Infinity and Hilbert’s Hotel Paradox
Named after mathematician David Hilbert, the Hilbert Hotel Paradox is a thought experiment demonstrating infinity in a humorous and approachable way.
Imagine a hotel with an infinite number of rooms with a bright and intelligent hotel manager.
On one night, the infinite hotel is completely booked with an infinite number of people. The hotel is fully booked.
Then, a new guest arrives and asks for a room.
You might think the hotel is completely booked, and the manager must turn him down.
But the manager is clever.
Here is what he does.
You can move each guest to the next room,
- the guest in room 1 moves to room 2,
- the guest in room 2 moves to room 3,
- the guest in room 3 moves to room 4, and so on.
And because there are an infinite number of rooms, every guest gets a new room.
Now, room number 1 is empty. So the new guest gets this room.
Infinity + any finite number = Infinity
Hence, you can fit in 1 new guest or 50 new guests. You can check in ANY finite amount of new guests.
Now, Imagine that a countably infinite number of guests arrive at the hotel all at once, requesting rooms.
You may think there are no rooms left, as all the infinite rooms have been given out.
However, this is not the case.
Here’s what he does:
You can move each guest to an even-numbered room,
- the guest in room 1 moves to room 2,
- the guest in room 2 moves to room 4,
- the guest in room 3 moves to room 6, and so on.
This leaves all the infinitely many odd-numbered rooms empty, so you can allot each of the infinite new guests to an odd-numbered room.
Infinity + Infinity = Infinity
This paradox demonstrates the concept of infinity.
Even when you think you have reached the maximum number of something (in this case, occupied rooms), there is always room for more.
It also highlights the idea that infinite sets can be divided into smaller, infinite subsets.
The Hilbert Hotel paradox is often used to illustrate the concept of Infinity in mathematics.
It can be a valuable tool for understanding this abstract and challenging concept.
It also serves as a reminder that the idea of Infinity is not always intuitive and can lead to counterintuitive conclusions.
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