# Hilbert's Hotel Room Numbers

I heard about Hilbert's Hotel and I was left with one Question open:

If I take a Picture of the infinite Floor and I want to label the rooms and I start with 1 on the very left side and label them to the right, what happens when I get over the middle of the image? What Number can I assign to the very right room? Is there some way to write this?

PS: I came here from the "How To Count Past Infinity" by Vsauce

EDIT1: I meant a floor with Doors on both sides and my camera being centered at the exact middle of the floor / the "infinity"

Well, if you take an image of the infinite floor, since your image has only finite size, it will look roughly like this:

So when you get over the middle of the image, you will get maybe to the fourth door. It obviously depends on how wide the doors are, and what their distance is; possibly you'll just reach the second door.

Almost all of the doors will, however, still be on the right hand side.

And note that there is no very right room; each room has another one to its right. That is because the rooms in the Hilbert hotel are numbered with natural numbers only.

Edit:

If you have two rows of doors, you cannot label them in the order you see them in the image with natural numbers. You can, however, label them for example with natural numbers from both sides, by using even numbers on the left, and odd numbers on the right.

If you insist to have them labelled strictly growing from left to right, you cannot do that with natural numbers, or even with ordinal numbers, as they violate an important condition for ordinals: They are well-ordered, that is, every nonempty set of ordinals has a minimal element. But for the doors on the right side of the floor you don't have a leftmost door (there's always a door left of it on the same side of the floor).

You can however label them in order if you don't insist on integers. For example, you could give the rooms on the left the numbers $1-1/n$, and the numbers on the right the doors $1+1/n$. Note however that there is still not a door exactly in the middle. However you could introduce an additional door in the middle (which so to say would be at the end wall of the infinite floor). In the suggested numbering scheme, it would then necessarily get the number $1$.

Here's the floor with the suggested numbering:

Another possibility to go beyond integers/ordinals is to number them using surreal numbers. The surreal numbers not only contain the real numbers and the ordinals, but also numbers like $\omega-1$.

Then you can label the doors on the left hand side with the natural numbers, and the right hand side with the numbers $\omega$, $\omega-1$, $\omega-2$, and so on.

However, there is no surreal number in between those sequences. But there is a gap in between the finite and the infinite surreal numbers; this gap is conventionally labelled $\infty$. So if you are willing to go beyond even the surreal numbers, $\infty$ would be an appropriate label for the door at the "infinite end" of the floor.

Actually you will not reach the middle. Because the middle of infinity is also infinity, if you can reach middle you can also reach the last room also(both are at infinity).

• Very very far away from the world I live in is the room between forever and here. – Simply Beautiful Art Sep 28 '16 at 17:20
• There is no last room. – celtschk Sep 28 '16 at 18:16