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.

## 70 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

1.) If there could be an eternal being capable of creation ex-nihilo, Hilbert's Hotel would be possible.

2.) If Hilbert's Hotel were possible, the GR Paradox could happen.

3.) The GR Paradox cannot happen.

4.)Therefore, there cannot be an eternal being capable of creation ex-nihilo.

Defense of 1:

Argument for (1): If there an eternal being capable of creation ex-nihilo, this being could create, out of nothing, an infinite number of hotel rooms none of which are destroyed. An infinite number of hotel rooms would then be the result.

If you don't accept the conclusion of this argument, you should understand why no rational person would accept the conclusion of Pruss's.

It's easy to explain why Pruss's first and second premises are flatly false, and his argument unsound as a result, but the illustration is, I think, even more effective.

Thanks for a good criticism.

My view has shifted since my post. I am happy with the possibility of Hilbert's Hotel. The difficulty lies not with a HH in general, but a HH that lies as a whole in the causal history of an event (in this case, the end of the life of the person who is to be killed by the reapers). The problem isn't with an actual infinite as such, as with an actual infinity of causes that causally impinge on one thing.

Anyway, if I'm right, then even God couldn't create an actual infinity of causes that impinge on one thing, since such an infinity would be impossible, and God can't do the impossible.

That's interesting to hear, and I do appreciate the response.

I would agree that idea of an infinite past sequence of causal events is more a subtle and interesting one--and, also, more relevantly tied to the possibility of past-complete spacetimes.

I do think a different argumentative approach would be required to demonstrate this alternate contention, though.

Anyway, thanks again for the response. Your argument was, clearly, thought provoking.

This is just too messy and is trying to prove something that we already know, quite easily, is impossible. The infinite reapers are superfluous as the first one will do simply to hint that time is NOT discrete(I don't know why someone thought this showed that) but continuous. I dont believe it adds weight to what's already obvious and only fought to avoid being created.

Examples like a book with no author are much easier. An infinite past allows a book to be passed around that has no author(everyone you ask says they got the book from someone else). If you remove Origins...you remove logic. The bottom line is the atheists have no hope and even if they cite some incredibly improbable loophole, it simply has no weight in reality.

Well, certainly seems to be the case that no-one can honestly claim to have proven a past-complete universe--or infinite temporal regression--impossible. This argument doesn't do it, and neither does any other approach I've ever seen.

Not that it has anything at all to do with theism or atheism, of course.

Perhaps there is another assumption, besides the discreteness/continuity of time, underlying this paradox.

It seems that the Grim Reaper Paradox implicitly assumes the A-Theory of Time. The action (or inaction) of each previous Grim Reaper provides a causal sequence which actualizes the action performed by the

nth Grim Reaper.What if this is the incorrect assumption? Perhaps it is wrong to assume that the actions of the Grim Reapers

canbe altered, in the first place.On the B-Theory of Time, there can be no Grim Reaper paradox, since the actions of each individual Reaper-- whether finitely or infinitely many-- are co-extant with the actions of all the other Reapers.

The problem is that you have asserted that Fred is alive at 11:00 but your inverted supertask entails that he will be dead at that time. So, your conditions are inconsistent.

Unknown, that's actually the point of his argument. He's constructed a scenario which seems to demand two incompatible outcomes: that Fred is dead and that Fred has not actually been killed by anything.

His proposal was that, since the scenario leads to inconsistent outcomes, an infinite past must be impossible. Of course, this reasoning is badly flawed: the scenario requires many different elements, and the most that the argument could actually obligate us to do is agree that the conjunction of those elements is impossible, which could be true even if even all of the elements are possible.

This failure in reasoning is easy to highlight in a more concrete way, however. One of the elements on which the scenario relies is that there can be infinitely many physical signals within a finite space. Hence, we can satisfy the demands of the argument simply by agreeing that "either a universe cannot have an infinite past, or it cannot have infinitesimal signals'"

However, we already know that our universe does not allow infinitesimal signals, which means that the argument tells us nothing at all about whether our universe has an infinite past.

Hence, the argument is a failure. Frankly, it's a pretty egregious failure, to the point where I would question the honesty and/or competence of anyone suggesting in seriousness that it does anything at all to prop up the Kalam argument, but (in Pruss's defense) this is a pretty old post. I am pretty sure Pruss knows by now that this argument is a failure, so it would not be fair to judge present-day Pruss on it.

