Does the impossibility of Hilbert’s Hotel rule out an infinite past?

0. Introduction

I recently went on the Unbelievable podcast and debated with a Christian philosopher called Andrew Loke. Following our conversation, we have had a lively email exchange where we have been continuing to debate the same issues. This has been a helpful exchange for me, as it has focused my attention on one or two very specific things and made me get clear about them. Here, I want to get some of it down into a blog post.

Hilbert’s Hotel is a hotel with infinite rooms in it. For convenience, I will talk of ‘a Hilbert’s Hotel’ (or HH) to refer to any infinite object like this (such as a library with infinite books in, or a book with infinite pages in, etc). An object, x, is ‘a HH’ if and only if both of the following are true:

a) proper parts of x can be put in a one-to-one correspondence with the whole of it, and

b) each part of x concretely exists in the present.

What I want to know is, does the impossibility of a HH in this sense mean that the past is finite? In order to answer this, let us assume that there is no metaphysically possible world with an HH in it; HH’s are not possible. Let’s call that our ‘original assumption’. Now the question for this post is, if we make the original assumption, is it possible that the past is infinite?

  1. New Kalam

To put it the other way round: if a HH is impossible in the present, does that rule out an infinite past? If it did, then we could formulate a new version of the Kalam as follows:

  1. If is possible that the past was infinite, then a HH would be possible (assumption)
  2. HH’s are impossible (original assumption)
  3. Therefore, it is not possible the past is infinite. (1, 2, modus tollens)

I think that this argument is valid, but unsound; the first premise is false. I will explain why I think that here.

2. a1-worlds and a2-worlds

Before we get there, I will motivate why someone might think differently to me. Here is one reason for thinking that the argument is sound, which Andrew endorsed. It requires thinking of two different ways that the past can be infinite, which I will call a1-worlds, and a2-worlds.

Imagine that every ten minutes God creates a grain of sand ex nihilo, and adds it to a heap of sand. This seems intuitively possible. Whenever the past is infinite, and also contains a continuous cumulative process, like adding grains of sand to a heap, or adding rooms to a hotel, we shall all this an ‘a1-type infinite past’, or an ‘a1-world’ for short.

If God had been doing this once every ten minutes forever, and if time had no beginning, then there would now exist an actually infinite object, i.e. a heap of sand with an actually infinite number of grains. So there would exist an HH in the present. Thus, it seems like if it was possible that the past were infinite like this, then it would also be possible that there is an HH in the present.

This motivates premise 1.

But now premise 2 comes in. It says that HH’s are not possible. If the antecedent of premise 1 (time is infinite) is true, then the consequent is true (a HH is possible). But because of premise 2 (a HH is not possible), the consequent of premise 1 is false. Therefore, the antecedent of premise 1 is false (by modus tollens). Thus, the past is finite. This is how the new Kalam works.

However, we can think of a different type of infinite past. Imagine God creates a grain of sand ex nihilo every 10 minutes (just like above), but instead of adding them to a pile, he immediately annihilates them. In this scenario, there is no infinite heap of sand, no HH, in the present. The past is infinite, but it involves no continuous incrementally increasing cumulative process. Call such a world an ‘a2-world’.

Because an a2-world doesn’t involve an HH existing in the present, the impossibility of an HH cannot rule out the possibility of such an infinite history. This makes it different to an a1-world in a directly relevant way. The impossibility of an HH in the present is compatible with the possibility of an infinite past (contra the new Kalam), just so long as it is an a2-type infinite past and not an a1-type infinite past (i.e. just so long as there is no continuous incrementally increasing cumulative process).

 3. Disambiguating the first premise

Given that there are two types of infinite past (ones with a continuous incrementally increasing cumulative process, and ones without) the original premise was ambiguous. It said:

  1. If it is possible that the past was infinite, then a HH would be possible

It doesn’t distinguish between the two ways that the past can be infinite (i.e. a1- or a2-type). If 1 is true on both its disambiguations, then it is a distinction without a difference. However, if it is true on one but false on another, then we need to disambiguate to know which one is which.

What we need to do is consider each as a separate interpretation of the above premise, to see how it plays out on both disambiguations. So we have:

i) If the past is an a1-world, then a HH would be possible

ii) If the past is an a2-world, then a HH would be possible

If both of these are true, then the new Kalam holds. If one is false, then its first premise is also false. No surprises, it is true on the first, but false on the second. Let’s go through them one by one to see how this works.

