[benjohnson]

I think this is a fair point - if there's a one person to one room correlation, then moving people around isn't exposing any additional empty rooms.

It seems we're using the 'hey-everybody-move' as a buffer to accommodate new guests.

Two questions:

Is it ok to think because there's an infinite amount of moving, this buffer is infinitely large?

Why can't we tell guests to just keep moving to random rooms? Would that also solve the problem? If you say no - random moves don't work - then I would propose that some of those random moves would fall into the outcomes of Hilbert's formula. Is that not sufficient?

[lupire]

"Moving" has nothing to do with the paradox. Being reassigned to a new room could take 0 time, with no buffer. Mapping n to n+1 creates a hole at 1.

It's an unphysical model, that only exists in abstract mathematics.

[rsj_hn]

It's just functions f: N -> N

For example you can assign an integer to each guest, and an integer to each room, and the "move" is just the function: guests --> rooms.

As long as this function is injective, then you do not put more than 1 guest in each room.

Then this is just the observation that you can put every guest in a room and still have some empty rooms (rooms not in the range of the function). For example, the deep and paradoxical function f(n) = n+1 corresponds to an assignment of every guest that used to be in room n to room n+1.

I guess if you want, you can think of a description of the function as a sequence of "moves" with people in an actual hotel, but then since the domain of the function is infinite you have to think of an infinite "number" of moves. Then you add time to the mix and carpet hallways or whatever, you start getting lost and saying the whole thing doesn't make sense, but that's only because you are stretching these metaphors beyond the breaking point when what is really being described by the "paradox" is that you can have an injective function from an infinite set to itself that is not surjective.

By the way, once you realize that, for example, when dealing with positive real numbers there exists functions of the form: f(x) = x + 1, then you see that this "hotel" doesn't even need countably infinite moves, it can have uncountably infinite moves. Just please stop trying to worry about the weight of the hotel or how the HVAC works and getting all tangled up in stretching metaphors past their breaking point.

[benjohnson]

Thanks for answering! I think I get it.

Any idea on if directing guests to move randomly would work as well as directing them to move in patterns?

[bmm6o]

If collisions can happen they will happen with probability 1, since there is infinite opportunity. There's also the issue of there not existing a uniform distribution on the integers, so it's hard to get the chances of collision low in the first place.

[rightbyte]

An interpretation could be that remapping infinite sets like that creating holes is an invalid or nonsense operation.