What is infinity plus 1?

Ellen Hammatt from Te Kura Mātai Tatauranga—School of Mathematics and Statistics explains.

We can explain this concept using Hilbert’s Hotel.

So, we consider an infinite hotel with rooms that are numbered by the natural numbers. For any number, there is a room with that number on it. They're all currently full, but when a new person shows up the hotel manager promises that there is room for them.

So how is this possible?

The hotel manager is going to free up room 1 for this new person by telling everyone to move one room to the right. So, we tell room 1 to go to 2, 2 to go to 3, 3 to go to 4, and so on. Now everyone who was currently in a room has a new room to go to, they know where they're going, and now room 1 is free. So, the new person can enter room 1.

We can sort of count the infinity even though we'll never get to the end. As we can see by just adding 1, we have this rule of still being able to fit into the hotel, which means just by adding 1 to something that's already infinite, we're not actually changing its size. And you can notice that if any kind of finite number of people came along, if 5 people came along, we could do the same thing. Everyone moves 5 along. So, we can see that just by adding a finite number, we're not actually changing the size.

Let's consider if infinitely many new people arrive at the hotel. Suppose there is countably infinite of them, so they're all coming in one by one, and the hotel manager is going to tell each person what room they have. And we promise again that actually this is somehow possible.

What we're going to do is move all the people who are already in the hotel. They're all going to move to even numbered rooms. So, all they're going to look at the room number and double it. 1 is going to move to 2, 2 will move to 4, 3 will move to 6, and N is going to move to 2N. So now you can notice that actually all of the odd numbered rooms are free. So, the first person who came to the desk, the hotel manager, will tell them to go to 1, and the next person they'll tell to go to 3. So, we can see that even by, in a sense, doubling the number of people, we're actually still at the same size. We can still actually fit them into this hotel.

Now that we've talked about lots of sets that are the same size, that can fit into the hotel, maybe you're wondering “is there only one notion of size for infinity? Is infinite just one size?” In fact, that is not the case.

There are some sets that will not be able to fit into Hilbert's Hotel. So that can be a challenge for you to think about how that would work, and also, how would you go about proving that there is no way to fit them into this hotel?

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