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## No. 2950: On the lawn of Hilbert's Hotel

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Strip by: Alien@System

{Garfield walks over a lawn with a "Stay off the grass" sign}
{Happily Garfield instead stands on the sign}
{He notices that there is a "Stay off the "Stay off the grass" sign" sign next to him and frowns}
{Next to that is a "Stay off the "Stay off the "Stay off the grass" sign" sign" sign. Garfield looks up in surprise}
{Full of joy, Garfield jumps up an infinite progression of "Stay off the ..." signs demanding him to stay off the previous one}

The author writes:

Based on a comment by Xindaris.

No matter how far Garfield jumps, he'll always be breaking the rules, because there is always a next sign. The title is a blatant excuse for me to add a scientific annotation:

Hilbert's Hotel is a mathematical thought experiment meant to introduce students into the peculiarities of infinite sequences and sets, and to show how to go about proving whether or not two infinitely large sets are in the same equivalence class or if one is "bigger" than the other.

The titular hotel is a rather cheap although still comfortable lodging, on the pleasant if infinitely long coast of Gosper Island. After some rather unexpectedly high demand for its quaint rooms, there were a few extensions added to the original structure, so that now the hotel has an infinite number of rooms, one for each natural number. Nonetheless, clientele remain numerous, so that at any given time, it's more-or-less guaranteed that Hilbert's Hotel is completely booked out.

Now Dame Jane, after a rather long trip, arrives at this hotel, as there is a conference on primates happening in the nearby town. After wiping some dust off her shoes on the decorative Sierpinski carpet, she inquires if there are any rooms free. Now the doorman has to unfortunately tell her that it isn't so. In fact, due to it being a well-attended conference, all hotels in the area are completely booked out. They even had to assign somebody to room 0, which is the loo. Given that Dame Jane is such a distinguished personality and one of the key speakers at the upcoming conference, he nonetheless tries to find a way to get a room for her, by reassigning all the guests that are currently in the hotel to new rooms.

Of course, to ensure that everyone can still get their sleep and that there is no argument or hassle, the only way to do so is to move all people simultaneously, by sending out the infinite number of porters serving the hotel to tell people their new room number to which they have to move, and have one of them accompany Dame Jane to her new, now empty room. Being a diligent worker, the doorman of course manages this.

He is unfazed as well when on the next day, a bus arrives with an infinite number of monkeys who want to use the conference as a chance to present their newly typed Shakespeare collection to an interested audience. Since they have found that humans often have trouble telling their simian brethren apart, and because they aren't that creative, every single monkey has a sign around their neck with a natural number on it, with exactly one monkey for every number. Now, given that the authors of such literary masterpieces shouldn't be out in the cold, the doorman once again asks for the infinite number of porters to go out and help the guests pack so they can move to new rooms. After some shuffling of feet and some complaints about the noise of the new arrivals, there is not a single guest or monkey left out in the rain.

It should be noted that due to the great demand for lodgings, the current owner of Hilbert's Hotel had some years back turned it into a chain, which after some rapid expansion now consists of an infinite number of equally infinite hotels all along the coast. The original retained its old name, but after that, the creativity of the CEO waned a bit, so he named the rest after natural numbers, with each natural number being the name of a hotel. However, it appears the CEO tried to dodge taxes on his infinitely high revenue, and has been fined an equally infinitely high amount of money. This has left the chain so destitute that it now has to close all but the original hotel. But this is the high season, so they're all completely booked out, and given the good reputation that the chain tries to maintain despite the shady business of the CEO, it would not do at all to cancel the booking of even a single guest. The only choice is to go and put all the current lodgers into the one remaining hotel and try to find rooms for them. Thankfully, the doorman once again just calls for his infinite number of porters and tells them which room of which hotel they would have to go to to bring the guests to their new room in Hilbert's Hotel, and once that rather lengthy moving is completed, there is indeed nobody left out in the rain, except for the ex-CEO, who has been summarily fired by the board, who have instead given the resourceful doorman the position.

It shall be left as an exercise to the reader to figure out how that doorman managed these three amazing feats.

Original strip: 2017-05-09.