Âéśš´ŤĂ˝

Hilbert’s Hotel

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“SORRY, how do you spell that?” asked the receptionist.

“S, m, i, t, h,” said the portly man in the tweed jacket. He was beginning to
wonder whether spending Christmas away from home was a mistake.

“We have a booking for a Mr Silverbung—no, he arrived two days ago and
there are four of them anyway . . . I’m sorry, Mr Smith, but we have no booking
for you on the computer.”

“But I’ve got a voucher,” protested Smith, passing it over. The receptionist
looked at it and sucked in her breath like a motor mechanic about to break the
news that the car needed a new gearbox and it would take at least two months to
order one. “That’s your problem, you see,” she said sweetly, pointing to one
computer-printed entry.

Smith’s eyes followed her finger. HILTON HOTEL, TENERIFE, it said. “So? This
is Tenerife, isn’t it?”

“In a manner of speaking, yes . . . but this isn’t the Hilton.”

“I told you that when we turned into the driveway,” said Mrs Smith, a
stick-like brunette wearing a hat that resembled a fruit bat that had dropped
dead in mid-flight. “The sign was wrong. But you said the signwriter probably
couldn’t spell . ..”

“This is the Hilbert Hotel, not the Hilton,” said the receptionist.

“I thought it was part of the same chain. I said as much to you just now,
didn’t I, Hermione . . . “

“Yes, and Washingbert is pretty much where the American president lives, too,
I suppose,” snapped Mrs Smith.

“Oh. Then we’d appreciate it if you could call us a cab to the Hilton,
ąčąôąđ˛š˛őąđ.”

The receptionist suppressed a smile. “I’m afraid you’ll need an aeroplane,
not a taxi.”

“P˛š°ůťĺ´Ç˛Ô?”

“This isn’t Tenerife, and there’s no Hilton within a hundred miles.”

Smith looked puzzled. “But just now you said it was Tenerife.”

“In a manner of speaking, sir. This is Tenner Reef in the Viagran Islands. So
called because the Duchess of Wapping bought it for ÂŁ10 in 1712. Tenerife
is roughly six thousand miles that way.” She pointed towards a large potted
cactus. “A lot of people make the same mistake, sir, if it’s any comfort.” The
Smiths’ visions of a leisurely Christmas break, with roast turkey lovingly
prepared by a top chef and no dishes to do vanished like a snowflake in Death
Valley . . . at noon on Midsummer’s Day . . . with a blowlamp trained on it . . .
turned on full. “The airport’s closed until the New Year, too,” added the
receptionist. “Shame, really.”

“Oh, well, then—I guess we’ll just have to pay you for a room and sue
the travel agent when we get home,” said Smith. “You take Armenian Excess?”

“That will do nicely. I’ll just check room availability.” Her varnished
fingernails tapped daintily at the keyboard while the Smiths held their breath.
“Oh dear.”

“There’s a problem?”

“We’re completely full, sir. Not even a dog kennel to spare. In fact, the dog
seems to have double-booked his kennel to a pack of huskies.”

Smith gestured at their huge pile of luggage. “Well, we’re not equipped for
camping on the beach. We’ve got to stay somewhere!”

The receptionist nodded in sympathy. “Mmm, . . . I’ve got an idea, sir. But
it will mean a bit of disruption . . . Let me check with the Manager.” She
hurried off. About ten minutes later she returned, smiling broadly. “You’re in
luck, Room 1 has just become vacant.”

“Brilliant.” A thought occurred to Smith. “Where are they going, then? You
said the airport’s closed.”

“Into Room 2.”

“But Room 2 is occupied.”

“No, they’re moving into Room 3.”

“And where are they going?”

“Room 4, of course. We’ve relocated the people in Room 4 to Room 5. In fact,
we’ve moved everybody up one room number.”

Smith felt his head spinning. “But…but what about whoever’s in the last
°ů´Ç´Çłž?”

“Ah, I see you’re labouring under a misconception. This is a Hilbert Hotel.
There is no last room. We have infinitely many rooms—1, 2, 3, and so on
forever. Right now, of course, they’re all full because we have infinitely many
guests. But infinity plus one is still infinity, so we can rearrange the guests
to create an extra room for you.”

