Checking Out of Hilbert’s Hotel (Kalam Cosmological Argument)

In Reasonable Faith, William Lane Craig seeks to show the absurdity of an actually infinity of things by appealing to surprises that await us at Hilbert’s Grand Hotel.

This is a thought experiment proposed by the German mathematician David Hilbert (1862-1943). In the thought experiment, Hilbert envisioned a Grand Hotel with an infinite number of rooms, all of which are full.

Hilbert then posed a series of scenarios that would allow the hotel to accept additional guests—both any finite number of guests, and even an infinite number of guests!

Craig poses four scenarios involving Hilbert’s Hotel, which I will summarize this way:

1. 1. A Single New Guest: The hotel is full and a new guest arrives. The manager has each current guest move to the next higher room, freeing up the first room for the new arrival.
   2. Infinite New Guests: The hotel is full and an infinite number of new guests arrive. The manager has each current guest move to the room whose number is double that of their current room. This puts all the existing guests in the even numbered rooms, leaving the odd numbered rooms free to accept the infinite number of new arrivals.
   3. Odd Numbers Check Out: The hotel is full and all the odd numbered guests decide to check out, leaving their rooms vacant. The manager then has the guests in the even numbered rooms move to the room whose number is half that of their current room. This fills up both the odd and the even numbered rooms, and the hotel is completely full again.
   4. Guests Above 3 Check Out: The hotel is full and all of the guests with room numbers above 3 check out, leaving only three guests in the hotel. Yet since the set {4, 5, 6, . . . } has the same “size” as the set {1, 3, 5 . . . }, it would seem that the same number of guests checked out in this example as in the previous one, but the resulting number of guests in the hotel was different.

Craig regards all of these as absurd and concludes both that such a hotel could not exist and that an actual infinity of things cannot exist.

How might we respond to this?

 

Yes, They’re Strange

My first response would be to say, “Yes, these are strange, surprising, and counter-intuitive. If you want, you can even call them absurd.”

But that’s not what I’m interested in. There are lots of true things that fit those descriptions.

What I’m concerned with is whether any of these situations entail logical contradictions that God could not actualize in some possible world, and it’s not obvious that they do.

 

What Infinity Means

It seems to me that part of the problem is that we have a persistent tendency to slip into thinking of infinity as if it is a particular, concrete, limited number. It’s not. By definition, it is unlimited. That’s what the term “infinite” means. But as long as we use labels like “infinity,” we tend to slip into thinking of this as if it were an ordinary, limited number rather than something of unlimited magnitude.

It can help to strip away the label and repose the question without it.

• • Suppose that I had an unlimited number of apples. Then you bring me a new apple, and I add it to my collection. How many apples do I have now? Well, it’s still an unlimited number. Adding a new item to an already unlimited collection isn’t going to change the status of the collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you bring me a whole bunch of apples—an unlimited number of them—and I dump them into my collection. How many are there now? Again, adding even an unlimited number of apples to my already unlimited collection won’t change the status of my collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you go through my apple collection and pull out all the odd numbered apples (let’s suppose that I’ve conveniently numbered them so I can always find the apple I want). How many are left in the collection? Pulling out every other apple from an unlimited number of apples would still leave an unlimited number there, so my collection is still of unlimited size. And I can re-number the apples I have left if I want.
  • Finally, suppose that you pull all of the apples out of my collection except the first three. How many do I have left? Three. “Why is the number different than when I pulled out the odd numbered ones?” you might ask. “Because of which ones you pulled out,” I reply. “The first time, you pulled out every other apple, but the second time you pulled out all of the apples but the first three. Of course, you’re going to get a different number left over at the end.”

What we’ve just done here is run through the same scenarios that Craig uses, only we’ve done it with apples instead of hotel rooms and we’ve done it using the intuitive term “unlimited” rather than less intuitive terms like “infinity” or “aleph-null,” both of which suggest a definite amount to our ears.

When we keep the fact that the amount is unlimited clearly in focus, the situation sounds quite a bit less counter-intuitive. The reason is that we aren’t losing sight of the not-limited amounts we are playing with.

 

But Could All This Really Exist?

Certainly not in our universe—not without God suspending the physical laws he has set up to govern it.

If a hotelier decided to build a hotel with an infinite number of rooms, he would be immediately confronted with the fact that there isn’t enough room on earth to house such a hotel, there aren’t enough construction workers to build it, and he doesn’t have an infinite amount of money.

Similarly, I can’t go to a store, or an orchard, or any number of stores and orchards, and buy an infinite number of apples. (Nor do I have an infinite amount of money—quite the opposite, in fact!)

So, both the hotelier and I would quickly discover that our projects are practically impossible (meaning: not possible given the practical restrictions we operate under).

Yet suppose the hotelier and I were to make go of our projects, somehow pulling in resources from off-world once earth’s supplies started to go low. At some point the hotel and the apple collection would collapse under their own mass.

