A Grim Reaper (GR) timed to go off at *t*_{0} is an entity which does the following at exactly *t*_{0}. If Fred is not alive at *t*_{0}, the GR does nothing at *t*_{0}. If Fred is alive at *t*_{0}, the GR instantaneously annihilates Fred. (If instantaneous action is not logically possible, one can complicate the situation by allowing shorter and shorter time intervals for these actions.) The GR Paradox then is this scenario. Fred is alive at 11:00 am today, and that he does not die today unless killed by a GR and he does not get resurrected today. There are infinitely many GRs, timed to go off in a staggered way at the respectively times *t*_{1},*t*_{2},... where *t*_{n} is equal to 11:00 am + 1/*n* minutes. Well, by 11:02 am, Fred is certainly dead, since it is impossible that he survive a time at which a GR is timed to go off. But when was he killed? He wasn't killed by the 11:00 am + 1 minute GR, because if he were alive just before 11:01 am, then he would have been alive at 11:00 am + 1/2 minute, when another GR went off, and he can't survive a GR going off. It seems that none of the GRs could have killed him, because before each, there was another. So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that's mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually *do* anything—and if they don't do anything, how can they kill Fred?

The Kalaam argument needs the premise that there couldn't be a backwards infinite sequence of events. Here is an argument for this:

- If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible.
- If Hilbert's Hotel were possible, the GR Paradox could happen.
- The GR Paradox cannot happen.
- Therefore, there cannot be a backwards infinite sequence of events.

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.

Argument for (2): If Hilbert's Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room *n* should set the GR to go off at 11 am + 1/*n* minutes. And that would result in the GR Paradox.

The argument for (3) was already given at the beginning of the paper.

For about two years, I've smelled this argument coming, but I think my vanity has kept me from seeing it. I still have to confess that I have a really hard time accepting the corollary that Hilbert's Hotel couldn't exist—that corollary seems extremely counterintuitive to me. I wish I had some good way out.

On the other hand, establishing a major premise of an argument for the existence of God is a very happy outcome.

## 26 comments:

Why does the idea that Hilbert's Hotel couldn't exist seem counter-intuitive to you (if such can be explained)? This seems an unusual intuition. But then you seem a fairly unusual person (to me!) :)

Well, we can coherently model a hotel with an infinite number of rooms as well as one with a finite number of rooms. We can come up with a fine causal story about how such a hotel could come into existence (say, by divine causality acting simultaneously on all the rooms). There seems to be no reason to think that when we talk of such a hotel, we are talking nonsense. (Though Lowenheim-Skolem might worry one in regard to the determinateness of "countably infinite." That's an argument against actual infinities I haven't heard.) And all the stuff people say about why Hilbert's Hotel is impossible--apart from the present argument--seems completely unconvincing.

There is perhaps an easy cultural explanation of my intuitions, in terms of my mathematical inculturation.

Perhaps I'm missing something here, but it seems to me that the GR paradox is a good argument in favor of time being discrete rather than continuous. If this is so, then I don't see how Kalam enters into it at all.

That's true, and I haven't noticed it. The argument has an additional premise, namely that time is not necessarily discrete.

Distinguish between two discreteness views:

Rigid discreteness: Time is like the integers, with fixed spacing between times.

Aristotelian discreteness: There are in fact finitely many moments of time, but any interval of time can be subdivided.

I think a discrete time hypothesis is going to be hard to get working mathematically in a relativistic framework. There would, it seems, have to be some spacelike hypersurfaces, or else space would also need to be discrete, and then, I suspect, all of relativistic physics would be a mess, besides us having the sorts of problems in Zeno's Stadium argument.

So, probably, only Aristotelian discreteness is an option.

But I think there is an argument from Aristotelian discreteness to the impossibility of an infinite regress of events. Here it is. If an infinite regress of events (events #-1, -2, -3, ...) is possible, then event #-1 could cause something happening at time t0 + 1, event #-2 could cause something happening at time t0 + 1/2, event #-3 could cause something happening at time t0 + 1/3, and so on. And there is no reason to rule out all of these things happening together. But if that were to happen, Aristotelian discreteness would be violated.

Another way to see the point, is that I think the inference from an infinite regress to Hilbert's Hotel works even if time is discrete. And if time is Aristotelian discrete, the inference from Hilbert's Hotel to GR Paradox cannot be blocked, and yet GR Paradox is incompatible with Aristotelian discreteness.

I like these sorts of arguments, but having seen several of them (since Jose Benardete's 1964 'Infinity: An Essay in Metaphysics') I'm not convinced that yours works any better. E.g. why cannot the mereological sum kill Fred at 11am? None of them do anything individually, but that is not the suggestion (of, e.g., John Hawthorne's 2000 Nous paper).

