The Edge of Counting

You can count to a hundred You can count to a thousand You can count to a billion If you don't sleep, if you don't blink

But there's no LAST number You add one — there's another You add one — there's another You add one — there's another

That's the first kind of forever The kind you can keep walking toward Mathematicians call it COUNTABLE Because you can list it, one by one

Hilbert ran a hotel With a room for every counting number 1, 2, 3, 4, 5 Every room is full

A traveler walks in Looking sad and tired "No room at the inn?" The clerk just smiled

Move every guest one room down Room one to room two Room two to room three Forever and forever

ALWAYS ONE MORE ROOM! ALWAYS ONE MORE ROOM! A FULL HOTEL THAT FITS THE WORLD — ALWAYS ONE MORE ROOM!

Then a BUS pulls up A bus with infinity people The clerk doesn't even flinch The clerk just smiles

Move every guest to double their room Room one to room two Room two to room four Room three to room six

Now every odd room is empty Infinity new rooms For an infinity of travelers And a sign that says VACANCY

ALWAYS ONE MORE ROOM! ALWAYS ONE MORE ROOM! A FULL HOTEL THAT FITS THE WORLD — ALWAYS ONE MORE ROOM!

That's a paradox A thing that's true and weird You CAN'T do this with a real hotel You CAN do this with the counting numbers

Because counting numbers go on And on And on And on

Some infinities are bigger than others There's a kind that won't fit in the hotel A kind so dense You can't list it at all

(quiet) The decimals between zero and one There's more of those Than there are rooms In Hilbert's whole infinite place

ALWAYS ONE MORE ROOM! ALWAYS ONE MORE ROOM! EXCEPT WHEN THE GUESTS ARE A BIGGER INFINITY — THEN EVEN HILBERT TURNS HIS LIGHTS OUT.

You can count And count And count And the edge of counting still keeps moving