An Argument Against Craig's Hilbert's Hotel Illustration and a Proposed Solution

Jon Wright pointed out to me that the prior post did not handle the particular type of transfinite subtraction Craig says cannot be done. Namely, subtraction involving infinite quantities. I spoke about this with my math professor, Matthew Weathers. I proposed my argument, and he helped clear up some concepts for me, and explained the nature of cardinalities in a helpful way. Generally drawing from that experience, while sitting here in a class called "Mission in Political Context", I here provide an undercutting argument against Craig's Hilbert's Hotel argument for the impossibility of the existence of an actually infinite number of things. Afterward, I propose an alterative argument that serves the same purpose in the overall Kalam.

In the broader context of the Kalam, Craig says,

Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.

-http://www.leaderu.com/truth/3truth11.html

Now, his immediate premises and sub-premises are,

2 The universe began to exist.

2.1 Argument based on the impossibility of an
actual infinite.

2.11 An actual infinite cannot exist.

That the universe began to exist I agree with. That an actual infinite temporal regression of physical members such as days is impossible I agree with. But I believe that there are a potentially infinite number of physical terms, meaning that the set that contains each day of time is a potentially infinite set. Specifically, it has a beginning (creation), but it will not have an end. Thus, the cardinality of the set of days that exist or have existed is finite (it includes however many days have passed since creation), but its cardinality (the number of days in the set) will always be increased as days pass. God has no intention that I know of to stop creating and/or sustaining the passing of days.

But this is to say nothing of an actual infinite. If by denying the possibility of an actual infinite Craig is required to believe that the number line does not contain members that are real objects (and informed by his other writings we can be confident of this), then I disagree. I think that numbers are real, that they are objects, and that the set of all whole numbers is infinite, in fact, there are many infinite sets of numbers.

At any rate, I can give an example of a meaningful, coherent subtraction of an infinite set from another infinite set. Let's subtract all whole negative numbers from all whole numbers:

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set Y {...-3, -2, -1}
= Set Z {0, 1, 2, 3..}

So the cardinality of X is aleph-null, and so is that of the sets Y and Z. Here is where I feel that Craig has misunderstood the nature of transfinite arithmetic. For by "the same number of guests" Craig really means 'the same cardinality of the set of guests'. Let me explain. I believe Craig's contrual is something like the following.

Aleph-null
- Aleph-null
= Aleph-null

Thus, the cardinality of the set produced by the difference in the cardinalities of the sets X and Y is the same as the cardinalities of each set. This would not be a problem, but for other instances of transfinite subtraction, and Craig gives a few. Consider another example,

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set X {...-3, -2, -1, 0, 1, 2, 3...}
= Set Z' { }

The cardinality of X is aleph-null, and when X is subtracted, the resulting set Z' is empty, and therefore has a cardinality of 0:

Aleph-null
- Aleph-null
= Zero

Furthermore, we can construct all kinds of examples of subtracting infinite sets from other infinite sets such that the difference may be virtually any number. Consider,

Set A {1, 2, 3...}
- Set B {2, 3...}
= Set C {1}

The cardinality of A is alpeh-null, and the cardinality of B is aleph-null, but the cardinality of C is 1. Thus:

Aleph-null
- Aleph-null
= One

Therefore, subtracting an infinite set with a cardinality of aleph-null from another
set with a cardinality of aleph-null is an operation that is not well defined, and it is useless to us.

But is this the whole picture? Hold that thought, and follow me while I retrace a famous logical contradiction drawn from the principle of square roots:

√25 = 5
√25 = -5
∴ 5 = -5
But 5 ≠ -5

The above fallacy lies in the incompleteness of the statement √25 = 5, and the incompleteness of the statement √25 = -5. Rather, the two should be resolved with the statement √25 = ±5.

