"There's plenty of room..."

One of my favorite songs ("Hotel California") by one of my favorite bands (The Eagles - see my post "I can't tell you why"), which apparently is the subject of a lawsuit right now in New York City, has a line that is very appropriate for today's post:

There's plenty of room at the Hotel California

Any time of year (any time of year), you can find it here.

I know of another hotel in which there is always plenty of room for guests - it's called Hilbert's Grand Hotel, named after the German mathematician David Hilbert.  Hilbert's Grand Hotel is yet another veridical paradox, a paradox that produces a result that is absurd but nevertheless true.  It also illustrates some interesting properties about infinite sets and the "number" infinity (technically infinity is not a number).  Here's the problem.  Imagine a hotel that with rooms numbered 1, 2, 3, and so on with no upper limit (i.e. continuing on to infinity).  Let's also imagine that initially, every room is occupied.  Essentially we have a hotel with an infinite number of rooms and an infinite number of guests.  What happens when a new guest shows up?  A normal hotel with a finite number of rooms and no vacancy could not accomodate any additional guests, but no so for Hilbert's Grand Hotel!  

How can we accomodate new guests?  Hilbert suggested that we could do so by moving every guest from their current room n to the next room, n+1.  We can do so because there are an infinite number of rooms!  Room 1 is now empty and can accomodate the new guest:

Hilbert showed that the same method could be used to accomodate as many as k new guests (we could just simply ask every guest to move from their current room n to the n+k room).  As it turns out, Hilbert showed that the Hilbert's Grand Hotel could even accomodate an infinite number of new guests!  Here's how - start by moving the guest in room 1 to room 2.  Next move the guest in room 2 to room 4, and in general, the guest occupying room n to room 2n, such that all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

As it turns out, infinity is a pretty cool concept!  Basically, we are saying that infinity plus infinity equals infinity!  Even when it's full, there's always plenty of room at Hilbert's Grand Hotel.  Any time of year...