A Room For All Infinity Guests

You’ve travelled for a long time, finally arriving at Hilbert’s famous hotel. It’s a very special and prestigious hotel because particular hotel has infinitely many rooms. They start from $1$ and are numbered like so:

$1, 2, 3, 4, 5, ..., 1,000,000, 1,000,001, ...$

going on forever. Standing outside you look up and it never ends. The top of the building just vanishes into a point. Its slogan is “A million guests under on roof” but perhaps it’s a misnomer. How can an infinitely tall hotel even have a roof?

You’re tired from your long journey so you hope this next step won’t take too long. You walk up to the reception desk and ask if there are any free rooms.

“No, every room is taken”, the receptionist tells you.

“But… it’s an infinite hotel! Are you telling me there are infinitely many guests staying here?” you say, scarcely able to believe her.

“Yes, exactly!”, she replies, not even batting an eyelid. You look out the lobby window and think back to your journey. Hilbert’s hotel is the only place to stay in the vicinity and you don’t fancy another trip out.

“So I can’t stay here tonight?”, you say desperately.

“Of course you can! Please, check in and we’ll let you know when your room is ready” she replies cheerfully.

“But… you just said that every room is taken…”

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A moment for reflection

How are you being given a room in a hotel in which every room is occupied?

Hint: Remember that it’s an infinite hotel

Once you’re ready, continue ahead with the story to see how you’re being given a room

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She explains, while handing you the check in form, “I did say that, yes! Every room is taken, but if we move everyone along to the next room then we’ll have one free room that you can move into!”

“So Room 1 moves to Room 2, Room 2 moves to Room 3 and so on? Leaving Room 1 empty for me to move into?” you reply, with the pieces of the puzzle clicking into place.

A mathematical aside

What we’ve shown in the story above is an important result in set theory.

In this case, we are referring to a certain type of infinity, called a countable infinity. What the above story has taught us is that the size of this $\infty$ is the same size as $\infty + 1$.

To describe it more carefully, we say that $\mathbb{N}$ is a set that contains all the numbers below:

$\mathbb{N} = \{ 1,2,3,4,5,... \}$

Since we have seen that we can add a new guest to the hotel without having to build any new rooms, we see that the size, or cardinality of $\mathbb{N}$ is the same

$\vert \mathbb{N} \vert = \vert \mathbb{N}+1 \vert.$

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This story is the first part in a series about set theory.