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.

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.

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.

Now you have to walk infinity to get to your room, which is by an infinite amount of ice machines and elevators.

But there's also infinite demand for rooms, otherwise the hotel would have long gone bankrupt.

what if there are babymen in all of them though

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.