Today, I remembered one of my most favorite thought experiments in mathematics. The Hilbert's paradox of the Grand Hotel. It goes like this:
Imagine a hotel with an infinite number of rooms, all of which are occupied by an infinite number of guests. Suppose a new guest arrives and wishes to be accommodated in the hotel. How do you fit them in? All rooms are occupied, but at the same time, the rooms are infinite!
You somehow have to find space. So what do you do? You move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into room 1
Thus it's not only possible to make room for 1 guest, but for any finite number of new guests. Which means that, infinity plus one, or infinity plus any number, equals infinity. It also shows that some infinities are greater than others i.e
∞ + 1 = ∞
∞ + n = ∞
∞(n) < ∞(n+1)
Ok, but what if an infinite number of new guests arrive? Again, the hotel has infinite rooms, so you better accommodate them. Just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n (2 times n)
This means that all the odd-numbered rooms (which are countably infinite) will be free for the new guests. This means that
∞ + ∞ = ∞
However, does this mean that
∞ - ∞ also equals ∞?
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