Hilbert’s Hotel

by Luke Muehlhauser on June 22, 2009 in Kalam Argument

Part 5 of my Mapping the Kalam series.

hilbert_hotelPremise 2 of Craig & Sinclair’s Kalam Cosmological Argument is:

The universe began to exist.

They support this premise with evidence from physical cosmology, and also from the philosophy of mathematics. For the latter, they provide this supporting argument:

2.11. An actual infinite cannot exist.

2.12. An infinite temporal regress of events is an actual infinite.

2.13. Therefore, an infinite temporal regress of events cannot exist.

Last time, I made some clarifications and pointed out that…

…a Realist might say that there is an actually infinite number of mathematical objects, and because mathematical objects really exist, this disproves premise 2.11. But to do this, the Realist is going to have to rebut the arguments for Anti-Realism coming from Conventionalists, Deductivists, Fictionalists, Structuralists, Constructibilists, and Figuralists.

Craig & Sinclair leave the Realist with this rather difficult task, and move on to support premise 2.11 “by way of thought experiments that illustrate the various absurdities that would result if an actual infinite were to be instantiated in the real world.”

The first such thought experiment is that of Hilbert’s Hotel.

Imagine a hotel with a finite number of rooms. Say, 500 rooms. And all the rooms are taken. A new guest arrives and asks for a room, and the owner says, “Sorry, all the rooms are taken.” End of story.

But now let’s imagine a hotel with an infinite number of rooms, and suppose once more that all the rooms are taken. A new guest asks for a room, and the owner says, “But of course! Come on in.” The owner then shifts the person in room #1 to room #2, the person in room #2 to room #3, and so on – into infinity. He then places the new guest in room #1.

How did this happen? All the rooms were full, and yet the guest checked in to room #1. What is more, we added a new guest, didn’t lose any guests, and yet there are the same number of guests! Their number is, specifically, infinite.

It gets stranger. The next day, an infinity of new guests arrive, asking for rooms. “But of course!” says the owner, who shifts the person in room #1 into room #2, the person in room #2 to room #4, the person in room #3 to room #6, and so on. He moves each person to the room that is numbered double his original room number. Because doubles of integers are always even, every person in the hotel is in an even-numbered room. The infinite number of new guests now check in to the odd-numbered rooms. And yet, before they came, all the rooms were occupied! And yet the number of guests in Hilbert’s Hotel is the same as before: their number is infinite.

And this can be repeated an infinite number of times. Each time, the hotel is full when new guests arrive, and yet the guests check in, and after they check in the number of guests in the hotel remains the same as before.

And we’re not done yet. Suppose the guest in room #1 checks out. Are there any fewer guests in the hotel? According to set theory, no. There are still an infinite number of guests in the hotel. Suppose an infinite number of guests check out – say, all those in odd-numbered rooms. After this, there are still an infinite number of guests in the hotel. But the owner doesn’t like a half-empty hotel – that looks bad! So he shifts each guest to the room that has a number half that of his current room, and the hotel is now completely full without adding any guests.

But the owner can’t always keep his hotel full with these maneuvers. Let’s say that the infinite number of guests in any room numbered higher than #3 checks out. Now Hilbert’s Hotel has an very finite number of guests: three! And yet, the same number of guests checked out this time as had checked out when everyone in an odd-numbered room checked out. Both times, the number of departing guests was infinite. And yet in the first case, the hotel still had an infinite number of guests, and in the second case it’s guest count was reduced to three.

Craig & Sinclair conclude:

Hilbert’s Hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible.1

Next time, we’ll look at some objections that have been raised to Craig’s use of Hilbert’s Hotel to support the proposition that “an actual infinite cannot exist.”

  1. Craig & Sinclair continue: “Partisans of the actual infinite might concede the absurdity of a Hilbert’s Hotel but maintain that this case is somehow peculiar and, therefore, its metaphysical impossibility warrants no inference that an actual infinite is metaphysically impossible. This sort of response might seem appropriate with respect to certain absurdities involving actual infinities; for example, those imagining the completion of a so-called supertask, the sequential execution of an actually infinite number of definite and discrete operations in a finite time. But when it comes to situations involving the simultaneous existence of an actually infinite number of familiar macroscopic objects, then this sort of response seems less plausible. If a (denumerably) actually infinite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert’s Hotel. Since nothing hangs on the illustration’s involving a hotel, the metaphysical absurdity is plausibly attributed to the existence of an actual infinite. Thus, thought experiments of this sort show, in general, that it is impossible for an actually infinite number of things to exist in reality.” []