If we kept loading material into them, they would become so massive that they would begin to fuse, and they would both turn into stars.

If we still kept loading material into them, they would become more massive yet and, eventually, millions of years later, go supernova and turn into black holes.

Such is the way of things, given the physical laws that operate in our universe.

Even if God were to intervene and miraculously allow the things to be built and not become black holes, we still wouldn’t be able to do the infinite amount of guest and apple shuffling involved in the thought experiments. We wouldn’t have the time!

So, I’m quite prepared to concede that, in addition to being practically impossible, the infinite hotel and the infinite apple collection are also physically impossible (meaning: not possible given the physical laws of our universe).

But that’s not our question. We’re not asking about practical possibility or even physical possibility. We’re asking about logical possibility.

 

How useful is a hotel metaphor?

Here I would like to introduce what I think is the main problem with Craig’s use of Hilbert’s Hotel to disprove the possibility of an infinite history: I just don’t think it’s relevant.

Even if you grant that Hilbert’s Hotel involves a logical impossibility, that doesn’t show that all actual infinities involve a logical impossibility.

We’ve already shown that there is an actually infinite set of mathematical truths, so that is logically possible.

And that actual infinity does not seem to be subject to the same kind of difficulties that Hilbert’s Hotel raises. Even if we were to start numbering the different mathematical truths the way the rooms in Hilbert’s Hotel are numbered, we couldn’t perform the kind of manipulations on them that the hotelier does.

We couldn’t have the mathematical truths move around the way the guests in the hotel do. We couldn’t make new mathematical truths appear or existing mathematical truths vanish.

We could change the numbers we apply to these truths, but that’s not the same thing. That’s just our labeling. It doesn’t affect the truths themselves.

The apple collection fares no better. We can’t move, add, or delete mathematical truths the way we can apples. Mathematical truths are just there.

And so it seems that the kind of puzzles we can generate with Hilbert’s Hotel simply do not arise with the actual infinity of mathematical truths.

This means, if we grant that Hilbert’s Hotel involves a logical contradiction, that we have to ask the question: How relevant is it to the idea of God creating an infinite history for the universe? Is the idea of an infinite history in the same category as Hilbert’s Hotel or in the same category as the set of mathematical truths?

 

Some Similarities

At first glance, you might think that an infinite history would go in the same category as Hilbert’s Hotel.

For one thing, Hilbert’s Hotel involves an infinite amount of space (the infinite rooms it contains), and an infinite history would involve an infinite amount of time. We could map the rooms of Hilbert’s Hotel onto the individual moments of time in the universe’s history.

Furthermore, the rooms in Hilbert’s Hotel have content (guests and the things they are doing in their rooms), and the moments of history have content (people and the things they are doing—or at least whatever God would have chosen to put in them).

But these similarities turn out to be rather superficial. In fact, you could map any countably infinite set onto any other countably infinite set. You could, thus, map the rooms of Hilbert’s Hotel onto something totally non-physical, like the mathematical truths that fit the form X + 1 = Y, like this:

• • Room 1 –> (1 + 1 = 2)
  • Room 2 –> (2 + 1 = 3)
  • Room 3 –> (3 + 1 = 4)
  • Room 4 –> (4 + 1 = 5)

And just as each room in the hotel has contents (guests), each slot in our sequence of mathematical truths has contents (the particular truth in question).

The real issue is whether the moments of history and their contents can be manipulated the way that the rooms and guests in a hotel can.

And they can’t be.

 

Time Can Be Rewritten?

Despite Doctor Who’s repeated assurances that time can be rewritten, this question has proved a real puzzler for philosophers and scientists.

Scientists are of different opinions about whether time travel to the past is physically possible. Einstein was startled when, in honor of his 70th birthday, Kurt Gödel gave him a proof showing that Einstein’s own equations would allow for the possibility of time travel if we are living in a certain kind of universe.[1]

So what would happen if you really were able to go back in time? Could you change history? Could you carry out the famous “grandfather paradox,” where you kill your own grandfather before the conception of your father, thus preventing your own birth? (And, by extension, preventing you from coming back in time to kill your grandfather.)

According to much of the thinking on the issue, there would be two possibilities:

1) You simply would not be able to kill your grandfather. According to this view, time has an as-yet undiscovered principle that prevents time paradoxes from happening. This was proposed in the 1980s by the Russian physicist Igor Novikov (b. 1935), and the proposed principle is known as the Novikov self-consistency principle.

2) You would be able to kill your grandfather, but in so doing you would create a new timeline. According to this view, you would be able to kill your grandfather, but this would not change your history. The timeline in which your father and you were born would still exist. That timeline has to still exist, because you need to leave at some point to come back in the past. What would happen if you kill your grandfather is you would cause a new timeline to branch off from the original one, and in the new timeline you would never be born. You would still be around, however, because your original timeline is still out there, undamaged.

