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

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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,

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!

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.

Imaginary numbers are often used to compute real things that really exist. They're integral to quantum physics, for example.

I remember from my days teaching symbolic logic, that some solutions involved assuming something conditionally, without regard to whether it was true (real) or not. Of course, the truth value of the assumption didn't matter once the proof was done because it had been factored out along the way.


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