Given that the definition of an ‘a1-type’ history includes as a clause that a HH is present, then it is obvious that whenever there is an a1-type history there is also a HH. That much is just a matter of definitions. But, because there are no possible worlds at all where there is a HH (because of premise 2 of the new Kalam), there are no a1-type worlds either. Both antecedent and consequent are necessarily false. And (somewhat counterintuitively for those not familiar with classical logic) that means that i) is a tautology, and so necessarily true.

The truth of ii) though is not so easy to determine. a2-worlds don’t themselves involve HH’s, so on their face it is not so obvious that they imply the possibility of HH’s. In order to rule out a2-worlds, the defender of the Kalam must find a way of arguing that a2-worlds are impossible. How might they do that?

Here is how. God can make a grain of sand, and he can then either destroy it or put it on a heap. Both creating grains to add to piles, and creating grains to destroy, are possible for God. If he can do one, he can also do the other. And the thought is that if he could make 1 grain and put it in a heap (which he surely can), then he could make 2, and if he could make 2 he could make 3, etc. By induction, it seems, he could make an infinite pile. So if an a2-world is possible, then so is an a1-world.

And that just means that if he can make and destroy grains of sand, then he can also do something impossible (make grains and put them in an infinite heap). In classical logic, anything that implies an impossibility is necessarily false.

If the possibility of an a2-type history implies the possibility of an a1-type history, and an a1-type history is impossible (which it is by the original assumption), then an a2-type history is also impossible (by modus tolens):

  1. If a2 is possible, then a1 is possible
  2. a1 is not possible
  3. Therefore, a2 is not possible (1, 2, modus tolens)

Premise 1 of this little argument is what I want to focus on. If it succeeds, it would collapse the disambiguation between a1-type and a2-type histories. It would mean that the first premise of the original argument (‘If the past was infinite, then a HH would be possible’) would be true. And if that were the case, then the impossibility of a HH would mean that time was finite, and it would be game over.

So the salient question becomes:

Does the possibility of an a2-type history imply the possibility of an a1-type history? In other words: if it is possible for God to spend forever creating grains of sand and then destroying them, does that mean that it is also possible for God to spend forever creating grains of sand and piling them up?

I think I can make a pretty strong case to say that the answer is ‘no’. It involves the realisation that if a HH doesn’t already exist, it cannot exist, and why that’s the case. If I’m right, then the impossibility of a HH doesn’t tell us whether past is finite or infinite. It might be, for all the premises of the new Kalam say, that the past is infinite.

4. Life in an a1-world and an a2-world 

To understand why the possibility of an a2-world doesn’t imply the possibility of an a1-world, it helps to consider what is true in a1-worlds and in a2-worlds.

Imagine, first, that an a1-world is actually the case. In that setting, an HH exists in the present. Maybe an infinite heap of sand, or an infinite hotel, whatever. The past in that world involves a continuously increasing incremental cumulative process, like God adding grains of sand to a heap.

The number of grains of sand there are in the infinite heap right now is at least equal to ω1. But how many grains were there in the pile just before God added the most recent grain? However many there were, it cannot be any finite number, because there is no finite number, x, such that ω1 – 1 = x. So before God added the latest grain of sand, there was already more than a finite number of grains of sand in the pile.

This point generalises. Before God added the latest, say, 1,000,000 grains of sand, there were still more than infinite grains of sand in the heap (because there is no finite number, x, such that ω1 – 1,000,000 = x). What this shows is that if there is a HH in the present, then there has always been a HH throughout the infinite past. If the heap is infinite now, then there is no point at which the pile had finite members.

We can also generalise this into the future too. If the heap has ω1 grains in it now, then if God started removing grains of sand from it one by one, there would always remain an infinite number of grains of sand left in the pile. It wouldn’t decrease below infinite, no matter how long he continued to take grains of sand away.

So on any a1-world, where there is a HH in the present, it follows that there must also be a HH throughout the past and throughout the future as well.

If there is a HH in an a1-world, it follows (trivially) that it is possible that there is a HH there too. Yet, our original assumption was that a HH is not possible. Thus, the a1-world is inconsistent; in it a HH is both possible (because it exists at every point in time), and also impossible (by the original assumption). They are not just inconsistent, but they are ‘full’ of contradictions, in the sense that there is a contradiction true at every time in every a1-world. Thus, a1-worlds are inconsistent, thus are not metaphysically possible worlds at all.

Now consider an a2-type history. In this case, there is no HH in the present. We know from the above reasoning that had there been a HH in the past at any point, then there would still be one now. If one existed any finite number of days ago, then not even God could have removed all the grains from it one by one at regular intervals such that there are none left today. And that is true no matter how long ago God started to remove grains.