I think there is still something to the argument. My current version of it is in my infinity book: https://smile.amazon.com/Infinity-Causation-Paradox-Alexander-Pruss/dp/0198810334

I would be very surprised if you had a formulation that avoids the class of objections from which my objection above is pulled, but I have to admit that I don't really care enough to buy a book to find out.

I certainly don't think any formulation of the GR argument against an infinite past that I have seen, including this one, does anything more than fail trivially.

If Pruss asserted that Fred is alive at 11:00:30 he would have asserted an obvious impossibility. Would there be anything interesting about an argument that asserts an obvious impossibility? Asserting that Fred is alive at 11:00 is to assert the same impossibility, it is less obvious but it differs only as a matter of degree.

The thing that is interesting about the GR scenario is that it is clear that Fred is dead by the given time (11:00, in thus case) but that it is impossible to identify which grim reaper actually kills him. Indeed, for each grim reaper, it is possible to confirm that that grim reaper didn't kill Fred.

So, we agree that Fred must be dead...but If none of the GRs killed him, then he must still be alive. That's precisely the paradox that the argument revolves around.

You're mistaken, we cannot confirm for each Grim Reaper, that that Grim Reaper didn't kill Fred, we can only do so for finitely numbered Grim Reapers.

The results of supertasks aren't derived from the finite cases, as has been known since Benacerraf's response to Thomson's lamp, and this can be demonstrated by constructing two supertasks with identical finite intermediary states but different infinite final states.

If Pruss were to conclude that Fred is alive at 11:00, rather than just asserting it, then he would have made Thomson's mistake.

Please feel free to dismiss this question since I'm not a philosopher, this topic is old, you may well have already thought of this, etc.

It seems to me that, if 'death' means the death of my body, that's a physical process, and no physical process like this is truly instantaneous (or even of arbitrarily short duration). For my body to be destroyed in the most rapid way possible, it seems some chemical bonds need to be broken. If this were a matter of moving all my atoms apart, my death would arise sometime soon after the bond lengths were doubled, and if this were brought about at the speed of light, the timescale would be around 1e-19 s.

If that's true, then it seems like the Grim Reapers 1e19 and up are jointly responsible for killing me.

Now, maybe I'm wrong and it is physically possible for me to die in an instant, or even an arbitrarily short amount of time. What is the evidence or argument for this?

Or maybe there is still a paradox about who first started the atoms in the motion that ultimately results in my death, and maybe there's no good answer to that, kind of like, "show me the first delta along an infinitely divided line segment." But the paradox seems to cut less deep. To me anyway. How would you rephrase this paradox, if there is a finite minimum amount of time needed to kill a person? Or do you think in this case the argument falls apart?

This paradox bothers me more for its implications about the continuity of time (though your comment about Aristotelian understanding of the continuum helps). I reject the Kalam argument for other reasons, and favor other arguments for God's existence, esp. arguments that make use of the PSR.

Not related to the above; your book on the PSR played a small role in rescuing me from Spinozism. So regardless of whether this observation and questions are worth a reply, at least let me take this opportunity to say "thank you" for the wisdom and help the ideas in your book gave to me.

The given material conditionals

"1. If there could be a backwards infinite sequence of events, Hilbert's Hotel would be possible."and"2. If Hilbert's Hotel were possible, the GR Paradox could happen."are incorrect.

The first one is actually a biconditional:

1. There could be a backwards infinite sequence of events, if and only if Hilbert's Hotel would be possible.⇔

1. There could not be a backwards infinite sequence of events, if and only if Hilbert's Hotel would not be possible.

The second one is reversed:

2. The GR Paradox couldonlyhappen, if Hilbert's Hotel were possible.⇔

(2. If Hilbert's Hotel were not possible, the GR Paradox could not happen.)&(2. If the GR Paradox could happen, then Hilbert's Hotel were possible.)

From a paradox alone, which can't be or is impossible, can't be concluded, that any other backwards infinite sequence of events, which might be not contradictory and therefore possible, are also contradictory and therefore any other backwards infinite sequence of events are all impossible.

Regarding the first point, if the biconditional is true, so is each unidirectional conditional.

