Infinite Hotel Paradox

Thing is, the message has to be passed along either by an intercom or by the person moving to the next room passing it on... Either way or travels at fastest at the speed of light, so you'll have people in the corridors moving to the next room for an infinite amount of time purely from the time it takes to propagate.
Given you're therefore committing to (at least on average) at least one person being without a room for the rest of time, why not just tell the chap in the first room to keep walking until he finds an available room? In terms of overall inconvenience (overall time spent without a room per person), it's the same as the original as both are infinite, but for the average person it goes from the time to walk from one room to the next to 0
- Fun prank.
  "How do you know they're all booked? Maybe someone didn't show up. Can you go check?"
- But if being without a room is a form of suffering, then wouldbt it make more sense to distribute that suffering equally, so that no one person has to bear an unendurable length of time without a room, instead each person is just momentarily inconvenienced as they shift.
  
  That's without considering the time to pack up your bags etc - ie there's a fixed cost as well as the cost per room moved
  To minimise the total societal cost, only one person has to leave their room, and by that (or any) one person not making the sacrifice, the average suffering increases across all of median, mean and mode...
  It's the opposite situation from where one person can get huge gains to the detriment of many others - eusocially it makes sense to do what's best for the average person
- Yes, but, also, logically you can never reach any destination because you can only ever halve the distance therefore everyone is trapped in an infinite loop of suffering.
  
  Theoretically, I guess... But my argument came when introducing the laws of physics into the world of the infinite hotel, but there comes a point where the movement is small enough that the electron orbits are unaffected when the atom next to them "moves" therefore there's functionally no movement.
  You're not a criminal who goes around breaking the laws of physics like the rest of those "mathematician" types are you?
- The rooms are all full by definition. You're literally telling him to walk until he dies. Just kick him out at that point, hell shooting him on the spot would be more merciful.
  But the weird thing about infinite sets is they don't process over time as we perceive it. Things happen all at once. There's a classic situation where an infinite tub is filled with an infinite amount of numbered balls, and for every number that goes in, it's square root is taken out. So when 1 goes in, 1 goes out, when 2 goes in nothing happens, same with 3, when 4 goes in, 2 gets taken out, when 9 goes in, 3 gets taken out.
  How many balls are in it at the end?
  Intuitively it seems there must be infinite balls, as balls are being taken out at a slower rate than they're entering. But the actual answer is 0. Because the process happens all at once. So the question becomes, which balls get removed in this process? Well, the numbers that can be squared. Which is every number. If every number can be squared, every number gets removed. So if this infinite process were to play out, there would be no balls left in it.
  But if we were to try to physically do this, it's impossible for it to actually play out. Infinity isn't a number at the end of a line, it's the concept of an entire unbound set. That's why things like the hotel are good to try to explain and visualise the concept, but break down if you try to imagine them as real world places that follow time and physics.

Luckily with infinite rooms there are an infinite amount of people checking out so I'll just wait until one comes avail-
Ah, my room is ready.
- Now you have to walk infinity to get to your room, which is by an infinite amount of ice machines and elevators.
  
  The closest one that left is not infinite though. It has a set number.
- But there's also infinite demand for rooms, otherwise the hotel would have long gone bankrupt.
  
  Nah, tax scheme keeps it open
  
  Good one. lol
  But being infinite, would this actually matter...?
  Somebody ask Matt Parker real quick.

I've always hated this supposed paradox, because how is it possible for a hotel with infinite rooms to be full? Even with infinite guests, there will always be room for more. Because, you know, there are infinite rooms.
- It has to do with countably infinite sets.
  The analysis on Wikipedia does a better job of explaining the concept: https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel#Analysis
  The whole point is that it's something we can prove mathematically that is highly unintuitive.
- It's mostly just a way of communicating the bizarre nature of infinite series and other problems related to infinity. Just fun thought experiments.
- It's not always discussed as such, but Hilbert's Hotel is a mathematically well defined topic and can be proved rigorously. An infinite set of rooms can be the set x1, x2,...xinf, and people can be y1, y2...yinf. You can pair every entry in these two sets. x1&y1, x2&y2,...xinf&yinf. You can't number a room without having a person in the room, and you can't find a person who doesn't have a room.
- If you have n rooms and n guests there are no empty rooms. Let n go to infinity and there should still be no empty rooms.
  The trick of the Hilbert hotel is that if you add a guest to a hotel with countably infinite guests the number of guests does not change.
- If there is no vacancy then the hotel has infinite full rooms and no empty ones.
- The hotel should really just fill its even numbers rooms first, then if another person turned up after you already have an infinity of guests, you just house this start of a new Infinity in room 1 etc
  
  What if they filled in multiples of 3? They get vacant space for infinitr people twice. Well just fill people as multiple of infinity, so that infinite times infinite people can fill it in

Couldn't you just skip that room?
- what if there are babymen in all of them though
  
  unless the chance of having a babyman is 100%, there'll be enough non-babyman to move around, you just need to grab the subgroup of non-babyman rooms and assign each of them a new number, then they can each move to the next number among their own subgroup

Can someone explain the punch line.
- The joke comes from the infinite hotel paradox, which is from a property of infinities that you can always add to them and they're still infinity. This leads to a paradox where if you have an infinite hotel room with no vacancies, every single room in the infinity is full, as a given. But if everybody moved over one room, there would be an empty room in position 1.
  The punchline is that in real life if you asked every guest to move over 1, they wouldn't want to, and some would be big babies for some reason.
  
  I thought it was that the guest in room one was doing some weird diaper/puppet sex shit and the prospective guest was no longer interested in the room.
  But maybe that's a me problem.

I’d just stay in that room.
- He looks like he'd give good hugs.