Not only would there still be grains left, there would still be infinite grains left, no matter when he started to remove grains. Needless to say, if there was a HH at some point in the past, it would still exist now.

So, if there is no HH today, there never has been.

There is a future tense version of this too. God could start putting grains of sand in a heap now, but no matter how long he continues to do that there will never be an infinite number of them.

So, if there is no HH now, then there never will be.

So on any a2-world, there is no HH; there never was, and there never will be. This makes a2-worlds the mirror image of a1-worlds. Rather than being full of contradictions, a2-worlds are ’empty’ of them (there is no contradiction at any time in any a2-world). Unlike in the case of the a1-worlds, the original assumption that HH’s are impossible does not cause any contradiction here, because it is always false that there is a HH in all a2-worlds.

5. Objections

Imagine that an a2-world is actual. That means that there is no HH in the present, and no continuous incremental process throughout the past. However, is it possible for God to make a HH in this world? If there is no HH in the present, is it possible for there to be one?

The answer is: not if he has to start with nothing and can only add grains one at a time at regular intervals. No matter how long he does that for, he will never get to there being more than finite grains. An a2-world cannot change into an a1-world. Not even God can do that, assuming he can only add them one at a time.

That’s also true on the assumption that he adds any arbitrarily large finite number of grains to the pile at regular intervals. It doesn’t matter if he adds one grain, or a million, or a billion billion billion; so long as it is at regular intervals, like every 10 mins, there will always only be a finite number of grains, no matter how long he has been doing it. Thus, it is not possible for God to make an infinite heap of sand if there isn’t already one there.

There are two possible objections to this.

We might question the condition that God can only add finite numbers of grains to the heap at any one time. Maybe God could create all the grains at the same time, instantaneously. Maybe he could just click his fingers and make an infinite pile of sand ex nihilo. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Alternatively, we could question the condition that God can only add grains at regular intervals. Maybe, God could do a supertask, and get infinite grains added in a finite amount of time. Instead of adding them at regular intervals, he could speed up the time it takes him to add them to the pile. If so, then it wouldn’t be impossible for God to change an a2-world into an a1-world.

Let’s take these one by one.

Firstly, let’s suppose that God could make infinite grains of sand appear instantly as one big heap of sand. The proposal is then that: if God existed, then a HH would be possible. But that would then contradict our assumption that a HH is metaphysically impossible. From that we could conclude that God doesn’t exist:

  1. If God could make a HH all at once, then a HH would be metaphysically possible (new assumption)
  2. A HH is not metaphysically possible (original assumption)
  3. Therefore, God cannot make a HH all at once (1, 2, modus tollens)

So the problem with the idea that God could have just made a Hilbert’s Hotel all in one go, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one in one go.

What about if God did a supertask? A supertask is where an infinite sequence is squeezed into a finite space or time. For example, imagine if you took one step to the door that got you half way, and your next step was only half the remaining distance, and the next half the remaining distance, etc. In this way, you could fit an infinite number of steps into a finite distance (between here and the door).

Similarly, maybe God could create the first grain of sand and add it to the heap in 10 minutes, and the next one in 5 minutes, and the next one in 2.5 minutes, etc. By the time 20 minutes had passed, he would have made an infinitely big heap of sand.

This would be an incremental process, but with an ever shorter interval between iterations. We might think that this is better than the previous idea, because it doesn’t require God doing anything infinite in extent; each iteration he adds a finite amount of sand to the heap (one grain), unlike in the last one where he did an infinite task in one jump.

However, the counter-argument is the same. If a supertask was possible (whether for God or for anyone else), then a HH would be possible. If our original assumption is that a HH is not possible, then a supertask is not possible either.

  1. If God could make a HH by a supertask, then a HH would be metaphysically possible (new assumption)
  2. A HH is not metaphysically possible (original assumption)
  3. Therefore, God cannot make a HH by a supertask (1, 2, modus tollens).

So the problem with the idea that God could have just made a Hilbert’s Hotel by a supertask, is that it is metaphysically impossible to do that (by the original assumption). Hilbert’s Hotels are impossible, so it is impossible to make one by a supertask.

The wider lesson is this: if HH’s are impossible, then there is nothing that God could do to make one.

What this shows is that if we hold fast to our assumption that HH’s are metaphysically impossible, then all God could do is add or subtract finite amounts of grains at regular intervals. He couldn’t do a supertask to build one, or make one with the snap of the fingers. And that means that if there is no HH now, then even God couldn’t make one.