Regarding to second point, I think the onus on someone who denies my conditional is to come up with a different explanation of why the GR story is impossible than that Hilbert's Hotel is impossible. It *seems* that if you could have HH, you could have the GR factory, and then you'd have the GR paradox.

That said, I have since found such a different explanation, given at length in my infinity book. HH is innocent as long as there is no causal cooperation between all the rooms. But in the GR factory case, there is such causal cooperation, since all the GRs are working together.

I have no problems with your propositions, but I have a lot of problems with your line of argumentations and their false conclusion of the impossibility of an infinite past couse of false or incorrect material conditionals.

It *seems* that if you could have HH, you could have the GR factory, and then you'd have the GR paradox.Again, this material conditional of yours is not correct, as it is stated.

If it *seems* to you that way, then in my opinion actually it is this way:

You could have the GR factory

only, if you could have HH.⇔ If you couldn't have HH, then you couldn't have the GR factory

⇔ If you could have the GR factory, then you could have HH.

It appears to me, that the possibility of GR necessitates the possibility of HH. In other words the possibility of GR factory demands the possibility of HH and not the other way around.

I think it's worth noting, several years after my first response to this post, that there is another serious problem with the GR scenario:

A set of physical apparatus capable of doing what the GRs are described as doing in this scenario is not possible. It doesn't matter how much time or space or stuff you have with which to create them, they can't be created. A physical GR has some finite, non-zero size. It is some finite, non-zero distance from Fred, and any signal between the two has some finite, non-zero travel time. Each GR has to receive a signal from Fred (to verify that Fred is alive) and then send a signal to Fred to annihilate him, if he is alive. For each GR, these two signals have to complete their round trip before the next set of signals for the next GR begin.

However, trivially, this requires there to be an infinite number of GRs within a finite distance of Fred. Since each GR has non-zero volume, this isn't physically possible.

The general idea of the argument is that, if one could have infinitely many physical rooms, one could build a GR in each room, ending with infinitely many GRs, thus allowing for the scenario described. But, in fact, even if you had infinitely many rooms and could build infinitely many GRs, the scenario as described would still be impossible.

What you actually need are not physical GRs, but rather magical murder-ghosts. Either you'd need to be able to pack infinitely many magical murder-ghosts into a finite space, or your magical murder ghosts would have to have the magical power to instantly survey Fred for life, at any distance as well as the magical power to instantly annihilate Fred from any distance.

My original objection to this argument, years ago, was that it isn't merely the possibility of an infinite past that entails the possibility of the GR scenario. Rather, the possibility of the GR scenario requires the conjunction between the possibility of an infinite past and a bunch of other factors--including the infinite divisibility of time, which is far from a given. But, of course, even that was being far too generous. In fact, an infinite past has nothing to do whatsoever with whether the GR scenario is possible. The GR scenario requires the possibility of infinite, magical, physics-defying murder-ghosts.

If infinite, magical, physics-defying murder-ghosts are possible, then the GR scenario is possible, regardless of whether an infinite past is possible or not.

If infinite, magicaly, physics-defying murder-ghosts are not possible, then the GR scenario is not possible, regardless of whether an infinite past is possible or not.

The suggestion, then, that the GR scenario tells us anything at all about whether an infinite past is possible...is basically just laughable.

It seems obvious that the speed of light limit is not metaphysically necessary.

It seems obvious that that is not really a defense against my objection.

Sure, we can imagine a universe in which instantaneous action is possible. Maybe (maybe, though your argument still certainly doesn't establish this) a universe *like that* can't have an infinite past, but so what?

This is the same basic problem, again. The problem with the GR scenario is not an infinite past: it's all of these other unstated factors.

What's more likely: that instantaneous action over physical distance is impossible, or that an infinite past is impossible? That both are possible, but not in conjunction? Oh, right, and that doesn't actually cover it: your GR scenario needs infinite stuff. It needs an infinite past without an information pinch-point, like our universe seems to have in its past. It needs infinitely subdivisible time, and machines capable of arbitrary chronological precision.

At *best,* the argument only points to the impossibility of the conjunction of all of these things. It remains that your suggestion, that this argument establishes the impossibility of an infinite past, is laughable.

It's kinda funny, too, that, while the *value* of the speed of light, or the maximum speed of physical signals, isn't metaphysically necessary, that doesn't really matter. Even if it is metaphysically necessary just that there is *some* upper limit on the speed of physical signals, your argument would fail in the same way.

