The Finite Hotel

Picture a hotel with one hundred rooms situated right off a busy highway. The hotel currently has one hundred guests, one in each room.

A rainstorm begins suddenly, and an anxious driver pulls off the road and runs into the hotel asking for a room. He’s devastated when reception turns him away. “Sorry, no vacancies.” All one hundred rooms are full.

If the hotel had one million rooms, and all were occupied by a total of one million guests, the same thing would happen. “Sorry, no vacancies.”

The same is true for five billion rooms with five billion guests, or a googolplex rooms with a googolplex rooms.

But what if there were an infinite number of rooms?

The Infinite Hotel

Let’s imagine the same scenario, but this time, the number of rooms is infinite, and they’re occupied by an infinite number of guests. And let’s clarify what we mean by infinite. An infinite number of rooms means that, if you counted them, you’d never stop counting. You could always add one more.

This time, the anxious driver arrives at the hotel begging for a room. But the infinite rooms are filled with an infinite number of guests. And we can’t just stick the guest in the last room, because there is no last room. “Sorry, no vacancies.”

That is, until reception comes up with a clever workaround.

What if, instead of searching for an empty last room (which doesn’t exist), we just freed up the first room? Here’s how we can do that: Every single guest in the hotel is told to move one room over. The guest in #1 moves to #2, the one in #2 moves to #3, the one in #6 moves to #7, the one in #googolplex moves to #googolplex-plus-one. Since, by our definition of infinite, we can always add another room, there will always be a “next room” to move to.

And if we do this for every single guest, room #1 will be vacated. And our anxious driver will be able to move in.

So, despite the infinite number of rooms having an infinite number of guests, we were still able to find an empty room.

In other words, we just proved this mathematical statement:

A More Complicated Scenario

Now what if we had our infinite number of rooms occupied by an infinite number of guests, but this time, instead of a single anxious driver coming in from the storm, an infinite number came in begging for rooms.

Could we find an infinite number of rooms in this hotel to accommodate them?

If you haven’t heard this before, I encourage you to pause here for a minute and think about how this could be done.

If the number of guests were finite—say 100—you could just use the same strategy as before to shift the room numbers of the existing guests by the number of new arrivals. But in this new scenario, that strategy won’t work, because the number of new arrivals is infinite, and infinity is not a number.

Yet here’s what you can do. Tell each guest to move to the room that is double their room count.

• The guest in room #1 moves to room #2

• The guest in room #2 moves to room #4

• The guest in room googolplex moves to room googolplex-times-two

We now have an infinite number of rooms available (the odd-numbered rooms). The new arrivals are all happy.

Here’s what we just proved:

This also shows another non-intuitive fact about infinity: The even-numbered rooms alone were sufficiently large to house the infinite number of existing guests. Put differently, there are just as many even numbers in existence as there are even numbers plus odd numbers.

How does that make any sense?

Hilbert

If you’ve never heard about this before, it all comes from a thought experiment by the German mathematician David Hilbert. For me, the exercise reveals three fascinating facts about infinity.

First, it’s an incredibly counterintuitive concept. The above logic should show why, and those examples are just a sliver of the many mind-boggling cases that exist.

Second, it’s incredible that insignificant, finite human beings such as ourselves are able to comprehend this concept to begin with. If that doesn’t showcase the power of the human mind, I don’t know what does.

Third, it hints at one of the most beautiful areas of pure math called the cardinality of infinite sets.

We were able to expand and manipulate the infinite number of rooms in Hilbert’s Hotel and still get the same total number of rooms. Infinity is the same as Infinity + 1, which is the same as Infinity + Infinity. They’re all the same “size” (or, mathematically speaking, the same “cardinality”).

That size is called aleph-0:

But did you know that there are other “sizes” of infinity? The next one up is called aleph-1. And no amount of aleph-0s will ever equal aleph-1. Even an infinite number of them!

In other words:

But that’s an article for another day…