does.

One can argue that numbering from 8:00 in forward temporal order is not just one way to do it, but the right one, on causal grounds: the decision as to whether to kill Fred is based on what happened before, not on what will happen in the future.

If you go beyond countably infinite reapers, however…

• You’re still thinking about this backwards. We are not talking about a sequence which begins at a finite value followed by a countably infinite number of reapers. We’re talking about a countably infinite set of reapers which ends at a finite value.

Think of it this way: what is the least rational number x such that 0<x≤1?

The answer is that there is no such number x. In exactly the same way, there is no first reaper.

• araybold on said:

I would say that you are thinking about it in backwards causal order – but who says there’s only one way to look at it, anyway?

With any infinite set, the probability that a given member will be assigned a finite number by any given numbering scheme is infinitesimal. What you have done is to show that a particular numbering scheme does not yield a finite ordinal number to the reaper that kills Fred. It does not follow that no reaper kills Fred; what’s actually happening here is that in your analysis, your process of induction never gets to that killer.

• You’re on the right track, in that this is certainly related to the causal ordering of the series. However the fact remains that there is no first reaper.

So, for any given reaper, if Fred had survived long enough for the previous reaper to activate, that previous reaper would have killed him and therefore the current reaper does not. Thus, for any given reaper, that reaper cannot have killed Fred.

• araybold on said:

Ah, I see the problem: you are begging the question in a rather subtle way. You write “if Fred had survived long enough for the previous reaper to activate…” but that is precisely the question that the induction is attempting to address. If the 100th. or the 1,000th. or any prior reaper has killed Fred, the induction does not get started from reaper 1 (the one activated temporally closest to 9:00.) You cannot justify an induction step by assuming the result of steps you have not yet reached; it is simply an invalid argument form.

Here’s an argument that Fred is killed not later than 9:00:

1) If Fred is alive when a reaper activates, Fred is killed (this is from the definition of the puzzle.)

2) If Fred is killed before time T, Fred is dead T (a fact about mortality.)

3) If a reaper activates before time T, Fred is dead at T (from 1 and 2).

4) At any time after 9:00, at least one reaper has activated (from the definition of the puzzle.)

5) (Conclusion) After 9:00, Fred is dead (from 3 and 4)

Putting aside the possibility of Fred dying of other causes, he must have been killed by a reaper.

• Ah, I see the problem: you are begging the question in a rather subtle way. You write “if Fred had survived long enough for the previous reaper to activate…” but that is precisely the question that the induction is attempting to address.

I’m actually not begging the question, at all. I’m following the direct implications of the premises of the thought experiment, which are as follows:

(1) Fred is alive at 8:00am.
(2) There are an infinite number of grim reapers in the room with Fred.
(3) If Fred is alive when a grim reaper activates, then that grim reaper instantaneously kills Fred. Otherwise, that grim reaper does nothing.
(4) Each grim reaper activates at a time exactly halfway between 8:00am and the time at which the subsequent reaper will activate with the sole exception of the final grim reaper, which activates at 9:00am. (Thus, the final grim reaper activates at 9:00, the previous one at 8:30, the one previous to that at 8:15, et cetera).

From these premises, we can draw a number of implications. Firstly, we can see that there is no first grim reaper in the sequence, thanks to (2) and (4). We can be certain, as you yourself noted, that Fred is dead at 9:00am from (3) and (4). However, by the exact same logic, we can be certain that Fred is dead by 8:30 from (3) and (4). And also that Fred is dead by 8:15 from (3) and (4). And that he is dead by 8:07:30 from (3) and (4). And, in general, that by the time any grim reaper activates, it would already have been dead from (3) and (4).

Thus, we have a paradox. None of the grim reapers can have killed Fred and yet Fred must be dead.

7. araybold on said:

You say that there was no begging the question in your earlier response, but now you have replaced it with a somewhat modified one that avoids that particular issue. Note that this argument, however, fails to find *any* time when Fred is *not* dead – and, by assumption, he did not die by any other cause but at a reaper’s hand.

It seems to be a paradox, I think, because this process for finding whodunit fails to terminate on the perpetrator, but that is because it does not terminate at all, and yet despite that, it never exhausts the list of suspects.

• I haven’t modified anything. I presented exactly the same argument, premises, and implications which I discussed in the original article.

There most certainly is a time when Fred is not dead. That is 8:00am, as per Premise (1).

Yes, the presumption of the thought experiment is that Fred will not die except by the activation of a grim reaper.

It is not the case that the “whodunit fails to terminate” or that “it never exhausts the list of suspects.” We quite clearly exhaust the entire set of suspects despite its infinitude by a fairly simple application of mathematical induction.

Again, this is a paradox because the terms of the thought experiment imply both that Fred must have been killed by a grim reaper by 9am AND that no grim reaper can have killed him.

