My friends at MathematiciansPictures.com have a terrific poster of Hilbert’s Hotel, with an explanation by me (for those who missed it in in my July 2005 Diary on NRO).
Now, if you’ll excuse me, I have to go back to the singing game I was teaching to the kids. OK, kids, where had we got to? Oh, right:
There are aleph-null bottles of beer standing on the wall,
Aleph-null bottles of beer standing on the wall;
And if just one bottle should accidentally fall,
There’ll be aleph-null bottles of beer standing on the wall.
There are aleph-null bottles of beer standing on the wall,
Aleph-null bottles of beer standing on the wall;
And if just one bottle should accidentally fall,
There’ll be aleph-null bottles of beer standing on the wall.
There are aleph-null bottles of beer standing on the wall,
Aleph-null bottles of beer standing on the wall…