The Hilbert Hotel

The Hilbert Hotel has an infinite number of rooms. All of them are full.

Somebody shows up at the front desk and asks for a room.

The manager says, "sure".

He moves the people from room one into room two. He moves the people in room two into room three. He moves the people in room n into room n + 1. When he's all done, everybody's got a place to stay, and there's one empty room ready for the new guy.

Now an infinite number of people show up at the front desk, all demanding rooms.

The manager says, "sure".

He moves the people from room one into room two. He moves the people in room two into room four. He moves the people in room n into room 2n.

Now everybody who had a room has a room and there are an infinite number of vacancies, just the right amount for the infinite number of weary travelers.