Similarly, I believe that Craig's fallacy is one of incompleteness. As I stated last time, one has not exhaustively defined a set if one has merely described its cardinality. Thus the operations within the ilk:

Aleph-null
- Aleph-null
= One

are not complete. While the unhelpful statement that 'aleph-null - aleph-null = 0 or 1 or 2 or n or aleph-null', isn't well defined, it is coherent. However, there is no reason to only perform transfinite arithmetic at the level of cardinalities. In fact, I can think of a very good reason not to perform transfinite arithmetic at the level of cardinalities...

In the last post I gave an application of the set theory paradigm that went like this

If the intersection of a set X and a set Y is empty, and the cardinality of X is 3, and the cardinality of Y is 2, then the cardinality of the union of X and Y is 5.

But, what if the intersection of X and Y is not empty? In such cases, even transfinite addition becomes undefined. For example, the intersection of a set X {1, 2, 3} and a set Y {2, 3, 4} is the set Z {2, 3} such that to add the cardinalities of X and Y would look like this:

3
+ 3
= 2

But we know that this is not a true equation. However, if the members of the set with a cardinality of 3 and the second set that also has a cardinality of 3 are defined, then the intersection of the two sets is coherent:

Set X {1, 2, 3}
∩ Set Y {2, 3, 4}
= Set Z {2, 3}

Thus, adding cardinalities is just as "absurd" as subtracting cardinalities. But this is exactly what Craig does when he subtracts 'an infinite number of guests' from 'an infinite number of guests' to get 'an infinite number of guests'. He calls an absurdity what is in reality an undefined expression. He should rephrase his expression to say that 'there is some infinite set D such that a proper infinite subset E of D may be subtracted from D, resulting in a third infinite subset F'.

I believe I have given an example of such an operation:

Set X {...-3, -2, -1, 0, 1, 2, 3...}
- Set Y {...-3, -2, -1}
= Set Z {0, 1, 2, 3..}

While Craig believes that Hilbert's Hotel is absurd, I can imagine such a hotel as existing. Picture this:

You are standing, facing the hotel. You look to the left, and you can see the building extending beyond sight. You look to the right and sure enough, the hotel rooms continue on beyond your visual range. God has constructed this hotel by the power of His word, and He sustains its existence. You walk straight ahead, which leads you to room #0, which is the clerks office. She tells you that all the room #'s are occupied, but she thinks she can squeeze you in. She makes an announcement over the loud speaker that every occupant of a room with a positive number is to move to the room with the number one above his. You walk outside and turn around to face the hotel again. Now you begin to see every occupant in the rooms to your right open their doors, walk farther right, and enter into the next rooms. This leaves room #1 open, and you walk in and place your luggage on the fold-out table you remove from the closet. Hilbert's 'Hotel is a strange place' you think, but you're sure they make their money.

In set theory terms, the cardinality of the set containing all room numbers is aleph-null. Moreover, the cardinality of the set containing all guests is aleph-null, and when you check in, the resulting set has an additional member (you), but it also has the cardinality aleph-null. And yet, just like in the examples I gave above, this phenomenon is coherent (and now even conceivable). But am I failing to handle his argument in its full force?

Let's explore Craig's illustration a little more. One example of absurdity he gives is:

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be... Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds!

I judge Craig to be accurate when he says that there is "one less person in the hotel". But he is mistaken when he accuses "the mathematicians" of disagreeing. What I think he means is that mathematicians would maintain (and I think rightly so) that the cardinality of the resulting set is the same as the initial set. But this does not mean that the initial set didn't lose a member! What Craig should say is something like 'in the set H containing all the hotel room numbers {-1, 0, 1, 2, 3...}, the guest in room #1 leaves, represented by the removal of the set I, {1} from H, resulting the set J {-1, 0, 2, 3...}. And yet, the cardinality of H is aleph-null, and the cardinality of the set J is also aleph-null, even though J contains one less member than H (J is a proper subset of H)'. Allow me to express this in a linear fashion:

