Infinite hotel has infinity guests. You have all the guests move down 10 rooms. Rooms 1-10 are now free. Zero to Infinity and 11 to infinity are equally infinity, since numbers extend into infinity.
In the same manner if you have one sit of infinite guests occupy all the even numbered rooms, you will still have an infinite number of rooms open, because the set of all odd (and even) numbers extends infinitely. You could have the first set of infinite guests take each hundredth room (100, 200, 300, etc), then the next set take 99, 199, 299, etc. in that way you could fit 100 sets of infinite guests.
ftbd@feddit.org 4 weeks ago
There are different “kinds” of infinity. For example, there is an infinite amount of natural numbers, and there is an infinite amount of real numbers. Still, natural numbers only make up a tiny part of real numbers, so while both are infinite, the set of real numbers is bigger. Hilbert’s Hotel is an analogy meant to convey how to deal with these different notions of infinity.
Klear@sh.itjust.works 4 weeks ago
Not really. The guests move to a room with double the number, freeing up an infinite number of rooms.
So the change is from natural numbers to even numbers, freeing up odd numbers. Those infinities are the same, but you can still do this because infinities are weird.