Infinite numbers are weird. They gnash their teeth against our intuition. They don’t behave how, deep down, we think numbers ought to behave.
That should not be surprising, since our intuition was formed around tiny little numbers like 3.1, 4000 or seven billion and two.
Hilbert told a story, in 1924, to illustrate three strangeness of infinite numbers. The story was popularized in this book.
Hilbert imagined a hotel with an infinite number of rooms. That’s quite a lot of real estate, but let’s imagine the hotel uses some hyperdimensional warp so it doesn’t occupy more than half of a large continent.
Anyway, one day, the hotel was full. This was very good for business, of course, but then a new traveller arrived, asking for a room.
Something like this happened to me once. We’d booked a room at a hotel, and when we arrived, the hotel was full. They’d lost our booking. Alas, the hotel was only a finite one, so we couldn’t stay. Full is full, isn’t it? They were good enough, at least, to find us a room at a different hotel and help us move there with our luggage.
When your hotel is infinite, though, full is no longer full. The manager of the infinite hotel was able to accommodate the traveller. The conversation went like this.
The phone in room 1 rings, and the occupant picks up the receiver. “Hello?”
The manager is on the line. “I’m terribly sorry, sir. I’m afraid we need to ask you to move to a different room.”
“Oh, really? Why? I just unpacked!”
“I understand it’s a terrible inconvenience, and I’m awfully sorry. As our way of saying how much we appreciate it, there won’t be any charge for tonight’s stay.”
The guest in room 1 is somewhat mollified. “Oh, well that’s all right I suppose. Which room will I be moving to?”
Of course, you and I know that room 10 is occupied. However, the manager can have the same conversation with the occupant of room 10, and move him or her to room 100. The occupant of room 100 goes to room 1000, and so on.
In a finite hotel, this method doesn’t work. Eventually, we run out of rooms to move people to. Not so in the infinite hotel.
After an infinite number of phone calls and discounts and tips to bellhops, every person in a power-of-ten room has been shifted up to the next power of ten, and room 1 is now empty. A quick clean, new sheets, a topped up bar fridge, and room is ready. The traveller moves in, just in time for a late afternoon coffee, a dip in the pool, and a selection from the room service menu.
How many guests were there at the start? Infinity. How many were there after one more guest was added? Exactly the same number, infinity – but also, infinity plus 1.
Infinity plus 1 is infinity. This example shows it. I did warn you infinite numbers were strange!
The next day, fifty new guests arrived. Of course, this was no problem for the hotel manager, who shifted other guests around to empty 50 rooms.
The day after that, though, a large tourist bus arrived, and an infinite number of guests poured out, seeking rooms in an already-full infinite hotel.
The book I mentioned earlier tells how the manager coped with this problem – Can you think what he might do? Hint: think about odd and even numbers.
Even the next day, when a convoy of infinitely many coaches, each containing infinitely many travellers arrived, the manager was able to accommodate them all.
Infinity plus infinity is infinity. Infinity times infinity is infinity.
Basic arithmetic involving infinite numbers seems very easy, even if it’s very counterintuitive.
As our story closes, the manager is organising a gala ball, and inviting all the hotel guests. His caterer only brought enough ice-cream for each guest to have one bowl, but somehow each guest came back for seconds, and still there was ice-cream to spare. This, surely, should come as no surprise.