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{ 31 comments… read them below or add one }

Ben June 22, 2009 at 11:27 pm

I still say that if the math works out, then there’s no reason to doubt the metaphysical possibility.  I can’t see how there can be a contradiction in reality while not a contradiction in the math.  If we aren’t adding an infinity of actual numbers when we circle odd numbers then we can’t be adding actual odd numbered rooms to the situation when we move everyone to the even rooms.  The absurdity only comes from our inexperience with dealing with infinity.  An infinite hotel by definition has an *unlimited* number of rooms so wondering where the “extra” rooms come from isn’t an acceptance of the terms to begin with.  Counter-intuitive, but that’s just the nature of the beast.
Ben

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Duke York June 23, 2009 at 2:03 am

It seems to me the problem with all Hilbert’s Hotel  is it presumes that the “infinite” hotel is actually a finite one! Look at the phrasing of it: All the rooms are occupied. This means that the Hotelier had to go through each of these infinite rooms, checking people in, putting mints on the pillows, getting fresh towels, etc.  If the hotel were actually infinite (which means, remember “without end”) he would still be checking people in. It’s only because of our experience with finite hotels that we can even think this is possible. Likewise, the operation of “moving the guest from #1 to #2, and so on” is actually a geologically long (cosmically long, longer than cosmically long!) procedure, so none of the new guests could ever move in!
This works more clearly with the infinite regress of past time. “If there were infinite past time,” I’ll paraphrase Craig and other apologists as saying, “we’d never be able to get from the start to here!” Of course, they’ve already defined the sequence as non-infinite by putting a start on it (every start must be an end or a fin, to use the Latin). Really, their argument boils down to “Something can’t be infinite and non-infinite at the same time! Nyah!”
Duke

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Jeremy Killian June 23, 2009 at 3:16 am

Duke, what you’re defending is the potential infinite, not the actual infinite.  Craig affirms that the potential infinite does exist, not the actualized one.  The Hotel is meant to represent an actual set of infinite rooms.

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Steven Carr June 23, 2009 at 7:20 am

I fail to see any logical contradiction in the various supposed scenarios.
 
Is Craig saying that anything which is counter-intuitive is absurd?
 
And that anything which is absurd is impossible?
Perhaps if Craig produced an actual argument, resulting in a real contradiction, rather than just saying that it looks very queer.
 
 
 

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Reginald Selkirk June 23, 2009 at 7:43 am

But the owner can’t always keep his hotel full with these maneuvers. Let’s say that the infinite number of guests in any room numbered higher than #3 checks out. Now Hilbert’s Hotel has an very finite number of guests: three! And yet, the same number of guests checked out this time as had checked out when everyone in an odd-numbered room checked out. Both times, the number of departing guests was infinite. And yet in the first case, the hotel still had an infinite number of guests, and in the second case it’s guest count was reduced to three.

It seems to me that if this were valid; that they had established that consideration of Hilbert’s Hotel with currently accepted mathematics led to the conclusion 0 = 3, that it would apply to theoretical infinites and not just actual infinities. This is a good clue that they may be doing something wrong.
Numerous infinities are being thrown around here. Note the bit about an infinite number of guests. That is above and beyond having a hotel with an infinite number of rooms. Suppose you considered only finite numbers of guests, would it then be conceptually acceptable to have an infinite number of rooms?
And all this checking people in and moving guests around will require an infinite span of time. The property taxes would be infinite. The payroll would be infinite. The construction time would be infinite. The construction costs would be infinite. The electricity bills required just to run the ice machines in the hallways would be infinite. And so on.
Many of those infinities being thrown around are infinities of space or matter. What Craig needs to do is to apply any conclusion drawn to an infinite past timeline, since this is the only infinity at issue in the Kalam argument. Keep an eye out for how successful he is at that in a future post.

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TK June 23, 2009 at 8:53 am

The real problem with Craig’s hotel argument is this, as far as I can see: Events are not like guests, or hotel rooms, or balls, or whathaveyou. I cannot pick them up and move them from place to place. I cannot add to or subtract from them. I cannot rearrange them. I cannot hold them in the palm of my hand.
What Craig’s argument really demonstrates is the absurdity (though not impossibility) of an actually infinite collection of physical objects. Events, on the other hand, are not physical objects. There is no reason to think this argument extends to them.

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Reginald Selkirk June 23, 2009 at 10:02 am

Steven Carr: I fail to see any logical contradiction in the various supposed scenarios.

