# 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?

Tags: Hilbert, hotel, infinity, mathematics

Views: 321

### Replies to This Discussion

Do you know the sound made by a car when part of the exhaust becomes unattached, clunks to the ground, and starts to be dragged around at 40 mph?  That's the sound my brain just made,

Hahahaa I just looked for a "like" button, but since there isn't one I have to tell you that I laughed out loud when I read this.  Same thing happening here.

Sadly I to had my moment of insane excess, I needed to hold back, I am in a library damn-it-shuss!

I LOVE it when that happens.  Poorly suppressed laughter is good for a person!

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

The universe may really be infinite.

-------------

"The density of the universe also determines its geometry. If the density of the universe exceeds the critical density, then the geometry of space is closed and positively curved like the surface of a sphere. This implies that initially parallel photon paths converge slowly, eventually cross, and return back to their starting point (if the universe lasts long enough). If the density of the universe is less than the critical density, then the geometry of space is open (infinite), and negatively curved like the surface of a saddle. If the density of the universe exactly equals the critical density, then the geometry of the universe is flat like a sheet of paper, and infinite in extent.

"The simplest version of the inflationary theory, an extension of the Big Bang theory, predicts that the density of the universe is very close to the critical density, and that the geometry of the universe is flat, like a sheet of paper.

"The WMAP spacecraft can measure the basic parameters of the Big Bang theory including the geometry of the universe. If the universe were flat, the brightest microwave background fluctuations (or "spots") would be about one degree across. If the universe were open, the spots would be less than one degree across. If the universe were closed, the brightest spots would be greater than one degree across.

"Recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe."
(Source: NASA)

I'm thinking, infinite as in the infinity of a moebius strip. I wonder if the mathematicians regard a moebius strip as an example of a real infinity(?). A little googling didn't yield a firm answer.

Yes, with some sources of 'light' that could be 'outside' the horizen of 13.7 billion LY.

Sadly I never really know, most likely...;p(

It's a stupid problem, unless the universe is infinite in size. An Infinite number of guests could not physically fit in a finite universe, with or without a hotel.

...and that hotel manager should be fired for moving people at all.

Many people feel infinity is simply a conceptual reality, like zero, imaginary numbers, or making conditional assumptions to solve formulae in symbolic logic.

I agree with them at this point (LOL) in time. A point is not real either.

(unless we find out the universe is infinite.)

I do remember, a question I asked in a HS Geometry class, which was not answered to my satisfation,'If points have no measurable size, then how do we use them as a helpful measure of anything? How do they establish a location?'

My answer would be that all they can do is define the ends of geometric lines. Either naked ends or ends meeting at a corner. For measuring distance, that is the sole contribution of a point. Lines are used to measure or express distances or dimensions.

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