this post was submitted on 23 Oct 2025
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You can't actually. If there's an infinite number of rooms and infinite guests occupying them then there are no open rooms.
Next-best strategy: make the set of guests who have to be moved arbitrarily sparse, so that 0% of the hotel's guests need to be bothered. Oh dear, that's still infinity of them.
This is a great plan, but it is critical to not be the person to notify each guest. Infinite calls to inconvenience infinite people sounds like hell.
Only if you mean ~0%. 0 is 0
Depends on the size of the infinities. If you have an infinite natural number of guests, but infinite real number of rooms, then you have more rooms than guests.
If you can see doors to enter each room, then they are countable.
How do you know the uncountable doors aren't there? You wouldn't be able to see them amyway
How would anyone get into one of those free, uncountable rooms if they can't see them.
Ever opened a door with the light off?
Yes, since I have hands to feel the door. Hands which, incidentally, are able to help me count things they touch :)
Okay, automatic doors with perfectly precise sensors based on shifts in gravity.
A single portal into all the different rooms might work if you are ok with your continuous room-choosing mechanism having a probability of zero to get you into your chosen room. Not a problem as long as you take everying with you since the probability of hitting some empty room is of course still one.
What about upon entering a room a new normal door is then constructed, allowing re-entry?
Wrong: ∞±n = ∞
The concept of infinity is well-defined in mathematics. It goes much deeper than that, with countability and differently-sized infinities, but if there are infinite rooms, you always have space.
Yes, but walking down the hallway you'll never find an open room. Space can be made but you can't just go to the ∞+1 room.