To visualize infinity, imagine a hotel. This is a thing one can read about in books about mathematics, which use this idea of the hotel, in order to help us see the concept of the infinite. So; to visualize infinity, imagine a hotel, a lodging place, a hostel — but one with endless rooms, a chain of endless rooms.
A stranger from out of town arrives; this stranger is you. Yes, you arrive at the hotel, carrying in your arms your two heavy suitcases. You are wearing a suit, a thick woolen suit and silken tie, as well as an old-time-y fedora. In the lobby, you set the suitcases down with a thunk. You wipe small beads of perspiration from your forehead.
You approach the front desk clerk; he is standing there attentive in his red uniform, with golden braids and tassels on the red uniform — really, it is a uniform fit only for a monkey, the monkey attendant to an organ grinder. But the clerk looks professional in his uniform, silly though the outfit is. The clerk is calm, deferential but not too deferential, his brown hair slicked back beneath his red cap. He takes you in, with his calm gaze; his gaze lingers for the exact right amount of time, for not a second too long.
“Hello there,” you say. “I need a room.”
“Oh, I am sorry, sir,” says the clerk. “It is an awkward thing, but all of our rooms are booked. Every single room, from Room 101 to Room Infinity.”
You are tired from your long journey, and the clerk’s statement is bizarre (Room Infinity? What’s all this now?); but it is still early in the evening, and you can find another place and so you say– “Oh. Well then. I guess I’ll look elsewhere.”
“No need, sir, no need. We have plenty of room for you.”
Now it is time for you to look puzzled. “But you said–”
“There are infinite rooms, sir. Even though all the rooms are filled, there is always an empty room. Shall I have one of our porters fetch your bags?”
And now the clerk explains precisely how the hotel works. The rooms are infinitely filled, but there is always room for one more, infinity being infinite like that. And so, the guest in Room 101 will be moved up to Room 102, 102 to 103, Room 19,881 to Room 19,882, and so on and so forth, on to the endless, the incalculable numbers. “So, shall we go?” the clerk says. A porter has appeared at his side. The porter takes your bag. As you travel up the hallway, he pauses to knock on Room 101, informing them that they will have to move to make way for the newest guest. Room 101 moves to Room 102; you see the man move, carrying in stages his scattered belongings — he is an elderly gentlemen, but he does not seem bothered at all by the move; he reacts as though it was expected, even though he was in the middle of doing his toiletries, his whiskers half-shaven, residue of shaving cream still on his cheeks. “How do you do?” he says to you, and he knocks on Room 102, where the process is repeated, with equal calmness and precision. “Time to move again?” says the guest in Room 102; a young man, with his wife or mistress standing behind him. “Hold on. We’ll be a second in gathering our belongings.”
Such order, such precision! So pleasant, all of the guests making room for you like that. Room 102 goes to Room 103, and on and on, and you are busy putting your belongings in the very first room — but you wonder, as the room transfers proceed, if everything will continue to go so smoothly. It is impossible for you to follow all of the transfers. Already, they are only at the fourth room, but you are losing track, and the transfers of numbers will continue all night, and not just all night, but forever, because you could never keep track, and you will have to move too, soon enough.
Who can anticipate what will happen, with all the room changes? They cannot all go so smoothly. There may be arguments, there maybe be fights. There may be births, there may be deaths, wars, reversals, catastrophes, for in time all of this will happen. “Time to move again?” says a distant voice down the hallway. “Oh well. Time to be moving on.”
This is all a strange thing to think about; a strange thing to happen. Still, you lie there awake at night and dream of it, in your new bed in a strange hotel. You lie there awake at night, your window a spot of light, and from this window, you see other windows glaring above the lamplit avenues of the unknown town; the avenues spread before you, a question without any resolution or answer.