And it isn't nearly so "obvious" that this isn't the case.

Indeed, fine tuning arguments generally go the opposite direction: this maximum speed is one of those parameters which must be given a finite value. I actually think if you look around, you'll find far more skepticism directed at the proposal that there could be a physical universe without a maximum speed for physical signals than at the proposal that there could be a physical universe with an infinite past. After all, actual physicists are pretty well on the fence about the latter, but we are all quite certain about the former. We have plausible models of hypothetical universes with infinite pasts, but none without finite maximum signal speeds.

Honestly, it gets worse even than that.

Let's imagine that it's possible for there to be a universe with infinite space and infinite physical stuff (both of which are required for your GR scenario). Let's imagine that this universe can also be one without a maximum speed for physical signals, since that is also required for your GR scenario. In this case, we can run into the GR scenario even with a finite past. After all, with infinite space and infinite stuff, we can have our infinitely many factories build our infinitely many GRs all at the same time. An infinite past isn't actually required at all!

So. Whatever factor or conjunction of factors is impossible, thanks to the impossibility of the GR supertask (assuming we even agree that the GR supertask is impossible, which we don't actually have to do at all) doesn't even include an infinite past.

An infinite past might be incidentally impossible, but (again) this argument absolutely fails to support that claim. It is flawed at every level of examination.

Huemer's book on paradoxes infinity, btw, embraces the solution to a lot of paradoxes by limiting parameters to the finite, including having a metaphysically necessary requirement that there be an absolute speed limit. That seems to me to be very implausible. It seems clear to me that teleportation is logically possible.

I agree that with infinite space and infinite stuff all causally connected and no speed limit one also gets the GR paradox. So that's a reason to reject the possibility of an actual infinity. I don't think it's a conclusive reason: what I end up rejecting in the end in my book on paradoxes of infinity is an actual infinity of things with certain kinds of causal interconnections.

We certainly have models of universes without an absolute speed limit: Newton's gravitational model.

It seems clear to me that an infinite past is logically possible. "Seems" isn't enough, is it?

I suppose I would be interested to see if you could actually produce an argument against an infinite past that doesn't fail trivially, as this one has, but "yeah, but read my book" doesn't really move me as a response to "your argument doesn't work."

I have very little incentive to spend time and money on a book, if this argument is any indication of the quality of the intellectual work therein.

I do actually think, that one could produce an argument against actual infinities of things with *certain* kinds of causal interconnections--indeed, I think if you had aimed your argument here differently, it could have been not-bad, but I don't believe for a second that you actually have a competent argument for causal finitism more generally, and I also don't care enough to pay money to find out.

I think "seems" is always evidence, but defeasible like most forms of evidence. That it seems that an infinite past is possible is evidence that it is possible. Indeed, this is why for decades before I came to the argument in this post, I thought that an infinite past was possible: it seemed to be! It *still* seems to me to be, but I have stronger seemings behind the arguments to the contrary. (And in a sense I think an infinite past is possible. Just not an infinite past *causal* chain.)

Thank you Dr. Pruss for your response and clarification.

I agree with most of the things, that you stated in your response and I like to add, that I for myself also think, that there is something deeply flawed in the general concept of causal chains. From a pure physical standpoint causal chains appear to be very deterministical and if actually there would be the Laplace's demon, then it would appear, that all causal chains would be already determined. That would be very off putting.

Hey Dr. Pruss,

A couple of questions. For one, do you subscribe to Phenomenal Conservatism? I saw your comment about "seemings" as evidence and was wondering. Also, does it still seem to you that a HH is possible? What about transfinite operations done on the occupants in the hotel, does that sway you in any way? Blessings!

Mr Nagy:

Causal chains need not be determinstic. Jill won $10 at a casino. Then she took that $10 and bought herself a drink. Then she drove and smashed up her car. We have a chain: going to a casino, winning $10, buying a drink, driving, smashing car. But all the links in the chain are indeterministic.

Unknown:

I am inclined to accept one direction of Phenomenal Conservativism: if something seems to me, that fact is evidence for it. I don't accept the other direction, that that's the only evidence there is.

