One to One Correspondence and Hilbert’s Grand Hotel

In the Understanding Hilbert’s Grand Hotel, we have discussed the brilliant schemes of a hotel manager in accommodating finite and infinite number of guests in a hotel with infinite number of rooms, where each room was occupied by one guest. In other words, the hotel was fully occupied. In this post, I will explain the mathematics behind these schemes. To be able to understand the explanation, it his highly recommended that you read first the post in the link above.

Finite Number Of Guests

In the Grand Hotel problem, during the first night, a guest arrived. The hotel was full, so there was no room available. However, to accommodate the new guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to room 4 and so on. This means that each guest had to move to the room whose number is 1 higher than the the current room number. This leaves the Room 1 vacant.

Now, how is this possible? » Read more

Understanding Hilbert’s Grand Hotel Paradox

Long ago, in a land far away, there was a grand hotel where there were infinitely many rooms. This hotel was attended by a brilliant manager.

One night, a guest arrived, but  the hotel was full — each room was occupied by one guest. The newly arrived guest asked if a spare room was available. “Of course we have, we are the Infinite Grand Hotel. There is always a room for everyone,” the manager said proudly.

Now since each room was occupied by a guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to Room 4, and so on. Basically, he told every guest in Room n to move to Room n + 1. Since the hotel had infinitely many rooms, there was no problem in moving, there was always a room to move to. This left Room 1 vacant, and therefore, the guest was accommodated. The guest was happy. The manager was happy.  » Read more

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