Room 1 wasn’t bad, actually, apart from the pickled onions on the wallpaper.
(No, not in the design. On the wallpaper.) The Smiths had just unpacked when the
phone rang. It was the receptionist.

“Mr Smith, I’m sorry to disturb you, but would you mind moving to Room
21?”

“Why, doesn’t that have pickled onions?”

“I’m sorry? Er, anyway, a coachload of people has arrived unexpectedly and we
need 20 more rooms. I’m asking everybody to move up 20.”

They could scarcely refuse, having just benefited from a similar
rearrangement. It was a nuisance repacking, though. When they got to Room 21, Mr
Smith began to unpack, but his wife stopped him. “Alfred, I am having one of my
premonitions—let’s wait half an hour.”

After 29 minutes, the phone rang. “I’m very sorry, sir, but it turns out that
the coach is a Cantor Coach which contains infinitely many passengers. We need
to create an infinite number of spare rooms.”

“I don’t see how you can manage that,” said Smith. “Surely infinity plus
infinity must be infinitely bigger than infinity!”

“Not at all, sir. We want you to move to Room 42. Everybody goes to the room
with twice the current number, leaving all the odd-numbered rooms empty. The new
people can go into those.”

Scarcely had the Smiths lowered their luggage to the floor of Room 42,
however, when there was a knock on the door. It was the bellboy. “I’ve been told
to help you move to Room 1723.”

“What the devil . . . ?”

“Didn’t the receptionist phone, sir? No, she’s a bit tied up—infinitely
many infinitely large coaches have arrived. Now, it’s all a bit complicated, but
apparently we can fit them all in if the people in Room 1 stay put, then one
group from coach 1 goes into Room 2, then the people from Room 2 go into Room 3.
Then another group from coach 1 goes into the next room followed by two groups
from coach 2, and the people from Room 3 go into Room 7. Then another group from
coach 1 goes into the next room followed by another group from coach 2 followed
by three groups from coach 3, and the people from Room 4 go into Room
13—sorry, I said it was complicated, but the upshot is that you need to be
moved to Room 1723, and that’s why I’m here.” The Smiths wearily picked up their
bags.

Night fell, and the influx of unexpected visitors seemed to have stopped.
They went down to dinner. The table was infinitely long—guests sat in
pairs facing each other at seats whose numbers matched their rooms.

The vegetables were served.

“Brussels ˛őąč°ů´ÇłÜłŮ˛ő!” came an anguished cry from seat 1. “I hate Brussels
˛őąč°ů´ÇłÜłŮ˛ő!”

“That’s OK, pass them to me,” said seat 2. “Oh, drat, I’ve just remembered, I
don’t like them either!”

“No worries,” said an Australian at seat 3. “Can’t stand the ruddy things
meself, mate, but if you give `em to me I’ll pass `em on. Yep, throw yours in
too. Get a move on, though, mate—we `aven’t got all night.” Within half a
minute he had passed the unwanted vegetables to seat 4, together with his own
sprouts. A quarter of a minute later they were at seat 5. An eighth of a minute
later at seat 6, and so on. A rapidly growing pile of Brussels sprouts whizzed
past the Smiths a split second before a minute had passed, accumulating theirs
as it did so, and that same split second later, the Brussels sprouts had
vanished from the dining room.

“Where did they go?” asked a puzzled Smith.

“Fell off the top end.”

“But there isn’t a top end. It just goes on forever.”

“I know. But nobody’s got any sprouts. Ask anyone, if you don’t believe
łžąđ.”

“Can’t be bothered,” said Smith. “But this sausage is excellent. I wish I had
˛š˛Ô´ÇłŮłóąđ°ů.”

“Allow me,” said seat 1724, passing over a sausage.

“But now you haven’t got any sausages yourself.”

“No matter. I’ll get two from seats 1725 and 1726. Then 1725 can get two from
1727 and 1728, and so on. If we hurry up, we can all have two sausages.” He
paused. “Or would you prefer three? Nineteen? A trillion?”

“No, no, thanks all the same,” said Smith. But he ended up with six puddings,
even though only one was served to each guest. Everybody ended up with six
puddings, except the person at seat 14 333 652, who was greedy and had
infinitely many.