And this means that if an a2-world is actual, then an a1-world (with an HH in it) is impossible.

6. Conclusion

Let’s circle back. Earlier, we had looked at the following implication:

  1. If a2 is possible, then a1 is possible

We saw a reason to think that this is plausible. After all, if it is possible for God to spend an infinite past making and then destroying grains of sand (i.e. if an a2-type history is possible), then it is possible for him to not destroy them, but pile them up instead (i.e. an a1-type history would be possible). Creating grains of sand is not beyond God’s power, and neither is putting grains of sand in a heap. Thus, surely, if an infinite past is possible, then God could have spent that whole time making an infinite heap of sand, as it just involves him doing things he can do. This is a compelling point. This is the best argument for thinking that an a1-type world is possible.

The problem, as we saw, is that it runs into our original assumption, that a HH is impossible. That’s because if God did accumulate infinite grains of sand, like in an a1-type world, then there would be a HH now. If the original assumption is true, then there can’t be a HH now. So there can’t be an a1-world.

And that feels right to me. An a1-world has a HH; but HH’s are impossible; so a1-worlds are impossible too. The impossibility of a1-worlds just is the impossibility of the HH’s that exist in them after all. The impossibility of a1-worlds is a logical consequence of the impossibility of HH’.

We looked at an inference, which was that the possibility of a2-worlds implied the possibility of a1-worlds. But if there is no HH in the present, and HH’s are impossible, then there is nothing God can do to make one. Not even God can change an a2-world into an a1-world, if HH’s are impossible. That makes the inference false. The possibility of an a2-world does not entail the possibility of an a1-world. Nothing does (because they are impossible).

This leads us all the way back to the disambiguation we started off looking at. There, we had disambiguated the following:

  1. If it is possible that the past was infinite, then a HH would be possible

The first disambiguation was a tautology, and so true. The remaining disambiguation to consider was the second one:

ii)  If it is possible that the past was an a2-world, then a HH would be possible

We know that the consequent is false (because of the original assumption). But it seems like the antecedent is true; it is possible that the past was an a2-world. That would make ii) false.

And that means that the original premise is ambiguous, and on one disambiguation, (the second one) it is necessarily false. Here is the premise again:

  1. If it is possible that the past was infinite, then a HH would be possible

If the past is infinite, then that means that it is an a2-world (because a1-worlds are impossible). At all a2-worlds, HH’s are not possible. Therefore, the antecedent is true and the conclusion is false. This means that the impossibility of HH does not mean that the past is finite, and our new Kalam argument is unsound.

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13 thoughts on “Does the impossibility of Hilbert’s Hotel rule out an infinite past?”

  1. Hey Alex,
    Thanks for your post and the podcast.

    Out of interest would you say the the set of integers doesn’t fit your definition of the hilbert hotel because it cannot be concretely realised in the real world?

    I would argue that that’s the type of HH we need in the case of time since there have been a countable number of minutes in the past for instance. (Assuming we can discretize time in some fashion another possibility would be to consider time as a continuum.)

    Liked by 1 person

    1. Hi Sam. If there were a HH, then we could number the rooms, so that there is a room number 1, room number 2, … etc. There would be a room for each integer. So, depending on what you mean by “concretely realised in the real world” it seems to me the integers are concretely realised in the real world by a hotel with infinite rooms. Also, if the past were infinite, we could assign integers to each interval of some given amount, such as a minute. There would be t-1, for the most recent past minute, and t-2 for the one just before that, … etc. There would be a time interval for each integer. So the two examples seem exactly the same to me. In fact, I think there is some super deep connection between time and numbers, although don’t ask me to explain it fully because I can’t!

      Liked by 1 person

  2. What does Loke say about God creating ex nihilo an actual infinite grains of sand timelessly (or simultaneously?) Presumably he thinks that there are potential infinities, such that there is no finite limit on things like the possible grains of sand God could create in the future. If so, then why can’t God simply actualise this potential infinity all in one go? If God took that potential infinity of grains of sand, put them conceptually into a one-to-one correspondence with the natural numbers, and said ‘let there be grain’ why wouldn’t that produce an infinite heap of sand? Does he think that God must have some maximal finite number of grains he could create all in one go? If so, why is it logically impossible to add another ‘potential’ grain to that act of creation? If it’s logically possible, which it seems to be, then what does it say to God’s omnipotence if he can’t create logically possible potential things?