If the first of these possibilities is correct then the Doctor is simply wrong. Time can’t be rewritten.

If the second of these possibilities is correct then the Doctor is right. –ish. Up to a point.

You could “change history” in the sense of causing a new timeline to branch, one in which you are never born. But you didn’t change your history. The timeline where you were born, grew up, and travelled back in time is still out there. It’s still real. It didn’t un-happen from an external perspective looking at the timelines.

How does this help us answer our question about an infinite history?

 

Forward, into the Past!

The ancients really don’t get enough credit, because since at least the time of the Greeks, people have been asking whether God could change the past.

The poet Agathon (c. 480-440 B.C.) wrote that:

For this alone is lacking even to God,
To make undone things that have once been done.

This statement is quoted and endorsed by Aristotle (Nichomachean Ethics 6:2).

The same view is taken up and endorsed by St. Augustine (c. 354-430), who wrote:

Accordingly, to say, if God is almighty, let Him make what has been done to be undone, is in fact to say, if God is almighty, let Him make a thing to be in the same sense both true and false [Reply to Faustus the Manichean 26:5].

And the same view is endorsed by St. Thomas Aquinas, who wrote:

[T]here does not fall under the scope of God’s omnipotence anything that implies a contradiction. Now that the past should not have been implies a contradiction. For as it implies a contradiction to say that Socrates is sitting, and is not sitting, so does it to say that he sat, and did not sit. But to say that he did sit is to say that it happened in the past. To say that he did not sit, is to say that it did not happen. Whence, that the past should not have been, does not come under the scope of divine power [Summa Theologiae I:25:4].

Notice that both Augustine and Aquinas identify the same reason for God not being able to change the past: It involves a logical contradiction.

Does it?

 

From Here to Eternity

The view that I’ve advocated in this paper—that God is outside of time and that, consequently, the B-Theory of time is true—indicates that God cannot change the past—at least not in the way relevant to our discussion.

To illustrate this, let’s think about a particular moment in my own past, which I have already referred to: the moment when I was five years old and, in my grandmother’s kitchen, reached for a broom, causing a Coke bottle to explode and injure my knee.

That moment is present to God in eternity.

In the eternal now, he is creating that moment in history, and, by my free will, I am grabbing the broom and causing the Coke bottle to explode.

Could God change history so that this doesn’t happen? Could he, for example, give my grandmother a sudden inspiration a few moments earlier so that she can intervene and prevent me from grabbing the broom?

No.

Why not?

Because that moment is present to him in the eternal now. It’s real. He can’t undo it, because there is no time where he is. He can’t first let it happen and then undo it. That would imply the passage of time for God. It would imply more than one moment in the eternal now, in which case God would not be in eternity but in time.

What God could do (I assume) is create a second timeline,[2] in which my grandmother does stop me from breaking the Coke bottle, but that would leave intact the original moment in which I did break it. That moment is still there, still present to God in eternity, and thus still real.

If Akin is right about this then so were Agathon, Aristotle, Augustine, and Aquinas: God cannot undo history.

Once an event has happened in time, it’s set in eternity.

This has important implications for the question of whether he could make an infinite history.

 

Not Like Hilbert’s Hotel

If God can’t change history then time is not like Hilbert’s Hotel.

God cannot, in the eternal now, cut up, rearrange, and reshuffle the moments of history the way Hilbert’s hotelier moved guests around. If Moment 2 follows Moment 1 in time, then that is the way they are arranged before God in eternity. He can’t swap their order because in eternity there is no time for him to swap them in.

God cannot, in the eternal now, create or delete additional moments in time beyond those he has already created, the way I could add or subtract apples from my collection.

There is no time in which God could do these things, because that is the way eternity works by definition. It is thus logically impossible.

That means that the kind of puzzles that arise with Hilbert’s Hotel simply do not arise with time from God’s perspective.

An actually infinite history for the universe thus would go in the same category as an actually infinite number of mathematical truths. It does not give rise to Hilbert paradoxes.

Neither does an actually infinite future for the universe, which is why God is able to create one of those—as he has.

Therefore, even if one grants that Hilbert’s Hotel involves a logical contradiction and could not be realized by an omnipotent God, it does not matter. Viewed from the eternal perspective, an infinite amount of time (past or future) does not generate the problems of Hilbert’s Hotel and so is not in the same category.

 

[1] This solution is known as the “Gödel metric.” For a good popular introduction to the subject of time travel from the perspective of contemporary physics, see The Physics of the Impossible by Michio Kaku, who is a professor of theoretical physics at City University of New York.

[2] Because God is in eternity, this second timeline, and any others he might create, would all exist simultaneously for him in the eternal now, just as our timeline does. From the eternal now, he would be simultaneously creating all of the histories that exist.