Note the paragraph in which they claim that infinity + 0 = infinity +3, therefore 0 = 3.
 

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Steven Carr June 23, 2009 at 11:23 am

William Lane Craig is a Molinist.

He preachers that there is a definite fact of the matter about what every conceivable person would choose from room service if they booked into any logically possible hotel room.

And that these ‘counterfactuals of freedom’ are free will decisions and that God knows every single one of them.

No matter how many rooms there are in a hotel, it is logically possible for a hotel to be built with a bigger number of rooms.

So if God knows for a fact what I will freely order from room X, God also knows what I will freely order if I had checked into room X + 1.

Soin Craig’s Molinistic world, there is an actual infinite number of definite facts about what I will freely choose from room service if I check into any room in Hilbert’s hotel – an infinite number of them.

There must be an actual infinite number of facts, because there can’t be a highest room number , after which there are no more facts.

Because there is always a logically possible higher room number, Craig’s Molinism means there is a definite fact about what I would choose from room service if I had checked into the room with the higher room number.

So how can there an actual infinite number of facts about my free will decisions?

Where did this actual infinite number of facts about my free will decisions come from? 

It is Craig’s Molinism which is absurd as it claims there is a definite fact about my free will decisions in every logically possible world.

That is an actual infinite number of facts (as the number of logically possible hotel rooms is not bounded) , about my free will decisions.

But how on earth can there be facts about *my* free will decisions in *Craig’s* thought experiment? How can I have genuinely free will decisions in somebody’s thought experiment?

It is like claiming I can make an actual infinite numbe of free will decisions in somebody else’s dream.

Craig’s Molinism is absurd.

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Haukur June 23, 2009 at 12:25 pm

Reginald Selkirk: And all this checking people in and moving guests around will require an infinite span of time.

Not necessarily. I’d imagine the hotel’s infinitely many employees can check in infinitely many guests in a finite time.

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Silas June 23, 2009 at 2:04 pm

I cannot imagine the universe being finite. First, there was nothing for eternity. Then the universe suddenly came from nothing. That doesn’t make sense to me. I think the universe has always existed, in some form.
But I don’t think Craig’s solution to how a finite universe could begin to exist is satisfying. An entity that is beginningless, changeless, immaterial, timeless, and spaceless, isn’t an entity at all. Craig basically says, “The universe came from nothing, but it’s still God.”

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Duke York June 23, 2009 at 2:55 pm

Jeremy Killian: Duke, what you’re defending is the potential infinite, not the actual infinite.  Craig affirms that the potential infinite does exist, not the a

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Duke York June 23, 2009 at 3:13 pm

Sorry for that last post. For some reason I can’t work this commenting system.
 

Jeremy Killian: Duke, what you’re defending is the potential infinite, not the actual infinite.  Craig affirms that the potential infinite does exist, not the actualized one.  The Hotel is meant to represent an actual set of infinite rooms.

 
Eh. The Hotel — which I picture is right next to Library of Babel — isn’t really that interesting. Our conception of numbers breaks down when we try to think of “infinity” as a number. So?
 
What I think is more interesting is what I see as a logical flaw of Craig’s Kalam argument. He says that, if the universe is infinite, you’d never be able to get from the start to the present day. To me, this seems ridiculous, because when Craig establishes a “start”, he is putting an end to this supposedly infinite series, leaving him asserting that an infinite thing is finite, which leads to all the paradoxes in his  understanding of it.
 
To put it another way, if you have an infinite series, you can’t deny that it would get somewhere, and that somewhere is where it’s gotten to.
 
To put yet another way; if you have an infinite series, the first number wasn’t zero, or one, or a million or 10^100, or 10^100^100, or whatever. The first number was was infinity.
 
Of course, I could be wrong about this.
 
Duke
 

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Lorkas June 23, 2009 at 4:56 pm

Duke York: To put yet another way; if you have an infinite series, the first number wasn’t zero, or one, or a million or 10^100, or 10^100^100, or whatever. The first number was was infinity.

That depends. If it is an infinite series that begins with negative infinity ({…-3, -2, -1, 0, …}), then there is no “first number”.
 
If it is an infinite series that begins {1, 2, 3, …}, then the first number is 1. There are infinite series that begin with a specific number. Presumably, this is what Craig argues is the nature of the universe, since Christians assert that the afterlife lasts for eternity. He’s just arguing against an infinite that has no beginning, really.