    As I see it, on their view, God created the universe in no time at all, so they are already committed to the view that God doesn’t need to take time to create things.

    And they are committed to God being able to draw down from ‘potential’ infinities (that never lapse into actual infinities, but only because these scenarios would require an infinity of time to do this).

    God can also create ex nihilo, so he doesn’t need to draw on a finite quantity of things like atoms to forge together an infinite amount of things like the potential infinite of grains that could exist. This just seems to me to produce a really weird inconsistency in their commitments.

    Liked by 2 people

    1. That’s a great question. I have been discussing this with Andrew a lot over the last month or so. His view is that God cannot make a big heap with infinite grains in it all in one go, because the impossibility of a HH is a constraint on what he can do – God can only do what is metaphysically possible, and a HH is not metaphysically possible.

      I don’t think that is a good answer though. Andrew thinks that God can make a hotel with one room in it ex nihilo. Also, he could make a hotel with two rooms in it. We can imagine that he makes them both with the click of his fingers next to each other. And we could imagine that instead of making just these two hotels, he makes a third one next to them, which has three rooms in, etc. He can make a hotel with x many rooms in for any finite x. So he could make a street of hotels, where each one has one more room in it than the previous one. None of these hotels on their own is impossible for him to make, even if HH’s are impossible, because each one only has finitely many rooms in (each is a finite hotel). But if there is a hotel for each natural number, then we can ask how many hotels there are on this street. The answer to this question is that the cardinality (or size) of the number of hotels on the street is the size of the set of all natural numbers, which is aleph 0 (https://simple.wikipedia.org/wiki/Aleph_null), which is an actual infinite. So the prohibition on him making an infinite hotel all at once doesn’t help, I think. In fact the concession that he can make every finite hotel is enough to entail that he can make an actual infinite.

      Liked by 1 person

      1. I agree that doesn’t sound like a good answer Alex. I think that none of the conditions described in forming an actual infinite, say from actualising a potential infinite timelessly in ‘one go’ seem problematic, so which one would Andrew reject and why? rejecting one condition or other seems to either comprise God’s capacity to create ex nihilo, or to create all potential things that could be actual (entailed by omnipotence) or that there are potential infinities that could be actualised, and that God can actualise potentials that can be actualised in no time at all. There’s going to have to be something wrong with one or more of those conditions, and I can’t see which of these conditions he would reject, or some problem with having them all obtain. If HH is impossible, then it seems to entail that God being a timeless creator who is omnipotent and capable of actualising infinite potentials is somehow impossible, something would have to give there. It almost becomes like a paradox of omnipotence here- where it should both be possible and impossible for God to create a HH given his attributes.

        I actually think that as HH don’t seem to be outright impossible anyway once you clarify what’s going on with notions such as subtraction/addition etc in these scenarios, the supposed absurdity can be handled, as I think you’ve written elsewhere, which I think is the more important sort of response to render- though it’s worth pointing out that the theistic position is one that entails that HH’s should be possible given their possibility follows from a supposedly consistent set of divine attributes and things like infinite potentials (a distinction invoked to get off the hook of an infinite future that doesn’t really work either). It’s probably better to give up on the absurdity of HH claim and let God be a possible creator of one!

        Liked by 1 person

  3. Hi Alex,

    I’m new to your blog, and I’m not sure if you’ve expressed your stance on the possibility of HH before. Do you think HH is possible? It seems that the impossibility of HH is the only basis for rejecting the statement, “If a2 is possible, then a1 is possible.” It seems much more likely to me that either HH is possible, or an infinite past is impossible. I think it’s more likely that the argument you outlined below is sound than it is that HH is impossible.

    “After all, if it is possible for God to spend an infinite past making and then destroying grains of sand (i.e. if an a2-type history is possible), then it is possible for him to not destroy them, but pile them up instead (i.e. an a1-type history would be possible). Creating grains of sand is not beyond God’s power, and neither is putting grains of sand in a heap. Thus, surely, if an infinite past is possible, then God could have spent that whole time making an infinite heap of sand, as it just involves him doing things he can do.”

    Like

  4. Alex,

    Curious if you’ve seen TJumps argument for objective morality fundamentally being “the ability to choose” even if the choice is ultimately subjective.

    It feels like a cop out in the same way that presuppositional arguments for God are-

    The best answer that I can think of is that this form of morality is uninformative because it literally allows for all subjective choices that don’t interfere with others ability to choose because the objective moral truth is their initial ability to choose?

    Keep up the good work!

    Like

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