We’ve been considering the concept of infinity as it relates to the Kalam Cosmological Argument (KCA) and the existence of God.
Imagine you’re stranded in a town in the middle of nowhere. It’s getting dark, and you need a place to stay for the night. You come to a 10 room hotel and ask the clerk for a room. He says, “Unfortunately, every room is booked. You might try Hilbert’s Hotel down the road. They’re full right now, but they always have room.”
You can feel your face wrench into a puzzled expression, and the clerk merely shrugs and goes back to his business. You figure, in any event, that Hilbert’s Hotel might be the only other place in town, and it might be worth suffering some word play in order to find a place to sleep.
As you go, you seem dimly that Hilbert’s Hotel is quite a long building. It seems to go on forever toward the horizon, or as much of the horizon as you can still make out. You step inside.
“Hi, I’d like a room. The gentleman down the road said you were full, but might have a room anyway?”
The proprietor smiles at you. “Yes, yes, come on in! We have an infinite number of rooms – and an infinite number of guests – but no problem! Will it just be you?”
“Yes,” you say, apprehensively, “but if you’re full, how will I-”
“It is nothing!” he says with unbounded enthusiasm. “Here, I will show you.”
He leads you to Room 1, and knocks. A woman answers, and he says, “Would you kindly move over to Room 2?”
The woman, having been afforded the same courtesy earlier, obliges. When she gets there, she passes on a similar request: “The manager has asked me to move to Room 2. Would you please move to Room 3, and pass it on?”
In just this way, every guest shifts to the next room up. You now have a room, and no one has to leave, since there are still an infinite number of rooms.
This is surely an eerie phenomenon, so you decide to explore the building a bit after settling into your room. And as you walk (Room 167…513…2,134…) you have no sense that the building will ever end. There is no sense that the architects grew tired of designing the building, no sense that the builders experienced fatigue and began to fail in their workmanship. It actually seems to continue forever, and now the quest of finding an end to this building has become decidedly futile. You are tired, and a little overwhelmed, and so you return to your room to retire.
Just as you get back, the smiling manager approaches you.
“Great news! We have a large party here seeking rooms for every member – it’s a party of infinity!”
You unconsciously shake your head, like you’ve been struck blind. And there certainly are a lot of people, running clear out the door and as far as you can tell, on down the street. Even if this is a thousand, how will they all fit?
“No problem!” says the manager, perhaps reading your mind. “Sir-” now he’s addressing you “-will you please move to Room 2?”
In a state of bewilderment, though certainly not belligerence, you move to Room 2. When you get there, you pass on the manager’s instructions – Move to the room number which is double your current room number. You also inform the guest in Room 3, and so the shift occurs as follows: Room 1 moves to Room 2 – Room 2 moves to Room 4 – Room 3 moves to Room 6…
Once it is complete, all of the odd numbered rooms are open. Not only that, but there are an infinite number of odd numbers, and so the entire party of infinity guests can be easily accommodated!
This is really too much, and so you decide to close your eyes and see if a night’s sleep will clear your mind and make sense of all this.
In the morning, you discover that guests have begun to check out.
First of all, that party of infinity has already left. Yet, though an infinite number of people have left, there are still an infinite number of people still staying at the hotel!
But the manager does not like the appearance of a half-empty hotel (all the odd-numbered rooms are empty, after all) and so he asks everyone to return to the rooms they occupied before they moved last night (last night, of course, they all moved to the room number which was double the number they occupied at the time the infinite party checked in. Now they move to the room number which is half of their current room number).
Then, you discover that before you had arrived, there was a previous party of infinity that had checked in. In fact, at the beginning of the previous day, there were only three guests, one in each of the first three rooms. There was no shifting required for that first infinite party!
Now, everyone from Room 5 and up is checking out. (You’ll recall, when you arrived, that everyone had to shift over one room). There are just four guests left.
This is a puzzle, you think. How could you have two infinite departures – both representing an infinite number of people checking out – and while the first time there remained an infinite number of people, now there remain only four?
Now here is the main point of this wild illustration, originally the brainchild of mathematician David Hilbert: To the extent that it is wild, and absurd, it is also unlikely to manifest in any way in reality. (I say unlikely, but I believe there are serious thinkers who would say “impossible.” I am only trying to be cautious).
This is not simply because the hotel is impossibly long, or because it’s impossible for us really to conceive of an actual infinite. It’s also because the math doesn’t make sense.
For example, consider the guests checking in.
When you checked in, there were already infinity guests, and we will represent infinity as N. You were alone. By this math, we have to say that:
N = N,
and N + 1 = N
Then, when infinity guests arrive:
N = N
and N + N = N
Then, when guests start checking out:
N – N = N
and N – N = 4.
But our equations are obviously true, if we take seriously what infinity means. If N stood for any number other than infinity, these equations would be puzzling, because they’d be false (except for the equations of identity, which only show how the subsequent equations are unusual).
There are yet more things to say about infinity, but we will move on. This post and the previous two serve as supports to Premise 2 of the KCA – The Universe began to exist – as a way of demonstrating that the Universe could not exist from the eternal past, but must have had a beginning a finite time ago.