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lukeprog June 23, 2009 at 5:30 pm

By the way, I will try to come back to all these counter-arguments once I have made it through Craig’s original (2009) article.

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Reginald Selkirk June 24, 2009 at 8:38 am

Silas: I cannot imagine the universe being finite…

Ah, the old argument from lack of imagination. Note that this is not the same as having shown an actual contradiction.
 

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Reginald Selkirk June 24, 2009 at 8:40 am

Haukur: Not necessarily. I’d imagine the hotel’s infinitely many employees can check in infinitely many guests in a finite time.

Well, they used to have infinitely many employees, but due to the recession, they laid off half of them.
 

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Reginald Selkirk June 24, 2009 at 8:43 am

Duke York: What I think is more interesting is what I see as a logical flaw of Craig’s Kalam argument..

I agree with you on this, but it may be more appropriate to go into it in a later post.
 

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Reginald Selkirk June 24, 2009 at 12:31 pm

Are there enough rooms in Hilbert’s hotel for all the politicians cheating on their spouses?
 

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Lorkas June 25, 2009 at 8:07 am

 

Reginald Selkirk: Are there enough rooms in Hilbert’s hotel for all the politicians cheating on their spouses?

They just fit.
 

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Steven Carr June 27, 2009 at 2:35 am

Lorkas:
 
They just fit.
 
Presumably the hotel is in Argentina.

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Evolution SWAT June 29, 2009 at 7:35 pm

I wonder what God was doing before he created the universe. Was he thinking? Did his thoughts have an order to them? He’s infinitely smart … so he would already think about anything he needed to in one nanosecond. Impressive.
Well, it’s nice to know that God existed before the universe. We know he must have, because an infinite regression would be impossible. Wait … but there was no time before he created the universe. So how did he exist before? Well, he ALWAYS existed, because … he did. He existed, even before there was time…. so that’s not an infinite regression … but he must have existed and been conscious …. so time stops, but something happened before it….!?!?!?
The universe couldn’t have started without God. If it existed forever, it wouldn’t have been tipped off to start. If there ever was a moment where there was absolutely nothing, it would have continued forever. God had to wait for the right time to start it … but wait … there was no time before the universe so God wouldn’t have had any time to wait… Is God so great he can think, even in no time?
Hmmm, Craig’s argument seems a little ridiculous and contradictory, but maybe I need to immerse myself in some long, boring books of theology in order to be an intellectually satisfied atheist / agnostic?

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lukeprog June 29, 2009 at 7:43 pm

Evolution SWAT,

I never recommend theology for intellectual satisfaction.

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Evolution SWAT June 30, 2009 at 12:36 pm

@Lukeprog
LOL yes. Well I can see the point of making a more detailed argument than my satire above. Keep up the great work.

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Steven Carr July 11, 2009 at 10:37 am

CRAIG
But the owner can’t always keep his hotel full with these maneuvers.
 
CARR
Why can’t the owner build more rooms?
 
If the number of guests can increase, despite being already infinite, why can’t even an all-powerful God increase the number of rooms in the Hotel?

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Draegur August 15, 2009 at 5:46 am

The only ridiculous assumption I see in this is supposing that the hotel could ever, ever be FULL. I’d go so far as to say that an infinitely large hotel can NEVER be full, even if an ‘infinite’ number of guests check in. When dealing with ‘quantities of infinity’.
You might as well assume that a glass of water is still a glass of water when it is empty–empty to the point of hard vacuum. Ridiculous. I’m sorry. I stopped listening when he made the supposition that a hotel with inifinite rooms was ‘full’ *rolls eyes*

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Ben August 15, 2009 at 7:51 am

Draegur: The only ridiculous assumption I see in this is supposing that the hotel could ever, ever be FULL. I’d go so far as to say that an infinitely large hotel can NEVER be full, even if an ‘infinite’ number of guests check in. When dealing with ‘quantities of infinity’. You might as well assume that a glass of water is still a glass of water when it is empty–empty to the point of hard vacuum. Ridiculous. I’m sorry. I stopped listening when he made the supposition that a hotel with inifinite rooms was ‘full’ *rolls eyes*

Make some sense, dude.  If you are allowing for an infinite hotel in the first place, why in the world can’t it be full of people?  Isn’t it full of beds?  haha
 
Ben

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Gavin Kirby January 5, 2010 at 3:25 pm

The analogy fails utterly: a set of infinitely many discrete objects (i.e. a countably infinite set) has cardinality aleph-nought, whereas time (a continuum, at least classically speaking) is not a set of discrete entities at all. If one views time as a set of “points”, then even a finite interval of time contains (uncountably) infinitely many such points (whether infinitesimal divisions of time are physically meaningful is a separate question), yet still has a well-defined “beginning” and “end”. The real problem is not with an infinite number of entities per se, but with a lack of beginning.

Hence another objection to an infinite past is that we could “never get to the present if the Universe lacks a beginning”. But this is completely erroneous: rather than saying “infinite” we should really say “unbounded”, since if the Universe did not begin (it makes no sense to argue as though it “began an infinite time ago”, as though there were actually a time when it began separated from the present by an infinite interval of time) then the past is unbounded: for any time t one can speak of a prior time t – delta*t, for arbitrarily small positive numbers delta. But this in no way implies that it is “impossible to get to the present”, any more than if the Universe were unbounded in spatial extent (no “beginning/end” in space) it would be impossible to get “here”. Position, unlike displacement, is an affine quantity – one does not need to define it with reference to something (although of course this is necessary for measurement of position).

[A couple of side remarks:

By symmetry, if the past cannot be infinite then neither can the future be infinite, but this implication is never drawn (that I am aware of).

It's never spelled out, but the Hilbert argument, if granted, would also imply that space could not be infinite in extent.]

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Steve Maitzen February 10, 2010 at 2:53 pm

I think Gavin Kirby may anticipate my objection to C&S’s use of Hilbert’s Hotel when he writes that “a set of infinitely many discrete objects (i.e. a countably infinite set) has cardinality aleph-nought, whereas time (a continuum, at least classically speaking) is not a set of discrete entities at all.”

In Lukeprog’s footnote, C&S say that “If a (denumerably) actually infinite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert’s Hotel.” I think “denumerably” is crucial in their claim. As C&S seem to recognize, if successive states of the universe are densely ordered then they can’t be manipulated in the way that the integers, or the rationals, can be. I’m not sure what overall damage this does to the KCA, but I think it does cast doubt on HH as a defense of the argument.

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Gavin Kirby April 9, 2010 at 4:21 pm

Interestingly, this argument against an infinite past was the first used by Kant in his Critique of Pure Reason as an example of an antinomy of pure reason. It’s similarly trivial to “prove” that it’s impossible for the universe to have begun a finite time in the past – all this meaningless exercise in semantics tells us is that cosmology cannot be done in a purely a priori way.

The sleight of hand in the arguments for the (in)finitude of the past consists in slipping in assumptions about the nature of time which cannot be determined a priori – they are necessarily empirical hypotheses about the nature of time. This was (crudely put) Kant’s point – there is no rationalistic cosmology, since pure reason cannot decide in such questions (cf. Zeno’s paradoxes, which similarly confound a purely analytic approach to physics).

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Steven Carr April 5, 2011 at 9:55 pm

William Lane Craig makes a very good point about infinity.

Infinity leads to paradoxes and downright contradictions.

I thought I had zero money in my pocket.

But then I realised that if zero existed, I could divide it by 2 and still have zero.

That is not a contradiction, but what happens if I divide zero by zero.

According to well-respected mathematicians, zero divided by zero is impossible. It cannot exist!

So zero cannot be something which exists. You can’t divide by it, as you can by any other number ,without producing absurdity.

Having proved that zero could not exist, I looked in my pocket, and guess what? There was some money there!

That is the power of logic.

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Matt October 10, 2011 at 4:03 pm

Hopefully someone can address my concern as it may simply be ignorance on my part that leads me to make the following conclusions. I only wish to try and add something to the discussion if I miss the mark I apologize for my folly. It seems to me that if Dr. Craig acknowledges the potentiality of an infinite he also acknowledges the possibility of it’s actuality. While it seems reasonable to say there is no actualized infinite it does not seem to follow that he then leads us to recognize an actual infinite. If there is no actualized infinite then what is Dr. Craig’s concept of God? How can we follow him in recognizing his argument as a proof for the existence of God if that is not the infinite? If there was a cause for existence then something had to have preceded all that currently exists and there by it would be an actual infinite. If an actual infinite exists then it seems to me that fully allows there to be an infinite regress as a necessity of what could be possible. Perhaps the possibility is all that we can know and if we can only know possibility then we can only assume actuality and never actually demonstrate it or measure it. So actually an infinite might possibly be or not be. There is nothing definitive that can be shown or concluded either way more specifically if the argument refutes itself.

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