# Hilbert's Hotel, where there's always a room even when every room is taken

How can that be?

(T)he Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them.  Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on.  That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).

Now suppose infinitely many new guests arrive, sweaty and short-tempered.  No problem.  The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on.  This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

What it adds up to is that infinity is a very strange concept indeed.

Mathematics can't exist without it, and yet both mathematicians and philosophers have been wrestling with this question:

Is infinity real? Or, to put it another way, is there any such thing as a real infinity?

Tags: Hilbert, hotel, infinity, mathematics

Views: 337

### Replies to This Discussion

In real life, one never actually touches the mouse button anyway. What happens is that the repulsive forces of the atoms in your finger are met by the repulsive forces of the atoms in the mouse button. If the atoms of the finger actually were propelled hard enough to make subatomic contact, I think there might be an explosion. Any physicists here?

Yeah, I guess if you "touch" fissionable material hard enough, BOOM!

Otherwise the more tightly bonded (harder) material just displaces the other.

@ RobertPiano Yeah, I guess if you "touch" fissionable material hard enough, BOOM!

They would only be fissionable atoms if they are heavier than iron. Otherwise they would be just smashing each other to bits - and there is a difference. Most likely the atoms would just be broken apart creating: radiation, positive and negative ions. He'd need to replace the mouse and have the rest of the tip of his finger amputated.

See my other reply for the other possibilities.

A point is a math construct that occupies no volume or space, so you never get there. Molecules do take up space. Just another example where the math is not an appropriate model for physical reality.

I zeno problem with that. ;-)

@RobertPiano Here, cut this cake into zero pieces. I dare you!

It's not possible to cut a cake into zero pieces since 1 =/= 0

@Unseen If the atoms of the finger actually were propelled hard enough to make subatomic contact, I think there might be an explosion.

Well, to propel those atoms that vigorously would be difficult and would remove them from your finger. Even assuming you did that, how many atoms are we talking about? If only 1 or 2 the energy released would not be macroscopically noticeable. If you use the entire tip of your finger, well, that's going to make a significant boom.

Well, to propel those atoms that vigorously would be difficult and would remove them from your finger. Even assuming you did that, how many atoms are we talking about? If only 1 or 2 the energy released would not be macroscopically noticeable. If you use the entire tip of your finger, well, that's going to make a significant boom.

I think if it were to happen, it would be hard to limit it to just 1 or 2 atoms, don't you?

If you use a high powered laser to remove them and have the hand locked into a device holding it absolutely still you might be able to get it down to 1 or 2, well, ok, 1 or 2 dozen if very careful and very precise. If you aren't perfect, maybe 1 or 2 hundred - but still not much.    :)

If infinity is n + 1 and you can always add a 1, that's great. But in reality nothing that needs any kind of measurement, or for any sort of practical math, for example, Graham's number as talked about in the link Reg posted, will have infinity as an answer. So I for one (just my opinion) do not see "infinity" as "real" in the same way my shoes are real. I see infinity as a yardstick to the possibilities that the universe holds, knowing that to everything there is a mathematical equation with an answer, and that answer will never be infinity, no matter how big the numbers are. And so instead of saying "infinity is real" I would say, infinity is our only concept of how enormous the possibilities could be.

I'd go with "yes" simply because there are examples of different infinities. Take the whole numbers - there is an infinity of them. Take all fractions - a larger infinity. All the real numbers is an even larger infinity yet.

So, yeah infinity is real, if mind-bending.

A Mobius strip is a finite construct so it is not infinite even though you've created and endless one-sided loop (and beginning-less one-sided loop). A paper loop is similarly endless but with two sides.

Is the surface area, on a mobius loop, infinite? Nah....

Won't the rooms eventually (the ones people are being moved to) fill up? Once the rooms are full, that is it, unless the rooms grow in size.

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