And, yes, it still seems to me that a HH is possible. And my current view is that a HH is possible, but that one cannot have an infinite number of things cooperating to a single effect, so the denizens of the rooms cannot cooperate for a single effect.

Hallo Dr. Pruss,

I agree to, that causal chains need not be deterministic. Yet why shouldn't

Laplace's demonbe capable of determining Jill's actions?It appears to me, that if Laplace's demon has all the knowledge about physics and has all the knowledge about all the actual physical objects and further has enough capacities and power to do the necessary computations, then Laplace's demon is capable of determining all of Jill's actions and further is capable of determining all of external actions, that might happen to Jill.

Best regards,

Zsolt Nagy

But our physics is not deterministic. And we are not purely physical anyway.

If our physics is not deterministic, then why would anybody care to shoot satellites into our or other planets orbits? Every nation, that can and does shoot satellites into orbits, does it, because our physics is deterministic.

Also why would Facebook, Google and Amazon gather so much information about everybody?

Sure, we might be not purely physical, yet I still think, that

Laplace's demonwould be capable of determining Jill's and anyone's actions.All we need for satellites is high probabilities, not determinism.

Then I guess, that by probability p=1 we got determinism and by probability p≠1 or 0≤p<1 we don't got determinism.

What about a very high probability with almost 1=100% (p≈1)? Is that determinism or not? I suggest we could say, that it is approximately deterministic.

Further it still appears to me, that all considered causes and effects only appear to be not deterministic with probability p=1 cause of our incomplete knowledge.

What about

Laplace's demonwith his completed knowledge? It seems to me, that it is capable of determining all causes and effects.Even if we have incomplete knowledge, approximated determinism is still achievable.

We don't know whether in our brains we even have approximate determinism. It depends on how much quantum-level stuff is relevant to our brain function.

On standard views of quantum mechanics, the indeterminism is NOT merely due to incomplete knowledge. In fact, given some assumptions such as locality (basically, that one cannot have causal influences that are faster than light), it can be proved that the quantum indeterminism cannot be merely due to incomplete knowledge.

Even if reality should present itself indeterministic on the fundamental level, this won't stop

Laplace's demonfrom calculating and determining, that can be calculated and determined.Even if Jill's brain or consciousness should be indeterministic, this won’t stop Facebook, Google and Amazon from gathering all the information about Jill and from that determine her actions. Jill is going to a casino, because Jill already has shown her tendencies for gambling.

Her winning 10 $ with a specific expectation from general understanding of gambling and stochastics and spending that money on a drink, because Jill already has shown her tendencies for alcoholism. From that it’s not far to conclude and to determine her carambolage with her car given her physical, her mental, weather and car state in combination with her route, that she takes regularly from casino to her home.

You see, even if consciousness might be indeterministic, the rest of the world still might be determinisltic or a lot of the times approximately deterministic. And one day we might even could approximate and simulate the indeterministic mind via quantum computers.

You have NOT demonstrated that there cannot be an infinite sequence of events. You have merely demonstrated that dealing with infinities is difficult and confusing. If your argument were valid, then you would have demonstrated that there isn't an infinite number of points in an interval on the numberline in which case you would have demonstrated a fundamental flaw in modern mathematics for which you deserve the most prestigious prize in mathematics. That your proof is found on a blog not some mathematics journal, and the fact that you mention God in a mathematical proof suggests you need to do a bit more studying of the mathematics of infinities.

"If your argument were valid, then you would have demonstrated that there isn't an infinite number of points in an interval"

No. He doesn't claim to do that. There are different kinds of infinities and this paradox gives evidence that there cannot be a infinite causally related sequence. Numbers are abstract objects and do not stand in causal relations. LOL @ you saying that this isn't already published works, instead of being grateful that important works in philosophy are being discussed by the author himself on a blog site. He's literally written a book on this published by Oxford and it seems you are the one who needs some reading on the philosophical implications of the existence of infinities. It's quite striking that you're ignorant of the fact that actual learned philosophers who disagree with Pruss still don't respond with the nonsense Muh Number Line Objection.

If you want to learn more read the published works. Btw, here's a post by an atheist philosopher who disagrees with Pruss and doesn't respond with these childish objections.

There is no "approximate determinism" there is only determinism. The moment you add randomness to a decision or a result, even one which is 99.99% on one side, it becomes undeterministic. A good way of understanding why this should be obvious is that there is no clear threshold probability p we can choose so that only upto p things are undeterministic.

