Numberland

Chapter III · the end of the line

Infinity

There is a hotel with infinitely many rooms, and every one of them is full. A traveller arrives, tired, hoping for a bed. "No problem," says the manager. "Everyone moves up one room."

Hilbert's Hotel is the perfect place to watch infinity misbehave. A single new guest? Everyone shifts up one room, and room 1 is free. A bus with infinitely many passengers? Everyone doubles their room number, and the odd rooms — 1, 3, 5, 7, forever — open up. Infinity plus infinity is still infinity. The infinite hotel is never full, even when it is.

But Georg Cantor found an infinity that refuses to behave. List the numbers between 0 and 1, read down the diagonal, change every digit: the number you build differs from every entry on the list, yet it is a perfectly good number between 0 and 1. The size of the counting numbers is aleph-null; the numbers between 0 and 1 are strictly, provably larger.

A full hotel makes room: watch the room numbers roll over as every guest moves up one. Then the corridor dissolves into Cantor's list, and the diagonal argument builds a number that was never on it.
Zoom into the number line — forever. Between any two numbers, however close, there are infinitely many more waiting to be found.