8. araybold on said:

I see that I did not make my point clearly enough, which is that your argument considers an unterminating sequence of times at which Fred is already dead, yet never exhausts the list of suspects – there are always more reapers (an infinity of them, in fact), closer to the last known time when Fred was alive. Therefore, your argument’s conclusion, that no reaper killed Fred, does not follow from it.

Ironically, this turns the tables on your claim that there is no first reaper, as your argument for that claim shows that the form of argument used here cannot be fixed. Failure to come up with an algorithm to identify the killer is not proof that the victim was not killed.

• I see that I did not make my point clearly enough, which is that your argument considers an unterminating sequence of times at which Fred is already dead.

It’s not an unterminating sequence of times. The sequence of times very clearly terminates at 9:00am. It is an uninitiated sequence of times– there is no first time at which a reaper activates.

yet never exhausts the list of suspects – there are always more reapers (an infinity of them, in fact), closer to the last known time when Fred was alive.

It absolutely exhausts the list of suspects. We are not stepping through an iterative process, one at a time, in order to eliminate the reapers from contention. We are using mathematical induction to show that none of those infinite reapers could have been the one which killed Fred.

If I want to prove that for every negative integer x, it is the case that x<0, I do not need to iteratively or algorithmically test -1, then test -2, then -3, et cetera, in an infinitely iterated test. I can simply prove that -1<0 and prove that (x-1)<x, generally. The combination of these two premises allows me to make a judgment about the whole infinitude of negative integers without needing to test each one separately.

In exactly the same way, we can use mathematical induction to show that the whole sequence of reapers– not simply one at a time– cannot have been Fred's killer.

Ironically, this turns the tables on your claim that there is no first reaper

No, it really doesn’t. Once again, it is very clear that there is no first reaper in exactly the same way that there is no least rational number x such that 0<x≤1.

9. araybold on said:

Here is the unterminating sequence of times, all being when Fred is already dead (as you point out!), and all after the times when an infinite number of reapers have already activated:

“We can be certain, as you yourself noted, that Fred is dead at 9:00am from (3) and (4). However, by the exact same logic, we can be certain that Fred is dead by 8:30 from (3) and (4). And also that Fred is dead by 8:15 from (3) and (4). And that he is dead by 8:07:30 from (3) and (4).”

Your generalization does not work, as generalization leads to the conclusion that, at the time of __any__ step, Fred is already dead, and that it is after an infinite number of reapers have already activated

One issue here is that you are not actually using mathematical induction. As you say your argument has not changed, I will examine the earlier version, as it is closest in form to induction:

“So, for any given reaper, if Fred had survived long enough for the previous reaper to activate, that previous reaper would have killed him and therefore the current reaper does not. Thus, for any given reaper, that reaper cannot have killed Fred.”

An inductive argument has a statement to be proved, a basis in which it is established to be true, and an induction step which proves that if the statement is true in the predecessor step, then it is true in the current step.

The statement you need to prove by induction is “Fred was not killed in this step”, but the statement you actually make is conditional: ” if Fred had survived long enough for the previous reaper to activate, that previous reaper would have killed him and therefore the current reaper does not.” For the condition to be true in the basis case, you already have to assume that it is true in all successor steps. That is how you are begging the question.

Next, the induction step: here, you are just repeating the basis, but for this step. There is no argument that it follows from the truth of the predecessor step; in fact, as in the basis case, you are depending on it to be true for all the __successor__ steps – begging the question again.

If you disagree with me, please state explicitly the statement you are proving, the argument for its truth in the basis, and that it follows from the truth of the predecessor in the induction step.

I would also like to know how to do markdown in this blog!

• Your generalization does not work, as generalization leads to the conclusion that, at the time of __any__ step, Fred is already dead, and that it is after an infinite number of reapers have already activated

Yes. That is rather the point. At the time of any step, we can logically conclude that Fred is already dead. This is why we have a paradox.

An inductive argument has a statement to be proved, a basis in which it is established to be true, and an induction step which proves that if the statement is true in the predecessor step, then it is true in the current step.

This is exactly what we have done. Fred is alive at 8:00. If Fred survives until 9:00, the reaper which activates at that moment will kill him. However, if he had survived until 9:00, then it follows that he had survived until 8:30, in which case the reaper which activated at that time would have killed Fred; thus, the 9:00 reaper cannot have killed him. However, that is generally true, as well: if Fred is to survive until any particular reaper activates, then he must have survived until the previous reaper had activated, in which case he must already have been killed.

This is precisely mathematical induction. We show the implications of a specific step, generalize to show that it holds true for any particular step, and draw the implications over the entire set.

Once again, this is PRECISELY analogous to the manner in which we can prove that there is no least rational number x such that 0<x≤1.

10. araybold on said:

Actually, the situation is worse than I described it in my previous post. If, in the basis, you assume that the conditional is true in its successor step, then Fred was already dead in the successor step, so the conditional is false in the basis. On the other hand, if the conditional is false in the successor step, then the induction fails on the successor step.