We collect information about people only because we are operating under some assumptions and models of determinism which are only approximations. Predictions of consumer behavior is based on machine learning models and if you've taken a course in ML you'd have learn how much these things are based on so many assumptions and giving different extents of probabilistic confidence.

And please don't randomly throw in pop culture speculations on quantum computing.

@Mole

I'll ignore your unnecessary rudeness for now.

Regarding infinite causal chains, all this is is rehash of Zeno's paradox (Achilles and the tortoise). So is it your claim that that paradox proves that time is not infinitely divisible?

Unfortunately he has NOT demonstrated that there cannot be an infinite causally related sequence. A sequence may be both infinite and bounded. He does not seem to understand this most basic finding in mathematics. It is this same issue that Zeno's paradox plays with. It presumes an infinite sequence that is bounded by but does not include a point then points out that the point is not part of the sequence. Interesting, but not a paradox. The paradox occurs when you demand that the point IS part of the sequence. But this is a problem with your demand, not with the mathematics.

Hallo Mole,

If educated assumptions work, why wouldn't you make those? Because there are "just" assumptions?

There is a fine difference between educated assumptions and "just" assumptions. The difference is, that educated assumptions tend to work most of the times. If they wouldn't work, then they would be adjusted, till they do work. You can't state that also for "just" assumptions.

Also here is an infinite causal-chain for you without any contradictions:

After a chicken comes another egg.After an egg comes another chicken.So where is the contradiction in that infinite causal-chain. Why would it be impossible, if there is no contradiction in that infinite causal-chain?

Best regards,

Zsolt Nagy

A slightly more formal analysis than my previous comments:

The basic apparent paradox is built on the fact that in an infinite sequence there is never an infinity between any two terms. But that does NOT prove the sequence is not infinite. So although I agree that there cannot be an infinitely long causal chain BETWEEN two events, it does NOT prove that causal chains cannot be infinite.

Of course the original post only refers to subdividing causality infinitely, it does not deal with the possibility of an infinite past timeline, nor does it deal with the possibility of a bounded timeline that is nonetheless infinitely divisible eg if time is the open set of times after 11:00am.

And finally, there is the assumption that causation is pointlike and thus countable ie there is a definitive list of a causes b causes c etc. I am sure that with some thought one could come up with concept of causation that did NOT require countability or exhibit pointlike behavior and could thus be uncountably infinite.

Food for thought:

The set of rational numbers is countable. That means that we can find a sequence such that any given rational number has a previous number and a next number in that sequence. The sequence is infinite. But it is apparently NOT possible to create such a sequence that follow size order. Between any two rationals you care to pick there is always another rational. The hard part is given one rational, can you find the next biggest rational in the sequence? Does no such number exist?

TW:

The paradox is only one of many I consider. The main thought in the book is that the hypothesis that there are no infinite causal histories kills *many* paradoxes in one fell swoop. You're right that if this paradox were the only one, it would at most establish that we cannot have an infinite causal chain between two events. But the latter hypothesis is insufficient to handle some of the other paradoxes in the book.

@Ross. Well, I have not read the book, nor is the book in the post above. I do not however consider the fact that subsequences between two points of sequences are never infinite even if the sequence itself is infinite to be a paradox at all. Its a rather trivial mathematical observation. But as I say above, it doesn't really prevent infinite causal sequences in finite time, nor does it deal with anything other than causal sequences ie causation that follows a countable sequence of distinct points in time. The very fact that we are talking about infinities over finite periods of time would suggest a continuum rather than discrete functions. There is nothing preventing a form of causation that is ordered in time and infinitely divisible over a finite period. In fact the equations of quantum dynamics are continuous not discrete so this would seem to be the expectation. The problem with bringing quantum dynamics into the picture however is that if you were to try to divide it up infinitely you would not be able to find out anything about it beyond a given resolution (plank length) even if, in reality, such an infinitely divisible causal function exists. Quantum mechanics suggests that the vast majority of events are either uncaused (random) or we simply cannot determine the cause due to the nature of the physics involved, and all other events are a combination of the consequences of these possibly uncaused events. If time is finite in the past then there are two possibilities, it is a closed set and had a single point in time where it started and there was a first event, or it is an open set with no first event.

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