Hilbert's Hotel and the Beginning of the Universe

Analytical philosopher and theologian William Lane Craig explains logically why the universe had to begin to exist:

Ghazali argued that if the universe never began to exist, then there have been an infinite number of past events prior to today. But, he argued, an infinite number of things cannot exist. This claim needs to be carefully nuanced. Ghazali recognized that a potentially infinite number of things could exist, but he denied that an actually infinite number of things could exist. Let me explain the difference. 

When we say that something is potentially infinite, infinity serves merely as an ideal limit that is never reached. For example, you could divide any finite distance in half, and then into fourths, and then into eighths, and then into sixteenths, and so on to infinity. The number of divisions is potentially infinite, in the sense that you could go on dividing endlessly. But you’d never arrive at an “infinitieth” division. You’d never have an actually infinite number of parts or divisions. 

Now Ghazali had no problem with the existence of merely potential infinites, for these are just ideal limits. But when we come to an actual infinite, we’re dealing with a collection that is not growing toward infinity as a limit but is already complete: The number of members already in the collection is greater than any finite number. Ghazali argued that if an actually infinite number of things could exist, then various absurdities would result. If we’re to avoid these absurdities, then we must deny that an actually infinite number of things exist. That means that the number of past events cannot be actually infinite. Therefore, the universe cannot be beginningless; rather the universe began to exist. 

He continues with an interesting thought experiment from German mathematician David Hilbert: 

Hilbert first invites us to imagine an ordinary hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. If a new guest shows up at the desk asking for a room, the manager says, “Sorry, all the rooms are full,” and that’s the end of the story. 

But now, says Hilbert, let’s imagine a hotel with an infinite number of rooms, and let’s suppose once again that all the rooms are full. This fact must be clearly appreciated. There isn’t a single vacancy throughout the entire infinite hotel; every room already has somebody in it. Now suppose a new guest shows up at the front desk, asking for a room. “No problem,” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room #2 into room #3, the person who was staying in room #3 into room #4, and so on to infinity. As a result of these room changes, room #1 now becomes vacant, and the new guest gratefully checks in. But before he arrived, all the rooms were already full! 

It gets worse! Let’s now suppose, Hilbert says, that an infinity of new guests shows up at the front desk, asking for rooms. “No problem, no problem!” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room #2 into room #4, the person who was staying in room #3 into room #6, each time moving the person into the room number twice his own. Since any number multiplied by two is an even number, all the guests wind up in even-numbered rooms. As a result, all the odd-numbered rooms become vacant, and the infinity of new guests is easily accommodated. In fact, the manager could do this an infinite number of times and always accommodate infinitely more guests. And yet, before they arrived, all the rooms were already full! 

As a student once remarked to me, Hilbert’s Hotel, if it could exist, would have to have a sign posted outside: “No Vacancy (Guests Welcome).” 

But Hilbert’s Hotel is even stranger than the great German mathematician made it out to be. For just ask yourself the question: What would happen if some of the guests start to check out? Suppose all the guests in the odd-numbered rooms check out. In this case an infinite number of people has left the hotel— indeed, as many as remained behind. And yet, there are no fewer people in the hotel. The number is just infinite! Now suppose the manager doesn’t like having a half-empty hotel (it looks bad for business). No matter! By moving the guests as before, only this time in reverse order, he converts his half-empty hotel into one that is bursting at the seams! 

Now you might think that by these maneuvers the manager could always keep his strange hotel fully occupied. But you’d be wrong. For suppose the guests in rooms # 4, 5, 6, … check out. At a single stroke the hotel would be virtually emptied, the guest register reduced to just three names, and the infinite converted to finitude. And yet it would be true that the same number of guests checked out this time as when all the guests in the odd-numbered rooms checked out! Can such a hotel exist in reality? 

Hilbert’s Hotel is absurd. Since nothing hangs on the illustration’s involving a hotel, the argument can be generalized to show that the existence of an actually infinite number of things is absurd.

And a TedEd talk explaining the paradox: