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.

Is your poor head swimming yet? Well, read this article about Hilbert's Hotel and you'll see how the manager creatively solves even tougher problems.

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?

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@ 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.    :)

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....

I hope I don't misunderstand your question, but the answer is no. Infinity is a paradoxical concept. 

The rooms are already full up. All of the rooms in the hotel are full. Always.

Infinity + 1 is still infinity. The hotel's rooms—if they are represented by the counting numbers—are Room 1, Room 2, Room 3...etc. To make room, you just tell the occupant of Room 1 that s/he will have to move to Room 2, and the occupant of Room 2 moves to Room 3...etc. Room 1 is now available for the new guest. You can do this because there is no end to the number of rooms.

It's a concept, not a reality. 

Zero, too, is a conceptual reality. Most of math is impossible with the number zero. The idea of nothing existed before the number zero, but it wasn't until nothing was given a numerical value that even simple arithmetic became possible.

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