Aleph-null
- One
= Aleph-null

As I demonstrated earlier, this is not a logical contradiction, for the subtraction of a finite number from aleph-null results in aleph-null. More specifically, the subtraction of a finite number of members from an infinite set with the cardinality of aleph-null results in an infinite set with the cardinality of aleph-null as well, even though the resulting set is a proper subset of the original set. In the example Craig gives:

Set H {...-3, -2, -1, 0, 1, 2, 3...}
- Set I {1}
= Set J {...-3, -2, -1, 0, 2, 3...}

So the guest in room #1 may leave, and this is not absurd. But that was just the subtraction of a finite number of members from an infinite set. Can we also meaningfully subtract an infinite number of members from an infinite set? Craig says

Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel

By this I understand Craig to be expressing a phenomenon that goes something like 'in the set H containing all the hotel room numbers {-1, 0, 1, 2, 3...}, the guests in rooms with odd numbers leave, represented by the removal of the set L {1, 3, 5...} from H, resulting in the set M {-1, 0, 2, 4, 6...}. And yet, the cardinality of H is aleph-null, and the cardinality of the set L is also aleph-null, and the resulting set M has a cardinality of aleph-null even though M contains less members than H (specifically, M is a proper subset of H). Thus, the difference between the infinite set H and the infinite set L is another infinite set'. Allow me to express this in a linear fashion:

Aleph-null
- Aleph-null
= Aleph-null

As I demonstrated earlier, this is not a logical contradiction either, for just as the square root of 25 is 5 or -5, so the subtraction of aleph-null from aleph-null may be 0 or 1 or 2 or n or aleph-null. And although this is coherent but not meaningful, meaning may be added by simply defining the members in each infinite set:

Set H {...-3, -2, -1, 0, 1, 2, 3...}
- Set L {1, 3, 5...}
= Set M {...-3, -2, -1, 0, 2, 4, 6...}

Thus, Craig's argument is demonstrated to be unsuccessful in revealing any absurdity. As far as a clear and persuasive argument for the fact that the universe had a beginning, permit a modest formulation of my own. I have offered such arguments before, but will briefly indulge once again.

The morning after your first night's stay in Hilbert's Hotel, you put on your pajamas and open your door to pick up the copy of the L. A. Times whose thump against your door woke you up. You look up, and with the hotel to your back, you look to your right and see a crowd of runners gathered in front of the clerk's office. You inquire as to the event, and your neighbor in room #2 explains that there is to be a race, whose participants will start the race in front of room #-∞. 'How can they ever get started?' you ask. 'They will never stop walking down the line of negative numbered rooms' you assert. 'And as soon as they decide to turn around and begin the race, the room in front of which they do so will necessarily be finite, not infinite!' you argue, beginning to get concerned. 'There is no room whose number is -∞. Each room has a finite number, but the set of all rooms is infinite! Even if God Himself allows them to run toward the clerks office where the finish line is while He constantly pulls them in the direction of the negatively numbered rooms, their running will be in vain, for whatever progress they make will be more than cancelled out by God's pull - they can neither start nor finish the race'. You try to warn the runners, but they ignore you and take off toward the infinite. You feel sorry for the runners, whose racetrack is the set of rooms with negative numbers {...-3, -2, -1}. And furthermore, to get started, they have to cross that same track while searching for the starting line! How can they cross a distance that consists of a set of rooms with the cardinality aleph-null? How can they cross it twice?

Such a mental exercise demonstrates the absurdity of positing a universe whose temporal past extends infinitely. While it is coherent to posit a universe with a beginning and no end, it is incoherent to say that the universe never got started - never had a beginning. For, with today as the finish line of the race (and indeed, today has come to pass), the cursor of history - the ontological "now" - will have to transgress a set of days whose cardinality is aleph-null. This is logically impossible.

Therefore, while Craig's Hotel argument is susceptible to an undercutting defeater, there is another argument that may take its place, rendering his overall cosmological argument stronger.