Alternatively, the GR could collectively cause Fred to be teleported out of spacetime and back in again after some arbitrary time, if spatiotemporal discontinuity is not ruled out explicitly (and QM and black holes seem to tell against such a ruling). Then either the next GR kills Fred or else he returns after 11:01 and is fine. The cause of such a teleportation is the mereological sum of the GR. If such teleportation is ruled out explicitly in the statement of the scenario, the sum could (logically) do something else. The more that is ruled out, the less plausible the scenario becomes. One could just say that the sum of the GRs causes something possible to happen, indeterministically (e.g. Fred's death at 11, or his jumping to 11:02), so as to avoid logical contradiction.

And that countable sums can have bizarre side-effects follows from allowing actual infinities in various quasi-physical settings, some of which seem relatively plausible. Leonard Angel's 2001 BJPS paper is a relatively plausible example. Aleph-null collisions of moving masses with one stationary mass are set up to occur in the future, in such a way that the stationary mass should already have been hit before each of them arrives. So the mass has to move before any arrive. But if you look at the scenario in spacetime, the mass is hit by the edge of the oncoming masses, which is surely no weirder than being hit by a mass with an open instead of a closed surface?

If no member of a collection even partly causes anything outside of itself, the mereological sum of the members causes nothing outside of itself. We can suppose that no GR causes anything outside of itself except when Fred is alive.

What if we simply stipulate that, in fact, there is nothing in the world other than an individual GR that can kill Fred?

Hawthorne does not, I think, share my Aristotelian picture of all causal interaction arising from the interactions between substances. I also don't believe in mereological sums--the idea of a mereological sum is preposterous. :-)

Alex,

I'm a little puzzled by your argument from Ar. discreteness to no infinite regress. If it worked, couldn't a similar argument be used to show that an infinite progress is also possible?

Could saying that the sum causes the weird effect be a facon de parler though? I think it was Yablo (in a 2000 Analysis article) who likened the problem to the difficulty of getting a lot of interacting units to work together. That is, why should we suppose that all the GR work as they are supposed to when we have so many of them so arranged?

We can suppose that each GR does nothing but kill Fred if Fred is alive. That seems possible. But why should we think that it is therefore possible for them to so behave when they are all so arranged? One could regard your scenario as a reason for not so thinking (especially if one is not presupposing theism, and so takes the evolution of life via the complex interactions of simple parts to be possible).

There certainly is something paradoxical here (and I personally disbelieve in countable Actual infinities), but the problem with the argument is that such scenarios are standardly taken to show that infinite structures are counter-intuitive (and our intuitions plausibly apply only to small numbers of ordinary objects).

If we stipulate that nothing but an individual GR could kill Fred then (i) Fred might teleport, but furthermore (ii) although Fred could (given answers to all objections like (i)) be incompatible with the other stipulations and Actual countable infinities, why should we regard Fred as really possible? Such a stipulation is intuitively unrealistic.

Rob:

Yes, and an infinite progress

ispossible, as long as there is nothing after it. What is, apparently, not possible is an infinite progress with something after it, as that would let one generate a GR paradox.Alex,

It remains to be shown that it is an infinite progress with something after it that generates the GR paradox. Indeed, it remains to be shown that the GR paradox, even if the GR scenario is contradictory, implies that there cannot be a backwards infinite sequence of events, which the theistically interesting Kalaam argument needs.

First you need to show that the GR contradiction implies that there cannot be a backwards infinite sequence of events that is itself preceeded by some events, and then you would need to show that the latter implies the desired result. And that such implications are not straightforward can be simply indicated by spatial arrangements (which are especially compelling if one is not a Presentist).

Consider billiard balls. One might think that given some number we could in principle have that number of billiard balls. But if the number is a googol then that many balls clumped together would give us a black hole, not a lot of billiard balls. Such a number of balls could be spread out in space of course. Indeed, an infinite number might be spread out in an infinite space. But there is at least the shadow of a doubt over what you first need to show.

Now, GR are fantastic creatures, not physical ones, but that only helps with finite numbers, not with infinite numbers. Consider idealised Newtonian billiard balls, for a simple example (cf. Laraudogoitia's 1996 Mind paper). Suppose there is a line of such billiard balls centred at n inches from some origin, for all natural numbers n. Another billiard ball hits that line from the opposite direction from the origin. It is halted by the first ball, which moves off to hit the second ball similarly. In effect the momentum travels down the line of balls.

Now suppose that the balls could in principle be smaller and smaller but of the same mass, approaching point masses (a bit like the GR, but spatially). Then the line of balls could be finite. What happens to the momentum? It disappears at the open end. If that is a violation of a Law of Conservation of Momentum, then such balls (so arranged) are impossible. But if no such Law applies to such bodies, then they are possible (such an infinitary physics is developed by Laraudogoitia in subsequent papers). The contradiction for some idealised balls does not show that the other sort of idealised balls are impossible. Rather, the latter sort cast a doubt over whether such a Law (which holds in the finite case)

shouldcarry over to the infinite case.Whether or not there is a contradiction depends upon how the GR are defined. It would be a considerable job to show that such a contradiction threw any doubt at all on the more general possibility of a backwards infinite sequence of events that is itself preceeded by some events, let alone gave the desired result. That fact was perhaps obscured by the form in which you put your argument, which even seemed to allow a corollary of the impossibility of Hilbert's Hotel, which (being spatial) would require far more argumentation to establish, I fear...

