This idea of cardinality and sets of infinity really come together in the famous Hilbert’s Hotel. Hilbert’s Hotel is a magic hotel, and happens to be very popular as well, where there are an infinite number of rooms, and each room has a current occupant. So the interesting question is how does the Hilbert Hotel accommodate any more guests?

Say that a person walks into the hotel and wants a room. Although each room is currently occupied, the hotel manager comes up with an idea. He tells each of the hotel occupants to move to the room with room number 1 greater than the room they currently occupy. So the person in room 1 moves to room 2, while the person in room 2 moves to room 3, and so on for all of the rooms. So now, room 1 is available for the new customer.

Now, a bus with infinite passengers arrives at the hotel and the passengers are looking for a room to stay in. How will the hotel occupy an infinite number of more guests? The hotel manager comes up with a new idea. He tells each of the current occupants to move to the room with the room number twice as big as their current room. For example, the person in room 1 goes to room 2, the person in room 2 goes to room 4, and so on. This will open up an infinite number of rooms, one for each of the new customers.

However, the hotel runs into problems when a bus with infinite passengers arrives at the hotel, where each of the passengers on the bus corresponds to a number between 0 and 1. The manager tries to find a way to accommodate the passengers of the bus, and starts by assigning room 1 to the passenger with the number 0.1, then assigns room 2 to the passenger with the number 0.01, and so on for an infinite amount of zeros for each passenger. This idea works for all passengers with a number like 0.00001, but there are also passengers with the number 0.2, 0.02, 0.002, and so on, and with 0.3, 0.03, not to mention numbers with combinations of numbers. So there is no way for the hotel manager to find a way to assign each passenger a room number, so even though it is a hotel with infinite rooms, it still can run out of space.

This is because not all infinities are equal. Although the Hilbert Hotel does have an infinite number of rooms, it is countably infinite. The numbers between 0 and 1 are uncountably infinite, which means that these two sets of numbers have different cardinalities , which means although they are both infinite, they are different sizes of infinity.