Hotel Aleph Zero

(2013, revised 2022, to the tune of Hotel California)

On a dark desert highway
Route 17 was the name
Problems in mathematics
Definitions to blame
Stretching into the distance
There's a shimmering light
A room and bed for the whole of N
But could I sleep for the night?

Cantor stood in the doorway
I knew his theorems well
And I was thinking to myself,
Is there a vacancy in the hotel?
Move from n to n plus one
I heard on the PA
As the patrons opened up their doors
I thought I heard them say

Welcome to the Hotel Aleph Zero
Not a lonely place (not a lonely place)
Such a lovely place
Plenty of room at the Hotel Aleph Zero
Any time of year (any time of year)
You can find room here

My mind was starting to miss the
Cantor Schroeder Bernstein
And then a lot of yellow, yellow pigs,
(They) all walked in.
Buses came to the courtyard,
Infinite twice...
Some sort to reorder,
Some sort to be nice.

So I called up the Kelly,
"Please bring me my swine",
He said, "If you dona't have one here yourself, I got two-eighty-nine."
And still those piglets are calling from far away,
Move you down to the double of your room, (n → 2n)
Just to hear them say

Welcome to the Hotel Aleph Zero
Not a lonely place (not a lonely place)
Such a lovely place
Fillin' it up at the Hotel Aleph Zero
Room for everyone (room for everyone)
And we're never done...

In the pigs are streaming,
All real numbers to fit,
Cantor said, "We are all just pigs-in-holes here
In a countable set"
And in Kelly's chambers,
They gather up a list,
They do their best to diagon'lize,
But there's always one they've missed.

Last thing I remember,
I was looking for a proof,
I had to find a way to show,
That this could not be the truth.
"Relax," said the math man,
"We're not trying to deceive,
You can prove it any time you like,
But you just won't believe!"