But there is no need to think of these as physical entities. Non-physical entities with precisely specified causal powers will do the job.

But surely physics is a good example of the general metaphysical case: why would non-physical entities be any different mathematically? The GR could, for example, be objectifications of aspects of the will of God; but even then there would be the question of whether the inconsistency implicit in the hypothetical will to destroy Fred via a reverse omega-sequence of GR implies an inconsistency in

anyhypothetical will to act via reverse omega-sequences.The point is that although the specification of the causal powers of the GR may well seem to be unproblematic, it is arguably only finite possible numbers of GR that enter into our intuitive assessment of the reasonableness of such a specification. There are good arguments (e.g. from real-world and idealised physics) that if infinite numbers of things were possible then the properties of those things that we could have such numbers of would be restricted in counter-intuitive ways. Even large finite numbers can give rise to such counter-intuitive restrictions (in the real world).

..incidentally I was wondering what you think it is that makes something with a specified causal power a physical thing, what it is that might be lacking?

This is a brilliant argument, one of the best arguments for "the universe began to exist" premise I've ever seen; certainly the most convincing one I've found against an actual infinite existing. Yet I haven't found it published anywhere. Do you plan to get it published? Or has this argument for the finitude of the universe's past already been published and I missed it?

Rob Koons is writing an article along these lines.

Thanks! With any luck, we'll (eventually, hopefully) be able to find more about it here.

I think the problem with this argument is that it cannot be made in analogy with the assumption of an infinite series of past events.

Even if the infinite set of past events is finite in time, there exists no time marking some state of affairs ("Fred is alive") temporally before ("at 11 AM") all other states of affairs ("the GR checks to see if Fred is alive and kills him if he is"), since such a case directly contradicts the assumption the argument makes, namely that the set of events is past-infinite - meaning there is no "starting point" temporally prior to every other point, by definition.

Also, on a technical note, you have not defined what the GR does beyond t0. At t1, t2, etc., he could simply do the cha-cha. It would be more exact to say that the GR checks at any time tk.

Prof. Pruss,

It looks to me like we need a couple more controversial assumptions than just the past-infinitude of time to generate a contradiction. At the very least, the arguments (in their current form) require time be arbitrarily divisible (I guess you call this Aristotelian discreteness). Furthermore, we must consider the alternative possibilities that it requires a fixed length of time e > 0 to kill Fred, in which case the killing events overlap infinitely near 11am, or else that it takes an arbitrarily small length of time, say 1/n^2 for the nth reaper, to kill Fred. In the first case it's not obvious that we can reach a contradiction, and in the second case we have tacked on a third controversial assumption. So your argument does work to show that those three assumptions are apparently incompatible, but as far as I can tell it doesn't show that any particular one of them is false, i.e. it doesn't show that time is past-finite.

I wrote a blog post about this issue here.

Oh, oops... I forgot to mention the second of the three "controversial assumptions" I mentioned above: it is that the real world has the spacial and material resources to support an infinite collection of physical objects (such as grim reapers).

I've tried to create a new argument against an infinite past that is somewhat similar to this Grim Reaper argument, with the aim of avoiding some of the potential issues with it:

Argument against an infinite past.How about he was killed at the infinitesimal time <11 hours 1 minute, 11 hours 1/2 minute, 11 hours 1/3 minute, 11 hours 1/4 minute, ....>

If you can believe in infinite numbers of thing (like Hilberts hotel) then the inverse -- infinitesimals -- could also exist. Now infinitesimals don't exist on the real number line as irrational numbers don't exist among the rationals. One only gets a contradiction if one assumes that they do. The same is true for the infinitesimal time that you mention. Now it would mean that *if* Hilberts hotel were possible, then infinitesimals times would be possible and we don't live in a universe that is describable using the just the real numbers... we need the hyper-real numbers with both infinite numbers and infinitesimals.

Dear prof. Pruss,

I'm really interested to hear your thoughts on Thomas Larsen's modified argument against the infinitude of the past:

http://tomlarsen.org/2011/11/14/argument-against-an-infinite-past

It seems to me this argument might be even better than the Grim Reaper argument, since it works with equal discrete intervals of time and so it doesn't presuppose that time can be infinitely divisible.

I've thought of things of this sort, though not quite as nice as this, and I don't find it as convincing, but I can't put my finger on why.

Apparently this type of paradox (the one that Thomas Larsen put forward) is similar to the so called 'Yablo Paradox'.

In the following article by philosopher Casper Storm Hansen he argues against the possibility of an actual infinity based upon such a paradox:

http://core.kmi.open.ac.uk/download/pdf/5849860

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