The next morning, one of the guests, a Mr Russell, decided they ought to form
clubs to help pass the time. A bridge club, a chess club, a club for underwater
mountaineers, a club for people who didn’t like clubs. In fact, every possible
club, finite or infinite in membership, should be represented—subject only
to an obscure hotel rule that if a guest belonged to a club, then everybody else
in their room was an honorary member. So the clubs were made up of roomfuls of
people, not individuals.

“The clubs must have meeting rooms,” said Russell.

“A˛ľ°ůąđąđťĺ.”

“But all rooms are occupied. Maybe we should move . . . “

“No!” yelled the Smiths. “But we don’t need our rooms during the daytime, not
if we’re all going to be at clubs anyway. Why don’t we just agree that each club
meets in one of the rooms?”

“Excellent idea,” said Russell. “We’ll assign each club to a room, and to
avoid wasting space, every room will host one and only one club.” And he went
away. Two hours later he returned. “Hmm, hit a bit of a snag,” he explained.
“Don’t think it can be done.”

“Why not?”

“Well, it was all going swimmingly until I got to the Absentees’ Club.”

“Which is?”

“The club formed by all the people who do not belong to the club that meets
in their room.”

â€ÂŮ´Ç?”

“Well, it has to meet in someone’s room—Fred’s, say. So does Fred
belong to the Absentees’ Club, or not?”

“Up to him, isn’t it?”

“Apparently not.”

Smith thought it through. “Well, if Fred does belong to the Absentees’ Club,
then he doesn’t belong to the club that meets in his room. Fair
ąđ˛Ô´ÇłÜ˛ľłó—o´Çąč˛ő.”

“P°ůąđłŚžą˛őąđąô˛â.”

“The Absentees’ Club meets in his room. So if he belongs to it, then he
doesn’t. OK, then he doesn’t belong to the Absentees’ Club—oh. That means
that he does belong to the club that meets in his room. Which is . . . “

“The Absentees’ Club.”

“So if he belongs to it he doesn’t, and if he doesn’t, he does.”

“Bit of a problem, isn’t it?” said Russell.

“Exactly. Which leads me to conclude . . . “

“That it’s a silly club and shouldn’t be allowed?”

“Perhaps. A deep philosophical point. The trouble is, if we go down that
route, which clubs do we allow? We might have to disband the bridge club, for
all I know, and that way lies disaster, I can tell you. No, I conclude that the
infinity of all clubs formed from infinitely many people is bigger than the
infinity of people.”

Smith looked worried. “I’m not sure you can have one infinity bigger than
another,” he muttered.

“Let me tell you about the even bigger infinity of associations of clubs,”
said Russell darkly.

The rest of the holiday passed festively, and the Smiths relaxed and put on
10 kilos apiece. During the day, the Synchronised Swearing Club met in their
room, but they were out at the Goat-scrubbing Club one day, the
People-Who-Didn’t-Arrive-in-a-Coach Club the next. They had a great time.

Finally, it was time to check out. After 1722 parties of guests had been
dealt with, the Smiths finally found themselves at the head of the queue.

“That will be VI$ 264 317.47,” said the receptionist. “Roughly
£6000 sterling. Our standard charge, the same for every room.” Smith
swallowed bravely. It was all right, the travel agent would have to pay up
eventually. He handed over his credit card. The check-out girl gave it the kind
of look normally reserved for strange mounds of fungus in dark corners. “Sorry,
sir, but your card expired at midnight on New Year’s Eve.”

Smith sighed. “I’m sorry, but that’s the only way I have to pay.” The
check-out girl glared at him. He looked round frantically, like a rabbit in the
headlights of a juggernaut laden with lead blocks . . . with a very fat driver .
. . who nursed an irrational dislike for rabbits. But Mrs Smith was having an
animated discussion with the people from Room 1724.

“Let’s go, Alfred,” she said, taking him by the arm.

“But—the bill, Hermione. Who’s paying the bill?”

“Room 1724 is.”

“But that’ll cost them a fortune.”

“No, because their bill will be paid by Room 1725. Whose bill will be paid by
1726, whose bill will be paid by 1727, and so on.”

“You mean, all bills are paid but we get the holiday free?”

Mrs Smith settled the dead fruit bat firmly on her head and started working
out how everybody could go by taxi to the airport and end up with twice as much
cash as they’d had to begin with, while giving every taxi driver a hefty tip.
She gave her husband a smile. “Alfred, after everything else that has happened,
are you surprised?”

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